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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
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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.
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395:) 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".
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319:. 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)
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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
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should it be renamed as 'explicit formulae relating prime numbers and riemann zeros ' ? since it's a relationship between prime numbers and
Riemann zeros
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Besides verifying mathematically, I have always seen the prime counting function before as including x - and it is defined that way on
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you stumbled indeed on an error in this article. If you want the explicit formula to hold at prime powers, you have to define
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Could someone who is familiar with the topic of this page move it to a more specific title? Thanks.
1518:{\displaystyle g(u)=\sum _{n=1}^{\infty }\Lambda (n)\left(\delta (u-\ln n)+\delta (u+\ln n)\right)}
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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In modern literature this "normalized" prime counting function is sometimes denoted by
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Some of the formulas under this topic don't look correct to me.
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a notation which I would also suggest for this article.
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297:)
294:h
288:p
285:(
279:+
276:)
273:h
270:+
267:p
264:(
258:(
253:0
247:h
236:=
233:)
230:p
227:(
188:(
143:.
42::
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