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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 ...
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.
The only solutions up to 10 10 to the equation τ(p) ≡ 0 (mod p) are 2, 3, 5, 7, 2411, and 7 758 337 633 (sequence A007659 in the OEIS). [11] Lehmer (1947) conjectured that τ(n) ≠ 0 for all n, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n up to 214 928 639 999 (Apostol 1997, p. 22).
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 ...
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 ...
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The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
In January, a Japanese author admitted that her award-winning book, “The Tokyo Tower of Sympathy,” had been written with the help of ChatGPT. Shortly after receiving the Akutagawa Prize, Rie ...