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The main armament of the original model M1 and M1IP was the M68A1 105 mm rifled tank gun firing a variety of armor-piercing fin-stabilized discarding sabot, high-explosive anti-tank, high explosive, white phosphorus rounds and an anti-personnel (multiple flechette) round. This gun used a license-made tube of the British Royal Ordnance L7 gun together with the vertical sliding breech block and other parts of the U.S. T254E2 prototype gun. However, it proved to be inadequate; a cannon with lethality beyond the range was needed to combat newer armor technologies. To attain that lethality, the projectile diameter needed to be increased. The tank was able to carry 55 105 mm rounds, with 44 stored in the turret blow-out compartment and the rest in hull stowage.
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The main armament of the M1A1 and M1A2 is the M256A1 120 mm smoothbore gun, designed by Rheinmetall AG of Germany, manufactured under license in the U.S. by Watervliet Arsenal, New York. The M256A1 is an improved variant of the Rheinmetall 120 mm L/44 gun carried on the German Leopard 2 on all variants up to the Leopard 2A5, the difference being in thickness and chamber pressure. Leopard 2A6 replaced the L/44 barrel with a longer L/55. Due to the increased caliber, only 40 or 42 rounds are able to be stored depending on if the tank is an A1 or A2 model.
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The M256A1 fires a variety of rounds. The primary APFSDS round of the Abrams is the depleted uranium M829 round, of which four variants have been designed. M829A1, known as the "Silver Bullet", saw widespread service in the Gulf War, where it proved itself against Iraqi armor such as the T-72. The M829A2 APFSDS round was developed specifically as an immediate solution to address the improved protection of a Russian T-72, T-80U or T-90 main battle tank equipped with Kontakt-5 explosive reactive armor (ERA).
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Later, the M829A3 round was introduced to improve its effectiveness against next generation ERA equipped tanks, through usage of a multi-material penetrator and increased penetrator diameter that can resist the shear effect of K-5 type ERA. As a counter to that, the Russian army introduced Relikt, the most modern Russian ERA, which is claimed to be twice as effective as Kontakt-5. Development of the M829 series is continuing with the M829A4 currently entering production, featuring advanced technology such as data-link capability.
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The Abrams also fires high-explosive anti-tank warhead shaped charge rounds such as the M830, the latest version of which (M830A1) incorporates a sophisticated multi-mode electronic sensing fuse and more fragmentation that allows it to be used effectively against armored vehicles, personnel, and low-flying aircraft. The Abrams uses a manual loader, who also provides additional support for maintenance, observation post/listening post (OP/LP) operations, and other tasks.
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The new M1028 120 mm anti-personnel canister cartridge was brought into service early for use in the aftermath of the 2003 invasion of Iraq. It contains 1,098 tungsten balls that spread from the muzzle to produce a shotgun effect lethal out to . The tungsten balls can be used to clear enemy dismounts, break up hasty ambush sites in urban areas, clear defiles, stop infantry attacks and counter-attacks and support friendly infantry assaults by providing covering fire. The canister round is also a highly effective breaching round and can level cinder block walls and knock man-sized holes in reinforced concrete walls for infantry raids at distances up to .
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Also in use is the M908 obstacle-reduction round. It is designed to destroy obstacles and barriers. The round is a modified M830A1 with the front fuse replaced by a steel nose to penetrate into the obstacle before detonation.
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The U.S. Army Research Laboratory (ARL) conducted a thermal analysis of the M256 from 2002 to 2003 to evaluate the potential of using a hybrid barrel system that would allow for multiple weapon systems such as the XM1111 Mid-Range munition, airburst rounds, or XM 1147. The test concluded that mesh density (number of elements per unit area) impacts accuracy of the M256 and specific densities would be needed for each weapon system
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In 2013 the Army was developing a new round to replace the M830/M830A1, M1028, and M908. Called the Advanced Multi-Purpose (AMP) round, it will have point detonation, delay, and airburst modes through an ammunition data-link and a multi-mode, programmable fuse in a single munition. Having one round that does the job of four would simplify logistics and be able to be used on a variety of targets. The AMP is to be effective against bunkers, infantry, light armor, and obstacles out to 500 meters, and will be able to breach reinforced concrete walls and defeat ATGM teams from 500 to 2,000 meters. Orbital ATK was awarded a contract to begin the first phase of development for the AMP XM1147 High-Explosive Multi-Purpose with Tracer cartridge in October 2015.
