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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.
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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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 ...
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 [3] by considering the Hausdorff dimension of the following function in the l-adic topology: P ℓ ( b ; z ) := ∑ n = 0 ∞ p ( ℓ b n + 1 24 ) q n / 24 . {\displaystyle P_{\ell }(b;z):=\sum _{n=0}^{\infty }p\left({\frac {\ell ^{b}n+1}{24}}\right)q^{n ...
Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x 2 + y 2 + 10z 2 represents all integers that it represents locally, with exactly 18 exceptions. In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture on cubic Gauss sums, under the assumption ...
The USA and Canada (both 2-0-1-0) lead Group A with seven points. Those teams play on New Year's Eve in the final game of the round robin in a game that will determine the group's top seed.
On the Ramanujan conjecture and finiteness of poles of certain L-functions, Annals of Mathematics, vol. 127, 1988, 547–584 doi:10.2307/2007005 The notion of norm and the representation theory of orthogonal groups, Inventiones mathematicae, vol. 119, 1995, 1–36 doi : 10.1007/BF01245173