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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.
Download as PDF; Printable version ... Redirect page. Redirect to: Ramanujan–Petersson conjecture; ... Text is available under the Creative Commons Attribution ...
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.
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 ...
Flicker, Yuval Z.; Kazhdan, David A. (1989), Geometric Ramanujan conjecture and Drinfeld reciprocity law, Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, 201-218 (1989).
A celebrated conjecture of Ramanujan asserted that when Δ(z) is expanded as a power series in q, the coefficient of q p for any prime p has absolute value ≤ 2p 11/2. This was confirmed by the work of Eichler, Shimura, Kuga, Ihara, and Pierre Deligne as a result of Deligne's proof of the Weil conjectures, which were shown to imply Ramanujan's ...
The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL 2 over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series ...
The condition that the real part of μ i be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μ i is negative. Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Peterssen conjecture holds, and have a functional equation, but do not satisfy the Riemann hypothesis.