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Theorem 3.2. If \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma \) then\n\n\[ \Delta \left( \frac{{a\tau } + b}{{c\tau } + d}\right) = {\left( c\tau + d\right) }^{12}\Delta \left( \tau \right) \] | Proof. Since \( \Delta \left( {{\omega }_{1},{\omega }_{2}}\right) \) is homogeneous of degree -12 we have\n\n\[ \Delta \left( {{\omega }_{1},{\omega }_{2}}\right) = {\omega }_{1}{}^{-{12}}\Delta \left( {1,\tau }\right) = {\omega }_{1}{}^{-{12}}\Delta \left( \tau \right) ,\]\n\nwhere \( \tau = {\omega }_{2}/{\omega }_{... | Yes |
Theorem 3.3. If \( \tau \in H \) and \( x = {e}^{2\pi i\tau } \) we have\n\n(6)\n\n\[ \Delta \left( \tau \right) = {\left( 2\pi \right) }^{12}{\eta }^{24}\left( \tau \right) = {\left( 2\pi \right) }^{12}x\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - {x}^{n}\right) }^{24}. \]\n\nConsequently,\n\n(7)\n\n\[ \matho... | Proof. Let \( f\left( \tau \right) = \Delta \left( \tau \right) /{\eta }^{24}\left( \tau \right) \) . Then \( f\left( {\tau + 1}\right) = f\left( \tau \right) \) and \( f\left( {-1/\tau }\right) = f\left( \tau \right) \) , so \( f \) is invariant under every transformation in \( \Gamma \) . Also, \( f \) is analytic an... | Yes |
Theorem 3.4 (Dedekind’s functional equation). If \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma, c > 0 \), and \( \tau \in H \), we have \[ \eta \left( \frac{{a\tau } + b}{{c\tau } + d}\right) = \varepsilon \left( {a, b, c, d}\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \t... | We will prove Theorem 3.4 through a sequence of lemmas. First we note that Dedekind's formula is a consequence of the following equation, obtained by taking logarithms of both members of (10), \( \log \eta \left( \frac{{a\tau } + b}{{c\tau } + d}\right) = \log \eta \left( \tau \right) + {\pi i}\left( {\frac{a + d}{12c}... | Yes |
Lemma 1. Equation (12) is equivalent to the relation\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\lambda \left( {-{in\tau }}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\lambda \left( {-{in}\frac{{a\tau } + b}{{c\tau } + d}}\right) + \frac{\pi i}{12}\left( {\tau - \frac{{a\tau } + b}{{c\tau } + d}}\right) \]\n... | We shall prove (15) as a consequence of a more general transformation formula obtained by Shô Iseki [17] in 1957. For this purpose it is convenient to restate (15) in an equivalent form which merely involves some changes in notation. | No |
Lemma 2. Let \( z \) be any complex number with \( \operatorname{Re}\left( z\right) > 0 \), and let \( h, k \) and \( H \) be any integers satisfying \( \left( {h, k}\right) = 1, k > 0,{hH} \equiv - 1\left( {\;\operatorname{mod}\;k}\right) \) . Then Equation (15) is equivalent to the formula\n\n(16)\n\n\[ \mathop{\sum ... | Proof. Given \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \), with \( c > 0 \), and given \( \tau \) with \( \operatorname{Im}\left( \tau \right) > 0 \), choose \( z, h, k \), and \( H \) as follows:\n\n\[ k = c,\;h = - d,\;H = a,\;z = - i\left( {{c\tau } + d}\right) . \]\n\nThen \( \op... | Yes |
Theorem 3.5 (Iseki’s formula). If \( \operatorname{Re}\left( z\right) > 0 \) and \( 0 \leq \alpha \leq 1,0 \leq \beta \leq 1 \), let\n\n(17)\n\n\[ \Lambda \left( {\alpha ,\beta, z}\right) = \mathop{\sum }\limits_{{r = 0}}^{\infty }\{ \lambda \left( {\left( {r + \alpha }\right) z - {i\beta }}\right) + \lambda \left( {\l... | Proof. First we assume that \( 0 < \alpha < 1 \) and \( 0 < \beta < 1 \) . We begin with the first sum appearing in (17) and use (14) to write\n\n(19)\n\n\[ \mathop{\sum }\limits_{{r = 0}}^{\infty }\lambda \left( {\left( {r + \alpha }\right) z - {i\beta }}\right) = \mathop{\sum }\limits_{{r = 0}}^{\infty }\mathop{\sum ... | Yes |
Theorem 3.8. The number \( {6k}\mathrm{\;s}\left( {h, k}\right) \) is an integer. Moreover, if \( \theta = \left( {3, k}\right) \) we have\n\n(a) \( {12hks}\left( {k, h}\right) \equiv 0\left( {{\;\operatorname{mod}\;\theta }k}\right) \)\n\nand\n\n(b) \( {12hks}\left( {h, k}\right) \equiv {h}^{2} + 1\left( {{\;\operator... | Proof. From (30) we find\n\n(36)\n\n\[ \n{6ks}\left( {h, k}\right) = \frac{6h}{k}\mathop{\sum }\limits_{{r = 1}}^{{k - 1}}{r}^{2} - 6\mathop{\sum }\limits_{{r = 1}}^{{k - 1}}r\left\lbrack \frac{hr}{k}\right\rbrack - 3\mathop{\sum }\limits_{{r = 1}}^{{k - 1}}r. \]\n\nSince \( 6\mathop{\sum }\limits_{{r = 1}}^{{k - 1}}{r... | Yes |
Theorem 3.9. The Dedekind sums satisfy the congruence\n\n\[ \n{12ks}\left( {h, k}\right) \equiv \left( {k - 1}\right) \left( {k + 2}\right) - {4h}\left( {k - 1}\right) + 4\mathop{\sum }\limits_{{r < k/2}}\left\lbrack \frac{2hr}{k}\right\rbrack \left( {\;\operatorname{mod}\;8}\right) .\n\]\n\nIf \( k \) is odd this beco... | Proof. From (36) we obtain\n\n\[ \n{12ks}\left( {h, k}\right) = {2h}\left( {k - 1}\right) \left( {{2k} - 1}\right) - {12}\mathop{\sum }\limits_{{r = 1}}^{{k - 1}}r\left\lbrack \frac{hr}{k}\right\rbrack - {3k}\left( {k - 1}\right)\n\]\n\n\[ \n= - {2h}\left( {k - 1}\right) + {4hk}\left( {k - 1}\right) - {12}\mathop{\sum ... | Yes |
Theorem 3.10. If \( k = {2}^{\lambda }{k}_{1} \) where \( \lambda > 0 \) and \( {k}_{1} \) is odd, then for odd \( h \geq 1 \)\n\nwe have\n\n(41)\n\n\[ \n{12hks}\left( {h, k}\right) \equiv {h}^{2} + {k}^{2} + 1 + {5k} - {4k}\mathop{\sum }\limits_{{v < h/2}}\left\lbrack \frac{2kv}{h}\right\rbrack \left( {\;\operatorname... | Proof. Since \( h \) is odd we can apply (40) to obtain, after multiplication by \( k \) ,\n\n\[ \n{12hks}\left( {k, h}\right) \equiv k\left( {h - 1}\right) + {4k}\mathop{\sum }\limits_{{v < h/2}}\left\lbrack \frac{2kv}{h}\right\rbrack \left( {\;\operatorname{mod}\;{2}^{\lambda + 3}}\right) .\n\]\n\nBy the reciprocity ... | Yes |
Theorem 4.1. Let \( {S\tau } = - 1/\tau \) and \( {T\tau } = \tau + 1 \) be the generators of the full modular group \( \Gamma \), and let \( p \) be any prime. Then for every \( V \) in \( \Gamma, V \notin {\Gamma }_{0}\left( p\right) \) , there exists an element \( P \) in \( {\Gamma }_{0}\left( p\right) \) and an in... | Proof. Given \( V = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) \) where \( C ≢ 0\left( {\;\operatorname{mod}\;p}\right) \) . We wish to find\n\n\[ P = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) ,\text{ with }c \equiv 0\left( {\;\operatorname{mod}\;p}\right) \]\n\nand an integer \( k,0 \l... | Yes |
Theorem 4.5. Iff is automorphic under \( \Gamma \) and if \( p \) is prime, let\n\n\[ \n{f}_{p}\left( \tau \right) = \frac{1}{p}\mathop{\sum }\limits_{{\lambda = 0}}^{{p - 1}}f\left( \frac{\tau + \lambda }{p}\right)\n\]\n\nThen \( {f}_{p} \) is automorphic under \( {\Gamma }_{0}\left( p\right) \) . Moreover, if \( f \)... | Proof. First we prove the statement concerning Fourier expansions. We have\n\n\[ \n{f}_{p}\left( \tau \right) = \frac{1}{p}\mathop{\sum }\limits_{{\lambda = 0}}^{{p - 1}}\mathop{\sum }\limits_{{n = - m}}^{\infty }a\left( n\right) {e}^{{2\pi in}\left( {\tau + \lambda }\right) /p}\n\]\n\n\[ \n= \frac{1}{p}\mathop{\sum }\... | No |
