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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 ...
As evidence, many provided Ramanujan's τ(p) (case of weight 12). 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
Extending the results of A. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give For example, his results give p ( 107 4 ⋅ 31 k + 30064597 ) ≡ 0 ( mod 31 ) . {\displaystyle p(107^{4}\cdot 31k+30064597)\equiv 0{\pmod {31}}.}
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 ...
In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x 2 + y 2 + 10z 2 with integral values for x, y and z. [ 1 ] [ 2 ] Srinivasa Ramanujan considered this expression in a footnote in a paper [ 3 ] published in 1916 and briefly discussed the representability of integers in this form.
Still, Trump's nomination of Scott Bessent to the top Treasury post raised hopes that tariffs will be more measured. And with only 21 trading days left in the year, analysts, investors, and market ...
U.S. smoking rates have fallen dramatically in the past six decades, from 42.6% of American adults in 1965 to 11.6% in 2022, according to the American Lung Association. But, according to the U.S ...
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.