Ramanujan-Petersson conjecture

In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12

where ${\displaystyle q=e^{2\pi iz}}$, satisfies

when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms.

The Riemann zeta function and the Dirichlet L-function satisfy the Euler product,

${\displaystyle L(s,a)=\prod _{p}\left(1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+\dots \right)}$

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