Lemma 1. If \( V \in {\Gamma }_{0}\left( p\right) \) and if \( 0 \leq \lambda \leq p - 1 \), let \( {T}_{\lambda }\tau = \left( {\tau + \lambda }\right) /p \) . Then there exists an integer \( \mu ,0 \leq \mu \leq p - 1 \) and a transformation \( {W}_{\mu } \) in \( {\Gamma }_{0}\left( {p}^{2}\right) \) such that\n\n\[... | Proof of LEMMA 1. Let \( V = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \), where \( c \equiv 0\left( {\;\operatorname{mod}\;p}\right) \), and let \( \lambda \) be given, \( 0 \leq \lambda \leq p - 1 \) . We are to find an integer \( \mu ,0 \leq \mu \leq p - 1 \) and a transformation \( {W}_{\mu } = \le... | Yes |
Theorem 4.6. Iff is automorphic under \( \Gamma \) and if \( p \) is prime, then\n\n\[ \n{f}_{p}\left( {-\frac{1}{\tau }}\right) = {f}_{p}\left( \tau \right) + \frac{1}{p}f\left( {p\tau }\right) - \frac{1}{p}f\left( \frac{\tau }{p}\right) .\n\] | To prove this we need another lemma. | No |
Lemma 2. Let \( {T}_{\lambda }\tau = \left( {\tau + \lambda }\right) /p \) . Then for each \( \lambda \) in the interval \( 1 \leq \lambda \leq p - 1 \) there exists an integer \( \mu \) in the same interval and a transformation \( V \) in \( {\Gamma }_{0}\left( p\right) \) such that\n\n\[ \n{T}_{\lambda }S = V{T}_{\mu... | Proof of LEMMA 2. We wish to find \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( {\Gamma }_{0}\left( p\right) \) such that\n\n\[ \n\left( \begin{array}{ll} 1 & \lambda \\ 0 & p \end{array}\right) \left( \begin{array}{rr} 0 & - 1 \\ 1 & 0 \end{array}\right) = \left( \begin{array}{ll} a & b \\ c &... | Yes |
Theorem 4.7. For a fixed integer \( q \), let\n\n\[ \varphi \left( \tau \right) = \frac{\Delta \left( {q\tau }\right) }{\Delta \left( \tau \right) }\;\text{ if }\tau \in H \]\n\nThen \( \varphi \) is automorphic under \( {\Gamma }_{0}\left( q\right) \) . Moreover, the Fourier expansion of \( \varphi \) has the form\n\n... | Proof. First we obtain the Fourier expansion. We have\n\n\[ \Delta \left( \tau \right) = {\left( 2\pi \right) }^{12}\mathop{\sum }\limits_{{n = 1}}^{\infty }\tau \left( n\right) {x}^{n} = {\left( 2\pi \right) }^{12}x\left\{ {1 + \mathop{\sum }\limits_{{n = 1}}^{\infty }\tau \left( {n + 1}\right) {x}^{n}}\right\} \]\n\n... | Yes |
Theorem 4.8. If \( \tau \in H \) we have\n\n\[ \varphi \left( \frac{-1}{q\tau }\right) = \frac{1}{{q}^{12}\varphi \left( \tau \right) }.\]\n\nHence \( \varphi \left( \tau \right) \rightarrow \infty \) as \( \tau \rightarrow 0 \). | Proof. Since \( \Delta \left( {-1/\tau }\right) = {\tau }^{12}\Delta \left( \tau \right) \) we have\n\n\[ \Delta \left( {-\frac{1}{q\tau }}\right) = {\left( q\tau \right) }^{12}\Delta \left( {q\tau }\right) \]\n\nso\n\n\[ \varphi \left( \frac{-1}{q\tau }\right) = \frac{\Delta \left( {q\frac{-1}{q\tau }}\right) }{\Delta... | Yes |
Theorem 4.9. Let \( q = 2,3,5,7 \), or 13, and let \( r = {24}/\left( {q - 1}\right) \) . Then the function\n\n\[\n\Phi \left( \tau \right) = {\left( \frac{\eta \left( {q\tau }\right) }{\eta \left( \tau \right) }\right) }^{r}\n\]\n\nis automorphic under the subgroup \( {\Gamma }_{0}\left( q\right) \) . | Proof. If \( q = 2 \) we have \( r = {24} \) and \( \Phi \left( \tau \right) = \Delta \left( {q\tau }\right) /\Delta \left( \tau \right) \) . In this case the theorem was already proved in Theorem 4.7. Therefore we shall assume that \( q \geq 3 \) . Let \( V = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) ... | Yes |
Theorem 4.10. If \( p \) is prime and \( \tau \in H \) then\n\n\[ \n{j}_{p}\left( {-\frac{1}{p\tau }}\right) = {j}_{p}\left( {p\tau }\right) + \frac{1}{p}j\left( {{p}^{2}\tau }\right) - \frac{1}{p}j\left( \tau \right) .\n\]\n\nHence if \( x = {e}^{2\pi i\tau } \) we have the Fourier expansion\n\n\[ \np{j}_{p}\left( {-\... | Proof. We have\n\n\[ \nj\left( \tau \right) = {x}^{-1} + c\left( 0\right) + c\left( 1\right) x + c\left( 2\right) {x}^{2} + \cdots ,\n\]\n\n\[ \n{j}_{p}\left( \tau \right) = c\left( 0\right) + c\left( p\right) x + c\left( {2p}\right) {x}^{2} + \cdots ,\n\]\n\n\[ \np{j}_{p}\left( {p\tau }\right) = {pc}\left( 0\right) + ... | Yes |
Theorem 4.11. Assume \( p = 2,3,5,7 \) or 13, and let\n\n\[ \Phi \left( \tau \right) = {\left( \frac{\eta \left( {p\tau }\right) }{\eta \left( \tau \right) }\right) }^{r},\;\text{ where }r = \frac{24}{p - 1}. \]\n\nThen there exist integers \( {a}_{1},\ldots ,{a}_{{p}^{2}} \) such that\n\n(9)\n\n\[ {j}_{p}\left( \tau \... | Proof. By Theorem 4.10 we have\n\n\[ p{j}_{p}\left( {-\frac{1}{p\tau }}\right) = {x}^{-{p}^{2}} - {x}^{-1} + I\left( x\right) \]\n\nand, since \( {12\alpha } = r/2 \), Theorem 4.8 gives us\n\n\[ {p}^{r/2}\Phi \left( {-\frac{1}{p\tau }}\right) = \frac{1}{\Phi \left( \tau \right) } = {x}^{-1} + I\left( x\right) . \]\n\nL... | Yes |
Theorem 4.12. The coefficients in the Fourier expansion of \( j\\left( \\tau \\right) \) satisfy the following congruences:\n\n\[ c\\left( {2n}\\right) \\equiv 0\\left( {\\;\\operatorname{mod}\\{2}^{11}}\\right) \]\n\n\[ c\\left( {3n}\\right) \\equiv 0\\left( {\\;\\operatorname{mod}\\{3}^{5}}\\right) \]\n\n\[ c\\left( ... | Proof. The previous theorem shows that for \( p = 2,3,5,7 \) and 13 we have\n\n\[ c\\left( {pn}\\right) \\equiv 0\\left( {\\;\\operatorname{mod}\\{p}^{\\left( {r/2}\\right) - 1}}\\right) ,\]\n\nwhere \( r = {24}/\\left( {p - 1}\\right) \) . Therefore we simply compute \( \\left( {r/2}\\right) - 1 \) to obtain the state... | Yes |
Theorem 5.1. Let \( F\left( t\right) = 1/\mathop{\prod }\limits_{{m = 1}}^{\infty }\left( {1 - {t}^{m}}\right) \) and let\n\n\[ x = \exp \left( {\frac{2\pi ih}{k} - \frac{2\pi z}{{k}^{2}}}\right) ,\;{x}^{\prime } = \exp \left( {\frac{2\pi iH}{k} - \frac{2\pi }{z}}\right) ,\]\n\nwhere \( \operatorname{Re}\left( z\right)... | Proof. If \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma \) with \( c > 0 \), the functional equation for \( \eta \left( \tau \right) \) implies\n\n\[ \frac{1}{\eta \left( \tau \right) } = \frac{1}{\eta \left( {\tau }^{\prime }\right) }\{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\exp \left\{... | Yes |
Theorem 5.2. If \( \left( {a/b}\right) < \left( {c/d}\right) \), their mediant \( \left( {a + c}\right) /\left( {b + d}\right) \) lies between them. | \[ \frac{a + c}{b + d} - \frac{a}{b} = \frac{{bc} - {ad}}{b\left( {b + d}\right) } > 0\;\text{ and }\;\frac{c}{d} - \frac{a + c}{b + d} = \frac{{bc} - {ad}}{d\left( {b + d}\right) } > 0. \] | Yes |
Theorem 5.3. Given \( 0 \leq a/b < c/d \leq 1 \) . If \( {bc} - {ad} = 1 \) then \( a/b \) and \( c/d \) are consecutive terms in \( {F}_{n} \) for the following values of \( n \) :\n\n\[ \max \left( {b, d}\right) \leq n \leq b + d - 1. \] | Proof. The condition \( {bc} - {ad} = 1 \) implies that \( a/b \) and \( c/d \) are in lowest terms. If \( \max \left( {b, d}\right) \leq n \) then \( b \leq n \) and \( d \leq n \) so \( a/b \) and \( c/d \) are certainly in \( {F}_{n} \) . Now we prove they are consecutive if \( n \leq b + d - 1 \) . If they are not ... | Yes |
Theorem 5.4. Given \( 0 \leq a/b < c/d \leq 1 \) with \( {bc} - {ad} = 1 \), let \( h/k \) be the mediant of \( a/b \) and \( c/d \) . Then \( a/b < h/k < c/d \), and these fractions satisfy the unimodular relations | Proof. Since \( h/k \) lies between \( a/b \) and \( c/d \) we have \( {bh} - {ak} \geq 1 \) and \( {ck} - {dh} \geq 1 \) . Equation (9) shows that \( k = b + d \) if, and only if, \( {bh} - {ak} = \) \( {ck} - {dh} = 1 \) . | Yes |
