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The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form.
Extending the results of A. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give For example, his results give p ( 107 4 ⋅ 31 k + 30064597 ) ≡ 0 ( mod 31 ) . {\displaystyle p(107^{4}\cdot 31k+30064597)\equiv 0{\pmod {31}}.}
| τ(p) | ≤ 2p 11/2 for all primes p. 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.
In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x 2 + y 2 + 10z 2 with integral values for x, y and z. [ 1 ] [ 2 ] Srinivasa Ramanujan considered this expression in a footnote in a paper [ 3 ] published in 1916 and briefly discussed the representability of integers in this form.
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George Andrews [14] showed that several of Ramanujan's fifth order mock theta functions are equal to quotients Θ(𝜏) / θ(𝜏) where θ(𝜏) is a modular form of weight 1 / 2 and Θ(𝜏) is a theta function of an indefinite binary quadratic form, and Dean Hickerson [15] proved similar results for seventh order mock theta ...