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In addition to these, the XM1111 (Mid-Range-Munition Chemical Energy) was also in development. The XM1111 was a guided munition using a dual-mode seeker that combined imaging-infrared and semi-active laser guidance. The MRM-CE was selected over the competing MRM-KE, which used a rocket-assisted kinetic energy penetrator. The CE variant was chosen due to its better effects against secondary targets, providing a more versatile weapon. The Army hoped to achieve IOC with the XM1111 by 2013. However, the Mid-Range Munition was cancelled in 2009 along with Future Combat Systems.
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The Abrams is equipped with a ballistic fire-control computer that uses user and system-supplied data from a variety of sources to compute, display, and incorporate the three components of a ballistic solution—lead angle, ammunition type, and range to the target—to accurately fire the main gun. These three components are determined using a laser rangefinder, crosswind sensor, a pendulum static cant sensor, data concerning performance and flight characteristics of each specific type of round, tank-specific boresight alignment data, ammunition temperature, air temperature, barometric pressure, a muzzle reference system (MRS) that determines and compensates for barrel drop at the muzzle due to gravitational pull and barrel heating due to firing or sunlight, and target speed determined by tracking rate tachometers in the Gunner's or Commander's Controls Handles.
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All of these factors are computed into a ballistic solution and updated 30 times per second. The updated solution is displayed in the Gunner's or Tank Commander's field of view in the form of a reticle in both day and Thermal modes. The ballistic computer manipulates the turret and a complex arrangement of mirrors so that all one has to do is keep the reticle on the target and fire to achieve a hit. Proper lead and gun tube elevation are applied to the turret by the computer, greatly simplifying the job of the gunner.
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The fire-control system uses this data to compute a firing solution for the gunner. The ballistic solution generated ensures a hit percentage greater than 95 percent at nominal ranges. Either the commander or gunner can fire the main gun. Additionally, the Commander's Independent Thermal Viewer (CITV) on the M1A2 can be used to locate targets and pass them on for the gunner to engage while the commander scans for new targets.
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If the primary sight system malfunctions or is damaged, the main and coaxial weapons can be manually aimed using a telescopic scope boresighted to the main gun known as the Gunner's Auxiliary Sight (GAS). The GAS has two interchangeable reticles; one for "high-explosive anti-tank" (HEAT) and "multi-purpose anti-tank" (MPAT) ammunition and one for "armor-piercing fin-stabilized discarding sabot" (APFSDS) and "Smart Target-Activated Fire and Forget" (STAFF) ammunition. Turret traverse and main gun elevation can be performed with manual handles and cranks if the "fire control" or "hydraulic" systems fail.
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The commander's M2HB .50 caliber machine gun on the M1 and M1A1 is aimed by a 3× magnification sight incorporated into the Commander's Weapon Station (CWS), while the M1A2 uses the machine gun's own iron sights, or a remote aiming system such as the Common Remotely Operated Weapon Station (CROWS) system when used as part of the Tank Urban Survival Kit (TUSK). The loader's M240 machine gun is aimed either with the built-in iron sights or with a thermal scope mounted on the machine gun.
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In late 2017, the 400 USMC M1A1 Abrams were to be upgraded with better and longer-range sights on the Abrams Integrated Display and Targeting System (AIDATS) replacing the black-and-white camera view with a color sight and day/night thermal sight, simplified handling with a single set of controls, and a slew to cue button that repositions the turret with one command. Preliminary testing showed the upgrades reduced target engagement time from six seconds to three by allowing the commander and gunner to work more closely and collaborate better on target acquisition.
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The M1 Abrams's powertrain consists of a Honeywell AGT1500 (originally made by Lycoming) multifuel gas turbine capable of at 30,000 rpm and at 10,000 rpm and a six-speed (four forward, two reverse) Allison X-1100-3B Hydro-Kinetic automatic transmission. This gives it a governed top speed of on paved roads, and cross-country. With the engine governor removed, speeds of around are possible on an improved surface. However, damage to the drivetrain (especially to the tracks) and an increased risk of injuries to the crew can occur at speeds above .
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The tank was built around this engine and it is multifuel–capable, including diesel, kerosene, any grade of motor gasoline, and jet fuel (such as JP-4 or JP-8). For logistical reasons, JP-8 is the U.S. military's universal fuel powering both aircraft and vehicle fleets. The Australian M1A1 AIM SA burns diesel fuel, since the use of JP-8 is less common in the Australian Army.