Theorem 5.5. The set \( {F}_{n + 1} \) includes \( {F}_{n} \) . Each fraction in \( {F}_{n + 1} \) which is not in \( {F}_{n} \) is the mediant of a pair of consecutive fractions in \( {F}_{n} \) . Moreover, if \( a/b < c/d \) are consecutive in any \( {F}_{n} \), then they satisfy the unimodular relation \( {bc} - {ad... | Proof. We use induction on \( n \) . When \( n = 1 \) the fractions \( 0/1 \) and \( 1/1 \) are consecutive and satisfy the unimodular relation. We pass from \( {F}_{1} \) to \( {F}_{2} \) by inserting the mediant \( 1/2 \) . Now suppose \( a/b \) and \( c/d \) are consecutive in \( {F}_{n} \) and satisfy the unimodula... | Yes |
Theorem 5.6. Two Ford circles \( C\left( {a, b}\right) \) and \( C\left( {c, d}\right) \) are either tangent to each other or they do not intersect. They are tangent if, and only if, \( {bc} - {ad} = \) \( \pm 1 \) . In particular, Ford circles of consecutive Farey fractions are tangent to each other. | Proof. The square of the distance \( D \) between centers is (see Figure 5.2)\n\n\[{D}^{2} = {\left( \frac{a}{b} - \frac{c}{d}\right) }^{2} + {\left( \frac{1}{2{b}^{2}} - \frac{1}{2{d}^{2}}\right) }^{2},\]\n\nwhereas the square of the sum of their radii is\n\n\[{\left( r + R\right) }^{2} = {\left( \frac{1}{2{b}^{2}} + ... | Yes |
Theorem 5.7. Let \( {h}_{1}/{k}_{1} < h/k < {h}_{2}/{k}_{2} \) be three consecutive Farey fractions. The points of tangency of \( C\left( {h, k}\right) \) with \( C\left( {{h}_{1},{k}_{1}}\right) \) and \( C\left( {{h}_{2},{k}_{2}}\right) \) are the points\n\n\[ \n{\alpha }_{1}\left( {h, k}\right) = \frac{h}{k} - \frac... | Proof. We refer to Figure 5.3. Write \( {\alpha }_{1} \) for \( {\alpha }_{1}\left( {h, k}\right) \) . The figure shows that \n\n\[ \n{\alpha }_{1} = \left( {\frac{h}{k} - a}\right) + i\left( {\frac{1}{2{k}^{2}} - b}\right) . \n\] \n\nTo determine \( a \) and \( b \) we refer to the similar right triangles and we get \... | Yes |
Theorem 5.8. The transformation\n\n\[ z = - i{k}^{2}\left( {\tau - \frac{h}{k}}\right) \]\n\nmaps the Ford circle \( C\left( {h, k}\right) \) in the \( \tau \) -plane onto a circle \( K \) in the \( z \) -plane of radius \( \frac{1}{2} \) about the point \( z = \frac{1}{2} \) as center (see Figure 5.6). The points of c... | Proof. The translation \( \tau - \left( {h/k}\right) \) moves \( C\left( {h, k}\right) \) to the left a distance \( h/k \) , and thereby places its center at \( i/\left( {2{k}^{2}}\right) \) . Multiplication by \( - i{k}^{2} \) expands the radius to \( 1/2 \) and rotates the circle through \( \pi /2 \) radians in the n... | Yes |
Theorem 5.9. For the points \( {z}_{1} \) and \( {z}_{2} \) of Theorem 5.8 we have\n\n(10)\n\n\[ \left| {{z}_{1}\left( {h, k}\right) }\right| = \frac{k}{\sqrt{{k}^{2} + {k}_{1}^{2}}},\;\left| {{z}_{2}\left( {h, k}\right) }\right| = \frac{k}{\sqrt{{k}^{2} + {k}_{2}^{2}}}. \]\n\nMoreover, if \( z \) is on the chord joini... | Proof. For \( {\left| {z}_{1}\right| }^{2} \) we have\n\n\[ {\left| {z}_{1}\right| }^{2} = \frac{{k}^{4} + {k}^{2}{k}_{1}^{2}}{{\left( {k}^{2} + {k}_{1}^{2}\right) }^{2}} = \frac{{k}^{2}}{{k}^{2} + {k}_{1}^{2}}. \]\n\nThere is a similar formula for \( {\left| {z}_{2}\right| }^{2} \) . This proves (10). To prove (11) we... | Yes |
Theorem 6.1. Let \( f \) be an entire modular form of weight \( k \) which is not identically zero, and assume \( f \) has \( N \) zeros in the closure of the fundamental region \( {R}_{\Gamma } \), omitting the vertices. Then we have the formula\n\n\[ k = {12N} + {6N}\left( i\right) + {4N}\left( \rho \right) + {12N}\l... | Proof. The method of proof is similar to that of Theorem 2.4 where we proved that a modular function has the same number of zeros as poles in the closure of \( {R}_{\Gamma } \) . Since \( f \) has no poles we can write\n\n\[ N = \frac{1}{2\pi i}{\int }_{\partial R}\frac{{f}^{\prime }\left( \tau \right) }{f\left( \tau \... | Yes |
Theorem 6.3. Let \( f \) be an entire modular form of even weight \( k \geq 0 \) and define \( {G}_{0}\left( \tau \right) = 1 \) for all \( \tau \) . Then \( f \) can be expressed in one and only one way as a sum of the type\n\n(7)\n\n\[ f = \mathop{\sum }\limits_{\substack{{r = 0} \\ {k - {12r} \neq 2} }}^{\left\lbrac... | Proof. If \( k < {12} \) there is at most one term in the sum and the theorem can be verified directly. If \( f \) has weight \( k < {12} \) the weight formula (5) implies \( N = N\left( {i\infty }\right) = 0 \) so the only possible zeros of \( f \) are at the vertices \( \rho \) and \( i \) . For example, if \( k = 4 ... | Yes |
Theorem 6.4. Every entire modular form \( f \) of weight \( k \) is a polynomial in \( {G}_{4} \) and \( {G}_{6} \) of the type\n\n\[ f = \mathop{\sum }\limits_{{a, b}}{c}_{a, b}{G}_{4}{}^{a}{G}_{6}{}^{b} \]\n\nwhere the \( {c}_{a, b} \) are complex numbers and the sum is extended over all integers \( a \geq 0, b \geq ... | Proof. If \( k \) is odd, \( k < 0 \) or \( k = 2 \) the sum is empty and \( f \) is 0 . If \( k = 0, f \) is constant and the sum consists of only one term, \( {c}_{0,0} \) . If \( k = 4,6,8 \) or 10 then each of the respective quotients \( f/{G}_{4}, f/{G}_{6}, f/{G}_{4}{}^{2} \) and \( f/\left( {{G}_{4}{G}_{6}}\righ... | Yes |
Theorem 6.5. Let \( f \) be an entire form of weight \( k \) and let \( {z}_{1},\ldots ,{z}_{N} \) denote the\n\n\( N \) zeros of \( f \) in the closure of \( {R}_{\Gamma } \) (omitting the vertices) with zeros of order\n\n\( N\left( \rho \right), N\left( i\right) \) and \( N\left( {i\infty }\right) \) at the vertices.... | Proof. The product\n\n\[ g\left( \tau \right) = \mathop{\prod }\limits_{{k = 1}}^{N}\left\{ {J\left( \tau \right) - J\left( {z}_{k}\right) }\right\} \]\n\nis a modular function with its only zeros in the closure of \( {R}_{\Gamma } \) at \( {z}_{1},\ldots ,{z}_{N} \) and with a pole of order \( N \) at \( i\infty \) . ... | Yes |
Theorem 6.6. If \( f \in {M}_{k} \) and has the Fourier expansion\n\n\[ f\left( \tau \right) = \mathop{\sum }\limits_{{m = 0}}^{\infty }c\left( m\right) {e}^{2\pi im\tau } \]\n\nthen \( {T}_{n}f \) has the Fourier expansion\n\n(15)\n\n\[ \left( {{T}_{n}f}\right) \left( \tau \right) = \mathop{\sum }\limits_{{m = 0}}^{\i... | Proof. From the definition in (13) we find\n\n\[ \left( {{T}_{n}f}\right) \left( \tau \right) = {n}^{k - 1}\mathop{\sum }\limits_{{d \mid n}}{d}^{-k}\mathop{\sum }\limits_{{b = 0}}^{{d - 1}}\mathop{\sum }\limits_{{m = 0}}^{\infty }c\left( m\right) {e}^{{2\pi im}\left( {{n\tau } + {bd}}\right) /{d}^{2}} \]\n\n\[ = \math... | Yes |
Theorem 6.7. In every equivalence class of \( \Gamma \left( n\right) \) there is a representative of triangular form\n\n\[ \left( \begin{array}{ll} a & b \\ 0 & d \end{array}\right) ,\;\text{ where }d > 0. \] | Proof. Let \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) be an arbitrary element of \( \Gamma \left( n\right) \) . If \( c = 0 \) there is nothing more to prove. If \( c \neq 0 \) we reduce the fraction \( - a/c \) to lowest terms. That is, we choose integers \( r \) and \( s \) such that \( s/r ... | Yes |
Theorem 6.8. A complete system of nonequivalent elements in \( \Gamma \left( n\right) \) is given by the set of transformations of triangular form\n\n\[ A = \left( \begin{array}{ll} a & b \\ 0 & d \end{array}\right) \]\n\nwhere \( d \) runs through the positive divisors of \( n \) and, for each fixed \( d \) , \( a = n... | Proof. Theorem 6.7 shows that every element in \( \Gamma \left( n\right) \) is equivalent to one of the transformations in (18). Therefore we need only show that two such transformations, say\n\n\[ {A}_{1} = \left( \begin{matrix} {a}_{1} & {b}_{1} \\ 0 & {d}_{1} \end{matrix}\right) \;\text{ and }\;{A}_{2} = \left( \beg... | Yes |
