Talk:Explicit formulae for L-functions

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Error?

"...Riemann found an explicit formula for the number of primes π(x) less than a given number x." Should this not be the number of primes less than or equal to a given number x?

I tried to verify mathematically that the formula gives this, but I am not adept enough to work with it much...I did find that the f(x) formula can work either way (if pi(x) includes x, then f(x) includes x; and if pi(x) excludes x, then f(x) excludes x). This was quite simple using induction.

Besides verifying mathematically, I have always seen the prime counting function before as including x - and it is defined that way on the Wiki page. —Preceding unsigned comment added by Cstanford.math (talk • contribs) 01:33, 30 October 2010 (UTC)[reply]

Reply to Error?

For most questions about the prime counting functions it is irrelevant, whether you define it to be continuous from the left or from the right, since you only change it on a set of measure 0 and the difference is bounded. However, in the case of the Riemann explicit formula this is no longer true and you stumbled indeed on an error in this article. If you want the explicit formula to hold at prime powers, you have to define

\pi (p)=0.5\lim _{h\to 0}(\pi (p+h)+\pi (p-h))

for all prime numbers p. You can find this on page four both in the german and english version of the transliteration of Riemann's original paper by David R. Wilkins (follow the link in the references). There it says (in the english version)

"Let F(x) be equal to this number (the number of primes < x) when x is not exactly equal to a prime number; but let it be greater by 1/2 when x is a prime number..."

In modern literature this "normalized" prime counting function is sometimes denoted by $\pi _{0},$ a notation which I would also suggest for this article.

Jpb101 (talk) 14:17, 24 August 2011 (UTC)[reply]