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The gas turbine propulsion system has proven quite reliable in practice and combat, but its high fuel consumption is a serious logistic issue. The engine burns more than per mile ( per hour) when traveling cross-country and per hour when idle.
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The high speed, high temperature jet blast emitted from the rear of M1 Abrams tanks makes it hazardous for infantry to take cover or follow behind the tank in urban combat. The turbine is very quiet when compared to diesel engines of similar power output and produces a significantly different sound from a contemporary diesel tank engine, reducing the audible distance of the sound, thus earning the Abrams the nickname "whispering death" during its first Reforger exercise.
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The Army received proposals, including two diesel options, to provide the common engine for the XM2001 Crusader and Abrams. In 2000, the Army selected the gas turbine engine LV100-5 from Honeywell and subcontractor General Electric. The new LV100-5 engine was lighter and smaller (43% fewer parts) with rapid acceleration, quieter running, and no visible exhaust. It also featured a 33% reduction in fuel consumption (50% less when idle) and near drop-in replacement. The Common Engine Program was shelved when the Crusader program was canceled, however Phase 2 of Army's PROSE (Partnership for Reduced O&S Costs, Engine) program called for further development of the LV100-5 and replacement of the current AGT1500 engine.
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An Auxiliary Power Unit (APU) was designed by the Army's TARDEC, replacing an existing battery pack that weighs about . It uses a high power density Wankel rotary engine modified to use diesel and military grade jet fuel. The new APU will also be more fuel efficient than the tank's main engine. Testing of the first APUs began in 2009.
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Although the M1 tank is not designed to carry riders easily, provisions exist for the Abrams to transport troops in tank desant with the turret stabilization device switched off. A battle equipped infantry squad may ride on the rear of the tank, behind the turret. The soldiers can use ropes and equipment straps to provide handholds and snap links to secure themselves. If and when enemy contact is made, the tank conceals itself allowing the infantry to dismount.
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Strategic mobility is the ability of the tanks of an armed force to arrive in a timely, cost effective, and synchronized fashion. The Abrams can be carried by a C-5 Galaxy or a C-17 Globemaster III. The limited capacity (two combat-ready tanks in a C-5, one combat-ready tank in a C-17) caused serious logistical problems when deploying the tanks for the first Persian Gulf War, though there was enough time for 1,848 tanks to be transported by ship.
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Marines transport their Marine Air Ground Task Force (MAGTF)-attached Abrams tanks by combat ship. A "Wasp"-class Landing Helicopter Dock (LHD) typically carries a platoon of 4 to 5 tanks attached to the deployed Marine Expeditionary Unit, which are then amphibiously transported to shore by Landing Craft Air Cushion (LCAC) at 1 combat-ready tank per landing craft.
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The Abrams is also transportable by truck, namely the Oshkosh M1070 and M1000 Heavy Equipment Transporter System (HETS) for the US Military. The HETS can operate on highways, secondary roads, and cross-country. It accommodates the four tank crew members. The Australian Army uses customised MAN trucks to transport its Abrams.
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The first instance of the Abrams being airlifted directly into a battlefield occurred in October 1993. Following the Battle of Mogadishu, 18 M1 tanks were airlifted by C-5 aircraft to Somalia from Hunter Army Airfield, Georgia.
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ISO 4217 is a standard published by the International Organization for Standardization (ISO) that defines alpha codes and numeric codes for the representation of currencies and provides information about the relationships between individual currencies and their minor units. This data is published in three tables:
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The first edition of ISO 4217 was published in 1978. The tables, history and ongoing discussion are maintained by SIX Group on behalf of ISO and the Swiss Association for Standardization.
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The ISO 4217 code list is used in banking and business globally. In many countries, the ISO 4217 alpha codes for the more common currencies are so well known publicly that exchange rates published in newspapers or posted in banks use only these to delineate the currencies, instead of translated currency names or ambiguous currency symbols. ISO 4217 alpha codes are used on airline tickets and international train tickets to remove any ambiguity about the price.
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In 1973, the ISO Technical Committee 68 decided to develop codes for the representation of currencies and funds for use in any application of trade, commerce or banking. At the 17th session (February 1978), the related UN/ECE Group of Experts agreed that the three-letter alphabetic codes for International Standard ISO 4217, "Codes for the representation of currencies and funds", would be suitable for use in international trade.
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Over time, new currencies are created and old currencies are discontinued. Such changes usually originate from the formation of new countries, treaties between countries on shared currencies or monetary unions, or redenomination from an existing currency due to excessive inflation. As a result, the list of codes must be updated from time to time. The ISO 4217 maintenance agency is responsible for maintaining the list of codes.