Theorem 6.9. If \( {A}_{1} \in \Gamma \left( n\right) \) and \( {V}_{1} \in \Gamma \), then there exist matrices \( {A}_{2} \) in \( \Gamma \left( n\right) \) and \( {V}_{2} \) in \( \Gamma \) such that\n\n\[ {A}_{1}{V}_{1} = {V}_{2}{A}_{2} \]\n\nMoreover, if\n\n\[ {A}_{i} = \left( \begin{matrix} {a}_{i} & {b}_{i} \\ 0... | Proof. Since \( \det \left( {{A}_{1}{V}_{1}}\right) = \det {A}_{1}\det {V}_{1} = n \), the matrix \( {A}_{1}{V}_{1} \) is in \( \Gamma \left( n\right) \) so, by Theorem 6.7, there exists \( {A}_{2} \) in \( \Gamma \left( n\right) \) and \( {V}_{2} \) in \( \Gamma \) such that (22) holds. To verify (23) we first note th... | Yes |
Theorem 6.10. If \( f \in {M}_{k} \) and \( V = \left( \begin{array}{ll} \alpha & \beta \\ \gamma & \delta \end{array}\right) \in \Gamma \) then \[ \left( {{T}_{n}f}\right) \left( {V\tau }\right) = {\left( \gamma \tau + \delta \right) }^{k}\left( {{T}_{n}f}\right) \left( \tau \right) \] | Proof. We use the representation in (21) to write \[ \left( {{T}_{n}f}\right) \left( \tau \right) = \frac{1}{n}\mathop{\sum }\limits_{{A}_{1}}{a}_{1}{}^{k}f\left( {{A}_{1}\tau }\right) \] where \( {A}_{1} = \left( \begin{matrix} {a}_{1} & {b}_{1} \\ 0 & {d}_{1} \end{matrix}\right) \) and \( {A}_{1} \) runs through a co... | Yes |
Theorem 6.11. If \( f \in {M}_{k} \) then \( {T}_{n}f \in {M}_{k} \) . Moreover, if \( f \) is a cusp form then \( {T}_{n}f \) is also a cusp form. | Proof. If \( f \in {M}_{k} \) the definition of \( {T}_{n} \) shows that \( {T}_{n}f \) is analytic everywhere in \( H \) . Theorem 6.6 shows that \( {T}_{n}f \) has a Fourier expansion of the required form and that \( {T}_{n}f \) is analytic at \( i\infty \) . And Theorem 6.10 shows that \( {T}_{n}f \) has the proper ... | Yes |
Theorem 6.12. If \( \left( {m, n}\right) = 1 \) we have the composition property\n\n(26)\n\n\[{T}_{m}{T}_{n} = {T}_{mn}\] | Proof. If \( f \in {M}_{k} \) we have\n\n\[ \left( {{T}_{n}f}\right) \left( \tau \right) = \frac{1}{n}\mathop{\sum }\limits_{\substack{{a \geq 1,{ad} = n} \\ {0 \leq b < d} }}{a}^{k}f\left( {A\tau }\right) \]\n\nwhere \( A = \left( \begin{array}{ll} a & b \\ 0 & d \end{array}\right) \) . Applying \( {T}_{m} \) to each ... | Yes |
Theorem 6.14. Assume \( k \) is even, \( k \geq 4 \) . If the space \( {M}_{k} \) contains a simultaneous eigenform \( f \) with Fourier expansion (32), then \( c\left( 1\right) \neq 0 \) . | Proof. The coefficient of \( x \) in the Fourier expansion of \( {T}_{n}f \) is \( {\gamma }_{n}\left( 1\right) = c\left( n\right) \) . Since \( f \) is a simultaneous eigenform this coefficient is also equal to \( \lambda \left( n\right) c\left( 1\right) \), so\n\n\[ c\left( n\right) = \lambda \left( n\right) c\left( ... | Yes |
Theorem 6.15. Assume \( f \in {M}_{k,0} \) where \( k \) is even, \( k \geq {12} \) . Then \( f \) is a simultaneous normalized eigenform if, and only if, the coefficients in the Fourier expansion (32) satisfy the multiplicative property\n\n(39)\n\n\[ c\left( m\right) c\left( n\right) = \mathop{\sum }\limits_{{d \mid \... | Proof. The equation \( {T}_{n}f = \lambda \left( n\right) f \) is equivalent to the relation\n\n(40)\n\n\[ {\gamma }_{n}\left( m\right) = \lambda \left( n\right) c\left( m\right) \]\n\nobtained by equating coefficients of \( {x}^{m} \) in the corresponding Fourier expansions. Since \( f \) is a cusp form so is \( {T}_{... | Yes |
Theorem 6.16. Assume that \( f \in {M}_{2k} \), where \( k \geq 2 \), and that \( f \) is not a cusp form. Then \( f \) is a normalized simultaneous eigenform if, and only if,\n\n\[ f\left( \tau \right) = \frac{\left( {{2k} - 1}\right) !}{2{\left( 2\pi i\right) }^{2k}}{G}_{2k}\left( \tau \right) . \] | Proof. In the Fourier expansion (32) we have \( c\left( 0\right) \neq 0 \) since \( f \) is not a cusp form. The relation\n\n\[ {T}_{n}f = \lambda \left( n\right) f \]\n\nis equivalent to the relation\n\n\[ {\gamma }_{n}\left( m\right) = \lambda \left( n\right) c\left( m\right) \]\n\nobtained by equating coefficients o... | Yes |
Theorem 6.17. If \( f \in {M}_{{2k},0} \) we have\n\n\[ c\left( n\right) = O\left( {n}^{k}\right) \] | Proof. The series in (45) converges absolutely if \( \left| x\right| < 1 \) . Since \( c\left( 0\right) = 0 \) we can remove a factor \( x \) and write\n\n\[ \left| {f\left( \tau \right) }\right| = \left| x\right| \left| {\mathop{\sum }\limits_{{n = 1}}^{\infty }c\left( n\right) {x}^{n - 1}}\right| \leq \left| x\right|... | Yes |
Theorem 6.18. If \( f \in {M}_{2k} \) and \( f \) is not a cusp form, then\n\n\[ c\left( n\right) = O\left( {n}^{{2k} - 1}\right) \] | Proof. If \( f = {G}_{2k} \) each coefficient \( c\left( n\right) \) is of the form \( \alpha {\sigma }_{{2k} - 1}\left( n\right) \) where \( \alpha \) is independent of \( n \) . Hence\n\n\[ \left| {c\left( n\right) }\right| \leq \left| \alpha \right| {\sigma }_{{2k} - 1}\left( n\right) \]\n\nNow\n\n\[ {\sigma }_{{2k}... | Yes |
Theorem 6.19. If the coefficients \( c\left( n\right) \) satisfy the multiplicative property\n\n(51)\n\n\[ c\left( m\right) c\left( n\right) = \mathop{\sum }\limits_{{d \mid \left( {m, n}\right) }}{d}^{{2k} - 1}c\left( \frac{mn}{{d}^{2}}\right) \]\n\nthe Dirichlet series will have an Euler product representation of the... | Proof. Since the coefficients are multiplicative we have (see [4], Theorem 11.7)\n\n(53)\n\n\[ \varphi \left( s\right) = \mathop{\prod }\limits_{p}\left\{ {1 + \mathop{\sum }\limits_{{n = 1}}^{\infty }c\left( {p}^{n}\right) {p}^{-{ns}}}\right\} \]\n\nwhenever the Dirichlet series converges absolutely. Now (51) implies\... | Yes |
Theorem 6.20. Let \( \varphi \left( s\right) \) be the function defined for \( \sigma > k \) by the Dirichlet series (50) associated with a modular form \( f\left( \tau \right) \) in \( {M}_{k} \) having the Fourier series (49), where \( k \) is an even integer \( \geq 4 \) . Then \( \varphi \left( s\right) \) can be c... | Proof. From the integral representation for \( \Gamma \left( s\right) \) we have\n\n\[ \n\Gamma \left( s\right) {\left( 2n\pi \right) }^{-s} = {\int }_{0}^{\infty }{e}^{-{2\pi ny}}{y}^{s - 1}{dy} \n\]\n\nif \( \sigma > 0 \) . Therefore if \( \sigma > k \) we can multiply both members by \( c\left( n\right) \) and sum o... | Yes |
Theorem 7.1. Given any real \( \theta \) and any positive integer \( N \), there exist integers \( h \) and \( k \) with \( 0 < k \leq N \) such that\n\n\[ \left| {{k\theta } - h}\right| < \frac{1}{N} \] | Proof. Let \( \{ x\} = x - \left\lbrack x\right\rbrack \) denote the fractional part of \( x \) . Consider the \( N + 1 \) real numbers\n\n\[ 0,\{ \theta \} ,\{ {2\theta }\} ,\ldots ,\{ {N\theta }\} \text{.} \]\n\nAll these numbers lie in the half open unit interval \( 0 \leq \{ {m\theta }\} < 1 \) . Now divide the uni... | Yes |
Theorem 7.2. Given any real \( \theta \) and any positive integer \( N \), there exist relatively prime integers \( h \) and \( k \) with \( 0 < k \leq N \) such that\n\n\[ \left| {{k\theta } - h}\right| < \frac{1}{N} \] | Proof. By Theorem 7.1 there is a pair \( {h}^{\prime },{k}^{\prime } \) with \( 0 < {k}^{\prime } \leq N \) satisfying\n\n(3)\n\n\[ \left| {\theta - \frac{{h}^{\prime }}{{k}^{\prime }}}\right| < \frac{1}{N{k}^{\prime }} \]\n\nLet \( d = \left( {{h}^{\prime },{k}^{\prime }}\right) \) . If \( d = 1 \) there is nothing to... | Yes |
