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Line 14: ''m'' runs over positive integers ''F'' is a smooth function all of whose derivatives are rapidly decreasing &~~psi~~phi; is a Fourier transform of ''F'': <math>\~~psi~~phi(t) = \int_{-\infty}^{\infty}F(x)e^{itx}dx</math> Φ(1/2 + it) = φ(t) Ψ(t) = -log(π) + Re(ψ(1/4 + it/2)), where ψ is the [[digamma function]] Γ′/Γ Line 26: :with the terms corresponding to the prime ''p'' coming from the Euler factor of ''p'', and the term at the end involving Ψ coming from the gamma factor (the Euler factor at infinity). The left hand side is a sum over all zeros of ζ<sup>*</sup> counted with multiplicities, so thepoles at 0 and 1 are counted as zeros of order −1. ==Applications== Riemann's original use of the explicit formula way to give an exact formula for the number of primes less than a given number. To do this, take ''F''(log(''y'')) to be ''y''<sup>1/2</sup>/log(''y'') for

Explicit formulae for L-functions: Difference between revisions