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The Riemann zeta function can be replaced by a [[Dirichlet L-function]] of a [[Dirichlet character]] χ. The sum over prime powers then gets extra
factors of χ(''p''<sup> ''m''</sup>), and the terms ''Φ''(0) and ''Φ''(0) disappear because the L-series has no poles.
More generally, the Riemann zeta function and the L-series can be replaced by the [[Dedekind zeta function]] of an algebraic number field or a [[Hecke L-series]]. The sum over primes then gets replaced by a sum over prime ideals.
▲More generally, the Riemann zeta function and the L-series can be replaced by the [[Dedekind zeta function]] of an algebraic number field or a [[Hecke L-series]]. The sum over primes then gets replaced by a sum over prime ideals.
==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
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