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Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL(2) as saying that the local components of automorphic representations lie in the principal series, and suggested this condition as a generalization of the Ramanujan–Petersson conjecture to automorphic forms on other groups. Another ...
Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.
Elliott–Halberstam conjecture; Functional equation (L-function) Chebotarev's density theorem; Local zeta function. Weil conjectures; Modular form. modular group; Congruence subgroup; Hecke operator; Cusp form; Eisenstein series; Modular curve; Ramanujan–Petersson conjecture; Birch and Swinnerton-Dyer conjecture; Automorphic form; Selberg ...
In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: (+) (), (+) (), (+) ().In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence
The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).
See Winnie Li's survey on Ramanujan's conjecture and other aspects of number theory relevant to these results. [ 5 ] Lubotzky , Phillips and Sarnak [ 2 ] and independently Margulis [ 6 ] showed how to construct an infinite family of ( p + 1 ) {\displaystyle (p+1)} -regular Ramanujan graphs, whenever p {\displaystyle p} is a prime number and p ...
Lafforgue's theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GL n (F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1.
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