Theorem 7.3. For every real \( \theta \) there exist integers \( h \) and \( k \) with \( k > 0 \) and\n\n\( \left( {h, k}\right) = 1 \) such that\n\n\[ \left| {\theta - \frac{h}{k}}\right| < \frac{1}{{k}^{2}} \] | Proof. In Theorem 7.2 we have \( 1/\left( {Nk}\right) \leq 1/{k}^{2} \) because \( k \leq N \) . | No |
Theorem 7.4. If \( \theta \) is real, let \( S\left( \theta \right) \) denote the set of all ordered pairs of integers\n\n\( \left( {h, k}\right) \) with \( k > 0 \) and \( \left( {h, k}\right) = 1 \) such that\n\n\[ \left| {\theta - \frac{h}{k}}\right| < \frac{1}{{k}^{2}} \]\n\nThen \( S\left( \theta \right) \) has th... | Proof. Part (a) is merely a restatement of Theorem 7.3. To prove (b), assume \( \theta \) is irrational and assume also that \( S\left( \theta \right) \) is finite. We shall obtain a contradiction. Let\n\n\[ \alpha = \mathop{\min }\limits_{{\left( {h, k}\right) \in S\left( \theta \right) }}\left| {\theta - \frac{h}{k}}... | Yes |
Theorem 7.5. Let \( \theta \) be a real algebraic number of degree \( n \geq 2 \) . Then there is a positive constant \( C\left( \theta \right) \), depending only on \( \theta \), such that for all integers \( h \) and \( k \) with \( k > 0 \) we have\n\n\[ \left| {\theta - \frac{h}{k}}\right| > \frac{C\left( \theta \r... | Proof. Since \( \theta \) is algebraic of degree \( n,\theta \) is a zero of some polynomial \( f\left( x\right) \) of degree \( n \) with integer coefficients, say\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{r = 0}}^{n}{a}_{r}{x}^{r} \]\n\nwhere \( f\left( x\right) \) is irreducible over the rational field. Since... | Yes |
Theorem 7.6. Every Liouville number is transcendental. | Proof. If a Liouville number \( \theta \) were algebraic of degree \( n \) it would satisfy both inequality (7) and\n\n\[ \left| {\theta - \frac{{h}_{r}}{{k}_{r}}}\right| > \frac{C\left( \theta \right) }{{k}_{r}^{n}} \]\n\nfor every \( r \geq 1 \), where \( C\left( \theta \right) \) is the constant in Theorem 7.5. Ther... | Yes |
If \( \theta \) is a given irrational number the sequence of numbers \( \{ {n\theta }\} \) is dense in the unit interval. That is, given any \( \alpha ,0 \leq \alpha \leq 1 \), and given any \( \varepsilon > 0 \), there exists a positive integer \( k \) such that\n\n\[ \left| {\{ {k\theta }\} - \alpha }\right| < \varep... | Proof. First we note that \( \{ {n\theta }\} \neq \{ {m\theta }\} \) if \( m \neq n \) because \( \theta \) is irrational. Also, there is no loss of generality if we assume \( 0 < \theta < 1 \) since \( {n\theta } = \) \( n\left\lbrack \theta \right\rbrack + n\{ \theta \} \) and \( \{ {n\theta }\} = \{ n\{ \theta \} \}... | Yes |
Theorem 7.8. Given any real \( \alpha \), any irrational \( \theta \), and any \( \varepsilon > 0 \), there exist integers \( h \) and \( k \) with \( k > 0 \) such that\n\n\[ \left| {{k\theta } - h - \alpha }\right| < \varepsilon \text{. } \] | Proof. Write \( \alpha = \left\lbrack \alpha \right\rbrack + \{ \alpha \} \) . By Theorem 7.7 there exists \( k > 0 \) such that \( \left| {\{ {k\theta }\} -\{ \alpha \} }\right| < \varepsilon \) . Hence\n\n\[ \left| {{k\theta }-\lbrack {k\theta }\rbrack - \left( {\alpha -\lbrack \alpha \rbrack }\right) }\right| < \var... | Yes |
Lemma 1. Let \( \\left\\{ {\\lambda }_{n}\\right\\} \) be a sequence of distinct real numbers. For each real \( t \) and arbitrary complex numbers \( {c}_{0},\\ldots ,{c}_{N} \) define\n\n\[ f\\left( t\\right) = \\mathop{\\sum }\\limits_{{r = 0}}^{N}{c}_{r}{e}^{{it}{\\lambda }_{r}} \]\n\nThen for each \( k \) we have\n... | Proof. The definition of \( f\\left( t\\right) \) gives us\n\n\[ f\\left( t\\right) {e}^{-{it}{\\lambda }_{k}} = \\mathop{\\sum }\\limits_{{r = 0}}^{N}{c}_{r}{e}^{i\\left( {{\\lambda }_{r} - {\\lambda }_{k}}\\right) t}. \]\n\nHence\n\n\[ {\\int }_{0}^{T}f\\left( t\\right) {e}^{-{it}{\\lambda }_{k}}{dt} = \\mathop{\\sum... | Yes |
Lemma 3. Let \( g = g\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be the polynomial in \( n \) variables given by\n\n\[ g = 1 + {x}_{1} + {x}_{2} + \cdots + {x}_{n} \]\n\nand write\n\n(10)\n\n\[ {g}^{p} = 1 + \sum {a}_{{r}_{1},\ldots ,{r}_{n}}{x}_{1}{}^{{r}_{1}}\cdots {x}_{n}{}^{{r}_{n}}, \]\n\nwhere \( p \) is a positiv... | Proof. Since \( 1 + \sum {a}_{{r}_{1},\ldots ,{r}_{n}} = {g}^{p}\left( {1,1,\ldots ,1}\right) = {\left( 1 + n\right) }^{p} \) this proves (11). Let \( 1 + N \) be the number of terms in (10). We shall prove that\n\n(12)\n\n\[ 1 + N \leq {\left( p + 1\right) }^{n} \]\n\nby induction on \( n \) . For \( n = 1 \) we have\... | Yes |
Theorem 7.10 (Second form of Kronecker’s theorem). If \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) are arbitrary real numbers, if \( {\theta }_{1},\ldots ,{\theta }_{n},1 \) are linearly independent real numbers, and if \( \varepsilon > 0 \) is given, then there exists an integer \( k \) and integers \( {m}_{1},\ldots ,{m... | Proof. We apply the first form of Kronecker's theorem to the system \( {\alpha }_{1},\ldots ,{\alpha }_{n},0 \) and \( \left\{ {\theta }_{1}\right\} ,\left\{ {\theta }_{2}\right\} ,\ldots ,\left\{ {\theta }_{n}\right\} ,1 \), with \( \varepsilon /2 \) instead of \( \varepsilon \), where \( \varepsilon < 1 \) . Then the... | Yes |
Theorem 7.11. For each fixed \( \sigma > 1 \) we have\n\n\[ M\left( \sigma \right) = \zeta \left( \sigma \right) \;\text{ and }\;m\left( \sigma \right) = \frac{\zeta \left( {2\sigma }\right) }{\zeta \left( \sigma \right) }.\] | Proof. For \( \sigma > 1 \) we have \( \left| {\zeta \left( {\sigma + {it}}\right) }\right| \leq \zeta \left( \sigma \right) \) so \( M\left( \sigma \right) = \zeta \left( \sigma \right) \), the supremum being attained on the real axis. To obtain the result for \( m\left( \sigma \right) \) we estimate the reciprocal \(... | Yes |
Theorem 7.12. Let \( {\omega }_{1} \) and \( {\omega }_{2} \) be periods of \( f \) such that the ratio \( {\omega }_{2}/{\omega }_{1} \) is real and irrational. Then \( f \) has arbitrarily small nonzero periods. That is, given \( \varepsilon > 0 \) there is a period \( \omega \) such that \( 0 < \left| \omega \right|... | Proof. We apply Dirichlet’s approximation theorem. Let \( \theta = {\omega }_{2}/{\omega }_{1} \) . Since \( \theta \) is irrational, given any \( \varepsilon > 0 \) there exist integers \( h \) and \( k \) with \( k > 0 \) such that\n\n\[ \left| {{k\theta } - h}\right| < \frac{\varepsilon }{\left| {\omega }_{1}\right|... | Yes |
Theorem 8.2. Assume the series \( \sum a\left( n\right) {e}^{-{s\lambda }\left( n\right) } \) converges for some \( s \) with \( \sigma > 0 \) but diverges for all \( s \) with \( \sigma < 0 \) . Then the number\n\n\[ L = \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{\log \left| {\mathop{\sum }\limits_{{k = 1... | Proof. We know from Theorem 8.1 that the series converges for all \( s \) with \( \sigma > L \) and that \( L \) cannot be negative. Let \( S \) be the set of all \( \sigma > 0 \) such that the series converges for some \( s \) with real part \( \sigma \) . The set \( S \) is nonempty and bounded below. Let \( {\sigma ... | Yes |
