Talk:Explicit formulae for L-functions

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 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)

Article name is too general
The name of this article, "Explicit formula", is way too general. I came to this article via a link that was intended to give an explanation that an explicit formula is a formula of the type x = function of something not involving x.

Could someone who is familiar with the topic of this page move it to a more specific title? Thanks. Duoduoduo (talk) 13:18, 30 July 2013 (UTC) I would note that the article was originally called "Explicit formulae (L-function)", which sounds perfect to me, but it was moved to "Explicit formula" on 23:32, 8 January 2009‎ with the edit summary "shorter title". Duoduoduo (talk) 13:39, 30 July 2013 (UTC)

should it be renamed as 'explicit formulae relating prime numbers and riemann zeros ' ? since it's a relationship between prime numbers and Riemann zeros — Preceding unsigned comment added by 82.130.159.27 (talk) 12:18, 23 February 2017 (UTC)

Weil's Explicit Formula
Some of the formulas under this topic don't look correct to me.

Question (1): In the formula $$\frac{d}{du} \left[ \sum\limits_{n \le e^{|u|}} \Lambda(n) + \frac{1}{2} \ln(1-e^{-2|u|})\right] $$$$= \sum\limits_{n=1}^\infty \Lambda(n) \left[ \delta(u-\ln n) + \delta(u-\ln n) \right] + \frac{d\ln(1-e^{-2|u|})}{du} = e^u - \sum{\rho} e^{\rho u} $$,

(1a) Should $$\left[ \delta(u-\ln n) + \delta(u-\ln n) \right]$$ be $$\left(\delta(u-\ln n)+\delta(u+\ln n)\right)$$?

(1b) Should $$\frac{d\ln(1-e^{-2|u|})}{du}$$ be $$\frac{1}{2}\frac{d\ln(1-e^{-2|u|})}{du}$$?

(1c) Should $$\sum{\rho} e^{\rho u} $$ be $$\sum\limits_{\rho}{\rho}\,e^{\rho u}$$ or $$\sum\limits_{\rho}\,e^{\rho u}$$?

Question (2): In the last paragraph should $$g(u) = \sum_{n=1}^\infty \Lambda(n) \left[ \delta(u-\ln n) + \delta(u-\ln n) \right] $$ be $$g(u)=\sum_{n=1}^\infty\Lambda(n)\left(\delta(u-\ln n)+\delta(u+\ln n)\right)$$?

StvC (talk) 22:30, 28 February 2019 (UTC)