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Concept
Ramanujan–Petersson conjecture
Summary
In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12
where , satisfies
when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms.
The Riemann zeta function and the Dirichlet L-function satisfy the Euler product,
and due to their completely multiplicative property
Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the modular discriminant satisfies the modified relation
where τ(p) is Ramanujan's tau function. The term
is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function.
Ramanujan conjectured the following:
τ is multiplicative,
τ is not completely multiplicative but for prime p and j in N we have: τ(p j+1) = τ(p)τ(p j ) − p11τ(p j−1 ), and
τ(p) ≤ 2p11/2.
Ramanujan observed that the quadratic equation of u = p−s in the denominator of RHS of ,
would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be α and β, then
which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the τ(n), namely for any ε > 0:
In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by . The formulations required to show that it was a consequence were delicate, and not at all obvious.
Official source
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