Theorem 8.3. Assume the series \( \sum a\left( n\right) {e}^{-{s\lambda }\left( n\right) } \) converges absolutely for some \( s \) with \( \sigma > 0 \) but diverges for all \( s \) with \( \sigma < 0 \) . Then the number \[ {\sigma }_{a} = \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{\log \mathop{\sum }\li... | Proof. Let \( A \) be the abscissa of convergence of the series \( \sum \left| {a\left( n\right) }\right| {e}^{-{s\lambda }\left( n\right) } \) . Then, by Theorem 8.2, \[ A = \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{\log \mathop{\sum }\limits_{{k = 1}}^{n}\left| {a\left( k\right) }\right| }{\lambda \left... | Yes |
Example 2. Let \( \Lambda = \{ \log n\} \) . Then \( B = \left\{ {\log {p}_{n}}\right\} \) is a basis, where \( {p}_{n} \) is the \( n \) th prime. It is easy to verify properties (a),(b) and (c). | For independence we note that\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{q}{r}_{k}\log {p}_{k} = 0\;\text{ implies }\;{p}_{1}{}^{{r}_{1}}\cdots {p}_{q}{}^{{r}_{q}} = 1\;\text{ so }\;{r}_{1} = \cdots = {r}_{q} = 0. \]\n\nTo express each \( \lambda \left( n\right) \) in terms of the basis elements we factor \( n \) and compu... | Yes |
Theorem 8.4. Every sequence \( \Lambda \) has a subsequence which is a basis for \( \Lambda \) . | Proof. Construct a basis as follows. For the first basis element take \( \lambda \left( {n}_{1}\right) \) , the first nonzero \( \lambda \) (either \( \lambda \left( 1\right) \) or \( \lambda \left( 2\right) \) ), and call this \( \beta \left( 1\right) \) . Now delete the remaining elements of \( \Lambda \) that are ra... | Yes |
Theorem 8.5. If \( \Lambda \) has two bases \( B \) and \( \Gamma \), then there exists a Bohr matrix \( A \) such that \( \Gamma = {AB} \) . | Proof. There exist Bohr matrices \( R \) and \( T \) such that \( \Gamma = {T\Lambda } \) and \( \Lambda = {RB} \) . Hence \( \Gamma = T\left( {RB}\right) = \left( {TR}\right) B = {AB} \) where \( A = {TR} \) . | Yes |
Theorem 8.6. Let \( B \) and \( \Gamma \) be two bases for \( \Lambda \), and write \( \Gamma = {AB},\Lambda = {R}_{B}B \), \( \Lambda = {R}_{\Gamma }\Gamma \), where \( A,{R}_{B},{R}_{\Gamma } \) are Bohr matrices. Then \( {R}_{B} = {R}_{\Gamma }A \). Note. If we write \( \Lambda /B \) for \( {R}_{B},\Lambda /\Gamma \... | Proof. We have \( \Lambda = {R}_{B}B \) and \( \Lambda = {R}_{\Gamma }\Gamma = {R}_{\Gamma }{AB} \). Hence \( {R}_{B}B = {R}_{\Gamma }{AB} \), so \( \left( {{R}_{B} - {R}_{\Gamma }A}\right) B = 0 \). Since \( {R}_{B} - {R}_{\Gamma }A \) is a Bohr matrix and \( B \) is a basis, we must have \( {R}_{B} - {R}_{\Gamma }A =... | Yes |
Theorem 8.7. Let \( B \) and \( \Gamma \) be two bases for \( \Lambda \) and write \( \Gamma = {AB} \) for some Bohr matrix A. Then\n\n\[ \n{F}_{B}\left( Z\right) = {F}_{\Gamma }\left( {AZ}\right) \n\] | Proof. By Theorem 8.6 we have\n\n\[ \n\Lambda = {R}_{B}B = {R}_{\Gamma }\Gamma ,\;\text{ where }{R}_{B} = {R}_{\Gamma }A. \n\]\n\nHence\n\n\[ \n{F}_{B}\left( Z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }a\left( n\right) {e}^{-{\left( {R}_{B}Z\right) }_{n}} = \mathop{\sum }\limits_{{n = 1}}^{\infty }a\left( n\rig... | Yes |
Theorem 8.8. If \( B \) and \( \Gamma \) are two bases for \( \Lambda \) then \( {U}_{f}\left( {\sigma ;B}\right) = {U}_{f}\left( {\sigma ;\Gamma }\right) \) . | Proof. Choose any value \( {F}_{B}\left( Z\right) \) in \( {U}_{f}\left( {\sigma ;B}\right) \), so that \( \operatorname{Re}Z = {\sigma B} \) . By Theorem 8.7 we have \( {F}_{B}\left( Z\right) = {F}_{\Gamma }\left( {AZ}\right) \), where \( \Gamma = {AB} \) . But\n\n\[ \operatorname{Re}{AZ} = A\operatorname{Re}Z = {A\si... | Yes |
Theorem 8.10. Two equivalent Dirichlet series have the same abscissa of absolute convergence. Moreover, the relation \( \sim \) just defined is independent of the basis \( B \) . | Proof. Equivalence implies \( \left| {b\left( n\right) }\right| = \left| {a\left( n\right) }\right| \) so the series have the same abscissa of absolute convergence.\n\nNow let \( B \) and \( \Gamma \) be two bases for \( \Lambda \), and assume that two series are equivalent with respect to \( B \) . We will show that t... | Yes |
Theorem 8.11. The relation \( \sim \) defined in the foregoing definition is an equivalence relation. That is, it is reflective, symmetric, and transitive. | Proof. Every series is equivalent to itself since we may take each \( {y}_{n} = 0 \) . The corresponding \( {x}_{n} \) will then be zero.\n\nIf \( b\left( n\right) = a\left( n\right) {e}^{i{x}_{n}} \) then \( a\left( n\right) = b\left( n\right) {e}^{-i{x}_{n}} \) . Since \( X = {R}_{B}Y \) we have \( - X = \) \( {R}_{B... | Yes |
Theorem 8.13. Let \( f\left( s\right) \) and \( g\left( s\right) \) be equivalent general Dirichlet series, each of which converges absolutely for \( \sigma = {\sigma }_{0} \) . Then\n\n\[ \n{U}_{f}\left( {\sigma }_{0}\right) = {U}_{g}\left( {\sigma }_{0}\right) \n\] | Proof. Let \( B = \{ \beta \left( n\right) \} \) be a basis for the sequence \( \Lambda \) of exponents. If \( f\left( s\right) = \sum a\left( n\right) {e}^{-{s\lambda }\left( n\right) } \) and \( g\left( s\right) = \sum b\left( n\right) {e}^{-{s\lambda }\left( n\right) } \) then there is a real sequence \( \left\{ {y}... | Yes |
Theorem 8.16 (Bohr’s equivalence theorem). Let \( f \) and \( g \) be equivalent Dirichlet series with abscissa of absolute convergence \( {\sigma }_{a} \) . Then in any open half plane \( \sigma > {\sigma }_{1} \geq {\sigma }_{a} \) the functions \( f\left( s\right) \) and \( g\left( s\right) \) take the same set of v... | Proof. Let \( {S}_{f}\left( {\sigma }_{1}\right) \) be the set of values taken by \( f\left( s\right) \) in the half-plane \( \sigma > {\sigma }_{1} \) . Then\n\n\[ \n{S}_{f}\left( {\sigma }_{1}\right) = \mathop{\bigcup }\limits_{{{\sigma }_{0} > {\sigma }_{1}}}{V}_{f}\left( {\sigma }_{0}\right) \n\] \n\nNow we prove t... | Yes |
Lemma 1 (Helly selection principle). Let \( \left\{ {\theta }_{m, n}\right\} \) be a double sequence of real numbers which is bounded, say\n\n\[ \left| {\theta }_{m, n}\right| < A\text{ for all }m, n\text{. }\n\]\n\nThen there exists a subsequence of integers \( {n}_{1} < {n}_{2} < \cdots \) with \( {n}_{r} \rightarrow... | Proof of LEMMA 1. Let \( {\theta }_{1} \) be an accumulation point of the first row and suppose the subsequence \( \left\{ {n}_{r}^{\left( 1\right) }\right\} \) has the property that\n\n\[ \mathop{\lim }\limits_{{r \rightarrow \infty }}{\theta }_{1,{n}_{r}\left( 1\right) } = {\theta }_{1} \]\n\nIn the second row, consi... | Yes |
Lemma 2 (Rouché’s theorem). Given two functions \( f\left( z\right) \) and \( g\left( z\right) \) analytic inside and on a closed circular contour C. Assume\n\n\[ \left| {g\left( z\right) }\right| < \left| {f\left( z\right) }\right| \;\text{ on }C. \]\n\nThen \( f\left( z\right) \) and \( f\left( z\right) + g\left( z\r... | Proof of LEMMA 2. Let \( m = \inf \{ \left| {f\left( z\right) }\right| - \left| {g\left( z\right) }\right| : z \in C\} \) . Then \( m > 0 \) because \( C \) is compact and the difference \( \left| {f\left( z\right) }\right| - \left| {g\left( z\right) }\right| \) is a continuous function on \( C \) . Hence for all real ... | Yes |
