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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.
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 ...
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 ...
Download as PDF; Printable version; In other projects Appearance. move to sidebar hide. ... Redirect page. Redirect to: Ramanujan–Petersson conjecture; Retrieved ...
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
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.
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