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In the case of national currencies, the first two letters of the alpha code are the two letters of the ISO 3166-1 alpha-2 country code and the third is usually the initial of the currency's main unit. So Japan's currency code is : "JP" for Japan and "Y" for yen. This eliminates the problem caused by the names "dollar, franc, peso" and "pound" being used in dozens of countries, each having significantly differing values. While in most cases the ISO code resembles an abbreviation of the currency's full English name, this is not always the case, as currencies such as the Algerian dinar, Aruban florin, Cayman dollar, renminbi, sterling and the Swiss franc have been assigned codes which do not closely resemble abbreviations of the official currency names.
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In some cases, the third letter of the alpha code is not the initial letter of a currency unit name. There may be a number of reasons for this:
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In addition to codes for most active national currencies ISO 4217 provides codes for "supranational" currencies, procedural purposes, and several things which are "similar to" currencies:
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The use of an initial letter "X" for these purposes is facilitated by the ISO 3166 rule that no official country code beginning with X will ever be assigned.
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The inclusion of EU (denoting the European Union) in the ISO 3166-1 reserved codes list allows the euro to be coded as EUR rather than assigned a code beginning with X, even though it is a supranational currency.
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ISO 4217 also assigns a three-digit numeric code to each currency. This numeric code is usually the same as the numeric code assigned to the corresponding country by ISO 3166-1. For example, USD (United States dollar) has numeric code which is also the ISO 3166-1 code for "US" (United States).
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A number of currencies had official ISO 4217 currency codes and currency names until their replacement by another currency. The table below shows the ISO currency codes of former currencies and their common names (which do not always match the ISO 4217 names). That table has been introduced end 1988 by ISO.
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Examples for the ratios of :1 and :1 include the United States dollar and the Bahraini dinar, for which the column headed “Minor unit” shows “2” and “3”, respectively. , two currencies have non-decimal ratios, the Mauritanian ouguiya and the Malagasy ariary; in both cases the ratio is 5:1. For these, the “Minor unit” column shows the number “2”. Some currencies, such as the Burundian franc, do not in practice have any minor currency unit at all. These show the number “0”, as with currencies whose minor units are unused due to negligible value.
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The ISO standard does not regulate either the spacing, prefixing or suffixing in usage of currency codes. According however to the European Union's Publication Office, in English, Irish, Latvian and Maltese texts, the ISO 4217 code is to be followed by a hard space and the amount:
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In Bulgarian, Croatian, Czech, Danish, Dutch, Estonian, Finnish, French, German, Greek, Hungarian, Italian, Lithuanian, Polish, Portuguese, Romanian, Slovak, Slovene, Spanish and Swedish the order is reversed; the amount is followed by a hard space and the ISO 4217 code:
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Note that, as illustrated, the order is determined not by the currency but by the native language of the document context.
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The US dollar has two codes assigned: USD and USN ("US dollar next day"). The USS (same day) code is not in use any longer, and was removed from the list of active ISO 4217 codes in March 2014.
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A number of active currencies do not have an ISO 4217 code, because they may be: (1) a minor currency pegged at par (1:1) to a larger currency, even if independently regulated, (2) a currency only used for commemorative banknotes or coins, or (3) a currency of an unrecognized or partially recognized state. These currencies include:
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Despite having no presence or status in the standard, three letter acronyms that resemble ISO 4217 coding, are sometimes used locally or commercially to represent currencies or currency instruments.
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Minor units of currency (also known as currency subdivisions or currency subunits) are often used for pricing and trading stocks and other assets, such as energy, but are not assigned codes by ISO 4217. Two conventions for representing minor units are in widespread use:
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A third convention is similar to the second one but uses an upper-case letter, e.g. ZAC for the South African Cent.
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Cryptocurrencies have "not" been assigned an ISO 4217 code. However, some cryptocurrencies and cryptocurrency exchanges use a three-letter acronym that resemble an ISO 4217 code.
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Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation: according to Hans Eysenck: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results.
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During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Of his thousands of results, all but a dozen or two have now been proven correct. "The Ramanujan Journal", a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death. He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as Euler and Jacobi.
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In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.
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A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said his family goddess, Namagiri Thayar, revealed his mathematical knowledge to him. He once said, "An equation for me has no meaning unless it expresses a thought of God."
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Ramanujan (literally, "younger brother of Rama", a Hindu deity) was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, in present-day Tamil Nadu. His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop. His mother, Komalatammal, was a housewife and sang at a local temple. They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.