Theorem 8.17. Let \( k \geq 1 \) be a given integer, and let \( \chi \) be any Dirichlet character modulo \( k \) . Let \( \mathop{\sum }\limits_{{n = 1}}^{\infty }a\left( n\right) {n}^{-s} \) be any Dirichlet series whose coefficients have the following property:\n\n\[ a\left( n\right) \neq 0\text{ implies }\left( {n,... | Proof. Since these are ordinary Dirichlet series we may use Theorem 8.12 to establish the equivalence. In this case we take \( f\left( n\right) = \chi \left( n\right) \) . Then \( f \) is completely multiplicative and condition (a) is satisfied. Now we show that condition (b) is satisfied. We need to show that \( \left... | Yes |
Theorem 8.18. For a given modulus \( k \), let \( {\chi }_{1},\ldots ,{\chi }_{\varphi \left( k\right) } \) denote the Dirichlet characters modulo \( k \) . Then in any half-plane of the form \( \sigma > {\sigma }_{1} \geq 1 \) the set of values taken by the Dirichlet \( L \) -series \( L\left( {s,{\chi }_{i}}\right) \... | Proof. Applying the previous theorem with \( a\left( n\right) = {\chi }_{1}\left( n\right) \) we have\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\chi }_{1}\left( n\right) }{{n}^{s}} \sim \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\chi }_{1}\left( n\right) \chi \left( n\right) }{{n}^{s}} \]\n\nfor every ... | Yes |
Theorem 8.19. Let \( \\lambda \\left( n\\right) \) denote Liouville’s function and let\n\n\[ \nC\\left( x\\right) = \\mathop{\\sum }\\limits_{{n \\leq x}}\\frac{\\lambda \\left( n\\right) }{n}\n\]\n\nThen if \( \\sigma > 1 \) we have\n\n\[ \n\\frac{\\zeta \\left( {2s}\\right) }{\\left( {s - 1}\\right) \\zeta \\left( s\... | Proof. By Abel's identity (Theorem 4.2 in [4]) we have\n\n\[ \n\\mathop{\\sum }\\limits_{{n \\leq x}}\\frac{\\lambda \\left( n\\right) }{n}\\frac{1}{{n}^{s}} = \\frac{C\\left( x\\right) }{{x}^{s}} + s{\\int }_{1}^{x}\\frac{C\\left( t\\right) }{{t}^{s + 1}}{dt}.\n\]\n\nKeep \( \\sigma > 0 \) and let \( x \\rightarrow \\... | Yes |
Theorem 8.20. Let\n\n\[ \n{\zeta }_{n}\left( s\right) = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{{k}^{s}} \]\n\nIf there exists an \( {n}_{0} \) such that \( {\zeta }_{n}\left( s\right) \neq 0 \) for all \( n \geq {n}_{0} \) and all \( \sigma > 1 \), then \( \zeta \left( s\right) \neq 0 \) for \( \sigma > \frac{1}{2... | Proof. First we note that the two Dirichlet series \( \mathop{\sum }\limits_{{k = 1}}^{n}{k}^{-s} \) and \( \mathop{\sum }\limits_{{k = 1}}^{n}\lambda \left( k\right) {k}^{-s} \) are equivalent because \( \lambda \) is completely multiplicative and has absolute\n\nvalue 1. Therefore, by Bohr’s theorem, \( {\zeta }_{n}\... | Yes |
Theorem 8.21 (Turán). Let \( C\left( x\right) = \mathop{\sum }\limits_{{n \leq x}}\lambda \left( n\right) /n \) . If there exist constants \( \alpha > 0, c > 0 \) and \( {n}_{0} \) such that\n\n(12)\n\n\[ C\left( x\right) > - c\frac{{\log }^{\alpha }x}{\sqrt{x}} \]\n\nfor all \( x \geq {n}_{0} \), then the Riemann hypo... | Proof. If \( \varepsilon > 0 \) is given there exists an \( {n}_{1} \geq {n}_{0} \) such that \( c{\log }^{\alpha }x \leq {x}^{\varepsilon } \) for all \( x \geq {n}_{1} \) so (12) implies\n\n\[ C\left( x\right) > - {x}^{\varepsilon - 1/2}. \]\n\nLet \( A\left( x\right) = C\left( x\right) + {x}^{\varepsilon - 1/2} \), ... | Yes |
Lemma 1. If \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma \) and \( c > 0 \), then for every integer \( m \) we have\n\n\[ \varepsilon \left( {A{T}^{m}}\right) = {e}^{{\pi im}/{12}}\varepsilon \left( A\right) \] | Proof. We have \( A{T}^{m} = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \left( \begin{matrix} 1 & m \\ 0 & 1 \end{matrix}\right) = \left( \begin{array}{ll} a & {am} + b \\ c & {cm} + d \end{array}\right) \), so\n\n\[ \varepsilon \left( {A{T}^{m}}\right) = \exp \left\{ {{\pi i}\left( {\frac{a + {cm} + d}... | Yes |
Lemma 2. If \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma \) and \( c > 0 \), then we have\n\n\[ \varepsilon \left( {AS}\right) = \left\{ \begin{array}{ll} {e}^{-{\pi i}/4}\varepsilon \left( A\right) & \text{ if }d > 0, \\ {e}^{{\pi i}/4}\varepsilon \left( A\right) & \text{ if }d < 0. \en... | Proof. We have\n\n\[ {AS} = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \left( \begin{array}{rr} 0 & - 1 \\ 1 & 0 \end{array}\right) = \left( \begin{array}{ll} b & - a \\ d & - c \end{array}\right) . \]\n\nIf \( d > 0 \), we represent the transformation \( {AS} \) by the matrix\n\n\[ {AS} = \left( \begin... | Yes |
Lemma 3. If Dedekind's functional equation\n\n(5)\n\n\[ \eta \left( {A\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) ,\]\n\nis satisfied for some \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \) with \( c > 0 \) and ... | Proof. Replace \( \tau \) by \( {T}^{m}\tau \) in (5) to obtain\n\n\[ \eta \left( {A{T}^{m}\tau }\right) = \varepsilon \left( A\right) {\left\{ -i\left( c{T}^{m}\tau + d\right) \right\} }^{1/2}\eta \left( {{T}^{m}\tau }\right) \]\n\n\[ = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + {mc} + d}\right) {\} }^{1/2}{... | Yes |
Theorem 1. Let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{V}\right) \) be a linear representation of \( \mathrm{G} \) in \( \mathrm{V} \) and let \( \mathrm{W} \) be a vector subspace of \( \mathrm{V} \) stable under \( \mathrm{G} \) . Then there exists a complement \( {\mathrm{W}}^{0} \) of \( \mathr... | Let \( {\mathrm{W}}^{\prime } \) be an arbitrary complement of \( \mathrm{W} \) in \( \mathrm{V} \), and let \( p \) be the corresponding projection of \( \mathrm{V} \) onto \( \mathrm{W} \) . Form the average \( {p}^{0} \) of the conjugates of \( p \) by the elements of \( \mathrm{G} \) :\n\n\[ \n{p}^{0} = \frac{1}{g}... | Yes |
Theorem 2. Every representation is a direct sum of irreducible representations. | Let \( \mathrm{V} \) be a linear representation of \( \mathrm{G} \) . We proceed by induction on \( \dim \left( \mathrm{V}\right) \) . If \( \dim \left( \mathrm{V}\right) = 0 \), the theorem is obvious \( (0 \) is the direct sum of the empty family of irreducible representations). Suppose then \( \dim \left( \mathrm{V}... | Yes |
Proposition 1. If \( \chi \) is the character of a representation \( \rho \) of degree \( n \), we have:\n\n(i) \( \chi \left( 1\right) = n \) ,\n\n(ii) \( \chi \left( {s}^{-1}\right) = \chi {\left( s\right) }^{ * }\; \) for \( s \in \mathrm{G} \) ,\n\n(iii) \( \chi \left( {{ts}{t}^{-1}}\right) = \chi \left( s\right) \... | We have \( \rho \left( 1\right) = 1 \), and \( \operatorname{Tr}\left( 1\right) = n \) since \( \mathrm{V} \) has dimension \( n \) ; hence (i).\n\nFor (ii) we observe that \( {\rho }_{s} \) has finite order; consequently the same is true\nof its eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) and so these have... | Yes |
Proposition 2. Let \( {\rho }^{1} : \mathrm{G} \rightarrow \mathrm{{GL}}\left( {\mathrm{V}}_{1}\right) \) and \( {\rho }^{2} : \mathrm{G} \rightarrow \mathrm{{GL}}\left( {\mathrm{V}}_{2}\right) \) be two linear representations of \( \mathrm{G} \), and let \( {\chi }_{1} \) and \( {\chi }_{2} \) be their characters. The... | Let us be given \( {\rho }^{1} \) and \( {\rho }^{2} \) in matrix form: \( {\mathbf{R}}_{s}^{1},{\mathbf{R}}_{s}^{2} \) . The representation \( {\mathrm{V}}_{1} \oplus {\mathrm{V}}_{2} \) is then given by\n\n\[ \n{\mathrm{R}}_{s} = \left( \begin{matrix} {\mathrm{R}}_{s}^{1} & 0 \\ 0 & {\mathrm{R}}_{s}^{2} \end{matrix}\... | Yes |
