Explicit formulae for L-functions: Difference between revisions

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==Riemann's explicit formula==

In his 1859 paper "[[On the Number of Primes Less Than a Given Magnitude]]" Riemann sketched an explicit formula (it was not fully proven until 1895 by [[Hans Carl Friedrich von Mangoldt|von Mangoldt]], see below) for the normalized prime-counting function {{math|&pi;₀(''x'')}} which is related to the [[prime-counting function]] {{math|&pi;(''x'')}} by<ref>{{Cite web |title=Explicit Formulae (L-function) |url=https://encyclopedia.pub/entry/32287 |access-date=2023-06-14 |website=encyclopedia.pub |language=encn}}</ref>

:<math>\pi_0(x) = \frac{1}{2} \lim_{h\to 0} \left[\,\pi(x+h) + \pi(x-h)\,\right]\,,</math>

which takes the [[arithmetic mean]] of the limit from the left and the limit from the right at discontinuities.{{efn|The original prime counting function can easily be recovered via <math>~\pi(x) = \pi_0(x+1)~</math> for all <math>~x \ge 3~.</math>}} His formula was given in terms of the related function