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On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.
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Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture. At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic with the best scores in the district. That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.
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A child prodigy by age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book written by S. L. Loney on advanced trigonometry. He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers. He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the quartic. In 1903, he tried to solve the quintic, not knowing that it was impossible to solve with radicals.
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In 1903, when he was 16, Ramanujan obtained from a friend a library copy of "A Synopsis of Elementary Results in Pure and Applied Mathematics", G. S. Carr's collection of 5,000 theorems. Ramanujan reportedly studied the contents of the book in detail. The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places. His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.
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When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum. He received a scholarship to study at Government Arts College, Kumbakonam, but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry for about a month. He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit. Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.
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In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras.
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On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994), a girl his mother had selected for him a year earlier and who was ten years old when they married. It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony. As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.
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After the marriage, Ramanujan developed a hydrocele testis. The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost.
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After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.
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In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College." After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control. In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town, Madras, where they lived for a few months. In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.
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In 1910, Ramanujan met deputy collector V. Ramaswamy Aiyer, who founded the Indian Mathematical Society. Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:
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I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.
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Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society. Rao was impressed by Ramanujan's research but doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud. Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance. When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the "Journal of the Indian Mathematical Society.
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He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.
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Using this equation, the answer to the question posed in the "Journal" was simply 3, obtained by setting , , and . Ramanujan wrote his first formal paper for the "Journal" on the properties of Bernoulli numbers. One property he discovered was that the denominators of the fractions of Bernoulli numbers are always divisible by six. He also devised a method of calculating based on previous Bernoulli numbers. One of these methods follows:
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In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures. His writing initially had many flaws. As "Journal" editor M. T. Narayana Iyengar noted:
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Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.
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Ramanujan later wrote another paper and also continued to provide problems in the "Journal". In early 1912, he got a temporary job in the Madras Accountant General's office, with a monthly salary of 20 rupees. He lasted only a few weeks. Toward the end of that assignment, he applied for a position under the Chief Accountant of the Madras Port Trust.
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I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.
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Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics". Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month. At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.
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In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes. He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians. Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.
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The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment. On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud. Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe". One of the theorems Hardy found amazing was on the bottom of page three (valid for ):
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The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals. After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them". Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power". One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".
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On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions". Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land". Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."
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To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge. As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan". The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras.
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While he was engaged as a research student, Ramanujan continued to submit papers to the "Journal of the Indian Mathematical Society." In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail. In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.
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Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England. Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn". Apparently, Ramanujan's mother had a vivid dream in which the family goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose". On 17 March 1914, Ramanujan traveled to England by ship, leaving his wife to stay with his parents in India.
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Ramanujan departed from Madras aboard the S.S. "Nevasa" on 17 March 1914. When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.
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Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs. Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi", while Hardy said he "can compare him only with Euler or Jacobi."
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Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy.
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Ramanujan was awarded a "Bachelor of Arts by Research" degree (the predecessor of the PhD degree) in March 1916 for his work on highly composite numbers, sections of the first part of which had been published the preceding year in the "Proceedings of the London Mathematical Society." The paper was more than 50 pages long and proved various properties of such numbers. Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.
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On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a Fellow of the Royal Society, the second Indian admitted, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.
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Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. In 1919, he returned to Kumbakonam, Madras Presidency, and in 1920 he died at the age of 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.
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Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an officer of the State Bank of India and raised a family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C. Berndt and Béla Bollobás made it a point to visit her while in India. She died at her Triplicane residence in 1994.
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A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, than tuberculosis. He had two episodes of dysentery before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established. At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease; British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.
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Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a simple life at Cambridge. Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in his work and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls of complex mathematical content unfolding before his eyes. He often said, "An equation for me has no meaning unless it expresses a thought of God."
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Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.
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Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.
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In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite series for , one of which is given below:
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This result is based on the negative fundamental discriminant with class number . Further, and , which is related to the fact that
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Ramanujan's series for converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate . Truncating the sum to the first term also gives the approximation for , which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places. See also the more general Ramanujan–Sato series.
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One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem:
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for all such that formula_9 and formula_10, where is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of , , and gives some deep identities for the hyperbolic secant.
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In 1918, Hardy and Ramanujan studied the partition function extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.
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In the last year of his life, Ramanujan discovered mock theta functions. For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.
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Although there are numerous statements that could have borne the name "Ramanujan conjecture," one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ("q"), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work.
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