Wikipedia:Reference desk/Archives/Mathematics/2014 January 23

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January 23

Zeta Function and floor functions

The article for floor and ceiling functions says that it is easy to prove;

${\sum _{a<n\leq b}\phi (n)=\int _{a}^{b}\phi (x)dx+\int _{a}^{b}\left(\{x\}-{\tfrac {1}{2}}\right)\phi '(x)dx+\left(\{a\}-{\tfrac {1}{2}}\right)\phi (a)-\left(\{b\}-{\tfrac {1}{2}}\right)\phi (b).}$

Using integration by parts. How is this?

It then also goes on to say

$\zeta (s)=s\int _{1}^{\infty }{\frac {{\frac {1}{2}}-\{x\}}{x^{s+1}}}\;dx+{\frac {1}{s-1}}+{\frac {1}{2}}.$

And also;

For s = σ + i t in the critical strip (i.e. 0 < σ < 1),

$\zeta (s)=s\int _{-\infty }^{\infty }e^{-\sigma \omega }(\lfloor e^{\omega }\rfloor -e^{\omega })e^{-it\omega }\,d\omega .$

Can anyone prove these statements? — Preceding unsigned comment added by 31.48.36.7 (talk) 00:13, 23 January 2014 (UTC)

The article cites [1] which explains it in more detail. --RDBury (talk) 10:23, 23 January 2014 (UTC)

Can anyone prove these statements? — No... Not anyone... :-) — 79.113.237.228 (talk) 13:19, 23 January 2014 (UTC)

Cheers but that article starts by stating the first equation without explanation. I can sort of see some of the logic behind it but i can't prove it still. — Preceding unsigned comment added by 31.48.36.7 (talk) 17:19, 23 January 2014 (UTC)

Not sure what you mean by "without explanation", there is one paragraph proof following the equation. The Euler–Maclaurin formula is similar and the same technique is used to prove error bounds on numerical integration formulas such as the trapezoid rule. The idea is to split the interval up at the integers, perform integration by parts on each piece and add up the result. The last equation isn't in Titchmarsh btw; the article cites Crandall & Pomerance but Google books doesn't preview that page, however there is a pdf [2] online. Presumably it has a proof but I haven't bothered to look. --RDBury (talk) 19:18, 23 January 2014 (UTC)

See also the derivation of the Euler-Maclaurin summation formula in Knuth's TAOCP volume 1. Gutworth (talk) 00:46, 24 January 2014 (UTC)