Proposition 3. Let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{V}\right) \) be a linear representation of \( \mathrm{G} \), and let \( \chi \) be its character. Let \( {\chi }_{\sigma }^{2} \) be the character of the symmetric square \( {\operatorname{Sym}}^{2}\left( \mathrm{\;V}\right) \) of \( \mathr... | Let \( s \in \mathrm{G} \) . A basis \( \left( {e}_{i}\right) \) of \( \mathrm{V} \) can be chosen consisting of eigenvectors for \( {\rho }_{s} \) ; this follows for example from the fact that \( {\rho }_{s} \) can be represented by a unitary matrix, cf. 1.3. We have then \( {\rho }_{s}{e}_{i} = {\lambda }_{i}{e}_{i} ... | Yes |
Proposition 4(Schur’s lemma). Let \( {\rho }^{1} : \mathrm{G} \rightarrow \mathrm{{GL}}\left( {\mathrm{V}}_{1}\right) \) and \( {\rho }^{2} : \mathrm{G} \rightarrow \mathrm{{GL}}\left( {\mathrm{V}}_{2}\right) \) be two irreducible representations of \( \mathrm{G} \), and let \( f \) be a linear mapping of \( {\mathrm{V... | The case \( f = 0 \) is trivial. Suppose now \( f \neq 0 \) and let \( {\mathrm{W}}_{1} \) be its kernel (that is, the set of \( x \in {\mathrm{V}}_{1} \) such that \( {fx} = 0 \) ). For \( x \in {\mathrm{W}}_{1} \) we have \( f{\rho }_{s}^{1}x = {\rho }_{s}^{2}{fx} \) \( = 0 \), whence \( {\rho }_{s}^{1}\chi \in {\mat... | Yes |
Corollary 1. Let \( h \) be a linear mapping of \( {V}_{1} \) into \( {V}_{2} \), and put:\n\n\[ \n{h}^{0} = \frac{1}{g}\mathop{\sum }\limits_{{t \in \mathrm{G}}}{\left( {\rho }_{t}^{2}\right) }^{-1}h{\rho }_{t}^{1} \]\n\nThen:\n\n(1) If \( {\rho }^{1} \) and \( {\rho }^{2} \) are not isomorphic, we have \( {h}^{0} = 0... | We have \( {\rho }_{s}^{2}{h}^{0} = {h}^{0}{\rho }_{s}^{1} \) . Indeed:\n\n\[ \n{\left( {\rho }_{s}^{2}\right) }^{-1}{h}^{0}{\rho }_{s}^{1} = \frac{1}{g}\mathop{\sum }\limits_{{t \in \mathrm{G}}}{\left( {\rho }_{s}^{2}\right) }^{-1}{\left( {\rho }_{t}^{2}\right) }^{-1}h{\rho }_{t}^{1}{\rho }_{s}^{1} \]\n\n\[ \n= \frac{... | Yes |
Theorem 4. Let \( \mathrm{V} \) be a linear representation of \( \mathrm{G} \), with character \( \phi \), and suppose \( \mathrm{V} \) decomposes into a direct sum of irreducible representations:\n\n\[ \mathrm{V} = {\mathrm{W}}_{1} \oplus \cdots \oplus {\mathrm{W}}_{k} \]\n\nThen, if \( \mathrm{W} \) is an irreducible... | Let \( {\chi }_{i} \) be the character of \( {\mathrm{W}}_{i} \). By prop. 2, we have\n\n\[ \phi = {\chi }_{1} + \cdots + {\chi }_{k} \]\n\nThus \( \left( {\phi \mid \chi }\right) = \left( {{\chi }_{1} \mid \chi }\right) + \cdots + \left( {{\chi }_{k} \mid \chi }\right) \). But, according to the preceeding theorem, \( ... | Yes |
Corollary 1. The number of \( {\mathrm{W}}_{i} \) isomorphic to \( \mathrm{W} \) does not depend on the chosen decomposition. | Indeed, \( \left( {\phi \mid \chi }\right) \) does not depend on the decomposition. | Yes |
Corollary 2. Two representations with the same character are isomorphic. | Indeed, cor. 1 shows that they contain each given irreducible representation the same number of times. | No |
Theorem 5. If \( \phi \) is the character of a representation \( \mathrm{V},\left( {\phi \mid \phi }\right) \) is a positive integer and we have \( \left( {\phi \mid \phi }\right) = 1 \) if and only if \( \mathrm{V} \) is irreducible. | Indeed, \( \sum {m}_{i}^{2} \) is only equal to 1 if one of the \( {m}_{i} \) ’s is equal to 1 and the others to 0, that is, if \( \mathrm{V} \) is isomorphic to one of the \( {\mathrm{W}}_{i} \) . | Yes |
Corollary 1. Every irreducible representation \( {\mathrm{W}}_{i} \) is contained in the regular representation with multiplicity equal to its degree \( {n}_{i} \) . | According to th. 4, this number is equal to \( \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle \), and we have\n\n\[ \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle = \frac{1}{g}\mathop{\sum }\limits_{{s \in \mathrm{G}}}{r}_{\mathrm{G}}\left( {s}^{-1}\right) {\chi }_{i}\left( s\right) = \frac{1}{g}g \... | Yes |
(a) The degrees \( {n}_{i} \) satisfy the relation \( \mathop{\sum }\limits_{{i = 1}}^{{i = h}}{n}_{i}^{2} = g \) .\n\n(b) If \( s \in \mathbf{G} \) is different from 1, we have \( \mathop{\sum }\limits_{{i = 1}}^{{i = h}}{n}_{i}{\chi }_{i}\left( s\right) = 0 \) . | By cor. 1, we have \( {r}_{\mathrm{G}}\left( s\right) = \sum {n}_{i}{\chi }_{i}\left( s\right) \) for all \( s \in \mathrm{G} \) . Taking \( s = 1 \) we obtain (a), and taking \( s \neq 1 \), we obtain (b). | Yes |
Proposition 6. Let \( f \) be a class function on \( \mathbf{G} \), and let \( \rho : \mathbf{G} \rightarrow \mathbf{{GL}}\left( \mathbf{V}\right) \) be a linear representation of \( \mathrm{G} \). Let \( {\rho }_{f} \) be the linear mapping of \( \mathrm{V} \) into itself defined by:\n\n\[ \n{\rho }_{f} = \mathop{\sum... | Let us compute \( {\rho }_{s}^{-1}{\rho }_{f}{\rho }_{s} \). We have:\n\n\[ \n{\rho }_{s}^{-1}{\rho }_{f}{\rho }_{s} = \mathop{\sum }\limits_{{t \in \mathrm{G}}}f\left( t\right) {\rho }_{s}^{-1}{\rho }_{t}{\rho }_{s} = \mathop{\sum }\limits_{{t \in \mathrm{G}}}f\left( t\right) {\rho }_{{s}^{-1}{ts}}. \n\]\n\nPutting \(... | Yes |
Theorem 6. The characters \( {\chi }_{1},\ldots ,{\chi }_{h} \) form an orthonormal basis of \( \mathrm{H} \) . | Theorem 3 shows that the \( {\chi }_{i} \) form an orthonormal system in H. It remains to prove that they generate \( \mathrm{H} \), and for this it is enough to show that every element of \( \mathrm{H} \) orthogonal to the \( {\chi }_{i}^{ * } \) is zero. Let \( f \) be such an element. For each representation \( \rho... | Yes |
Theorem 7. The number of irreducible representations of \( \mathrm{G} \) (up to isomorphism) is equal to the number of classes of \( \mathbf{G} \) . | Let \( {\mathrm{C}}_{1},\ldots ,{\mathrm{C}}_{k} \) be the distinct classes of \( \mathrm{G} \) . To say that a function \( f \) on \( \mathrm{G} \) is a class function is equivalent to saying that it is constant on each of \( {\mathrm{C}}_{1},\ldots ,{\mathrm{C}}_{k} \) ; it is thus determined by its values \( {\lambd... | Yes |
Proposition 7. Let \( s \in \mathrm{G} \), and let \( c\left( s\right) \) be the number of elements in the conjugacy class of \( s \) .\n\n(a) We have \( \mathop{\sum }\limits_{{i = 1}}^{{i = h}}{\chi }_{i}{\left( s\right) }^{ * }{\chi }_{i}\left( s\right) = g/c\left( s\right) \) . | Let \( {f}_{s} \) be the function equal to 1 on the class of \( s \) and equal to 0 elsewhere. Since it is a class function, it can, by th. 6, be written\n\n\[ \n{f}_{s} = \mathop{\sum }\limits_{{i = 1}}^{{i = h}}{\lambda }_{i}{\chi }_{i},\;\text{ with }{\lambda }_{i} = \left( {{f}_{s} \mid {\chi }_{i}}\right) = \frac{... | Yes |
(ii) The projection \( {p}_{i} \) of \( \mathrm{V} \) onto \( {\mathrm{V}}_{i} \) associated with this decomposition is given by the formula:\n\n\[ \n{p}_{i} = \frac{{n}_{i}}{g}\mathop{\sum }\limits_{{t \in \mathrm{G}}}{\chi }_{i}{\left( t\right) }^{ * }{\rho }_{t} \n\] | We prove (ii). Assertion (i) will follow because the projections \( {p}_{i} \) determine the \( {\mathrm{V}}_{i} \) . Put\n\n\[ \n{q}_{i} = \frac{{n}_{i}}{g}\mathop{\sum }\limits_{{t \in \mathrm{G}}}{\chi }_{i}{\left( t\right) }^{ * }{\rho }_{t} \n\]\n\nProposition 6 shows that the restriction of \( {q}_{i} \) to an ir... | Yes |
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