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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.
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.
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
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.
The remaining five Australians from the infamous “Bali Nine” drug gang are “relieved and happy” to be home after Canberra struck a deal with Jakarta to end their two decades of imprisonment.
One quarter was all Ohio State really needed to prove it was better than Oregon. The Buckeyes were already up 7-0 after the first minute and the Ducks had back-to-back three-and-outs to start.
Zhang's contributions to mathematics include a generalization of an important conjecture in Vinogradov's Mean-Value Theorem, using novel techniques to solve Carleson's problem on pointwise convergence of solutions to the Schrödinger equation and solving the two-dimensional case of Sogge's conjecture for wave equations. [2] [3]