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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.
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 Aiyangar [a] (22 December 1887 – 26 April 1920) was an Indian mathematician.Often regarded as one of the greatest mathematicians of all time, 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 ...
Download as PDF; Printable version; ... move to sidebar hide. From Wikipedia, the free encyclopedia. Redirect page. ... Ramanujan–Petersson conjecture; Retrieved ...
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 ...
Technically, anything over 20 years old can be coined "vintage." But when you truly think of items worth this title, your brain doesn't go to Beanie Babies. Instead, it conjures up images of vinyl...
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 ...