Revision as of 03:51, 9 January 2009 edit R.e.b. (talk \| contribs) Extended confirmed users, Pending changes reviewers 42,907 edits →Explicit formula for the Riemann zeta function: typos ← Previous edit		Revision as of 04:01, 9 January 2009 edit undo R.e.b. (talk \| contribs) Extended confirmed users, Pending changes reviewers 42,907 edits →Applications: number fields Next edit →
Line 27: 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. ==Generalizations== 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. ==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