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Related to this and this questions.

Basically got definite integral that experimentally equals $\zeta(s)$ both numerically and symbolically.

Closed form for the indefinite integral is known, but I can't use the closed form for general $s$ to compute $\zeta(s)$ due to problem with limits.

Define $$ F(x) = \{ x \} = x -\lfloor x \rfloor=-\frac{\arctan\left(\cot\left(\pi x\right)\right)}{\pi} + \frac{1}{2}$$

Where the principal branch of $\arctan$ is taken. (Source mathworld).

For $\Re(s) > 0$,

$$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \int_1^\infty \frac{1/2-\{x\}}{x^{s+1}}\,\mathrm{d}x$$

Using $F(x)$, $$ Z(s) = \dfrac{s}{s-1} - \frac12+s \int_1^\infty \frac{1/2-F(x)}{x^{s+1}}\,\mathrm{d}x$$.

Unless multivalued functions cause problems, we should have $Z(s)=\zeta(s)$.

Symbolically $\zeta(n)=Z(n)$ at positive integers, infinite sums for rational $s$ are found too.

E.g. $$ Z(3) =1+3\,\lim _{x\rightarrow \infty }1/6\,{\frac {2\,{\it arccot} \left( \cot \left( \pi \,x \right) \right) +\pi \,x-\pi -2\,\pi \,{x}^{3}+2 \,\zeta \left( 3 \right) \pi \,{x}^{3}}{\pi \,{x}^{3}}} = \zeta(3)$$

Numerically with mpmath $Z(s)$ is very close to $\zeta(s)$ (unless $t$ is large, which might be explained by numerical instability with the integral).

For general $s, s \ne 0, s \ne 1$, Maple found closed form for $\int \frac{1/2-F(x)}{x^{s+1}}\,\mathrm{d}x$

I doubt the general closed form can be used to compute $\zeta$ and Maple returns "undefined" for the limit at $1$, which confuses me.

Why when specializing $s$ to integers and rationals $Z(s)=\zeta(s)$ but the general closed form doesn't appear to work?

The closed form for the indefinite integral: $$ {\frac {-ix\ln \left( {{\rm e}^{i\pi \,x}} \right) }{\pi \,s{x}^{s+1} }}+1/4\,x \left( 2-2\,s+4\,x-s{\it csgn} \left( i{{\rm e}^{2\,i\pi \,x }} \right) \left( {\it csgn} \left( i{{\rm e}^{i\pi \,x}} \right) \right) ^{2}+2\,s \left( {\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) \right) ^{2}{\it csgn} \left( i{{\rm e}^{i\pi \,x}} \right) +s \left( {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e} ^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( {\frac {i}{ {{\rm e}^{2\,i\pi \,x}}-1}} \right) +s \left( {\it csgn} \left( { \frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) +s \left( {\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( { \frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) + {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) {\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) { \it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) -s{ \it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i \pi \,x}}-1}} \right) {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x }}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) -s \left( {\it csgn} \left( { \frac {2\,i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) +s \left( {\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i \pi \,x}}-1}} \right) \right) ^{2}+ \left( {\it csgn} \left( {\frac { 2\,i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) -{\it csgn} \left( {\frac {2\,i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) {\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) +s \left( {\it csgn} \left( {\frac {2\,i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}+s{\it csgn} \left( {\frac {2\,i}{{{\rm e}^{2\,i\pi \,x}} -1}} \right) {\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) -s{\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) {\it csgn} \left( i{{\rm e}^{2\,i \pi \,x}} \right) {\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}} -1}} \right) +{\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) {\it csgn} \left( {\frac {i{{\rm e} ^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) -s \left( {\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) \right) ^{3}-s \left( { \it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{3}-s \left( {\it csgn} \left( {\frac {2\, i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{3}-s \left( {\it csgn} \left( {\frac {2\,i}{{{\rm e}^{2\,i \pi \,x}}-1}} \right) \right) ^{3}-2\, \left( {\it csgn} \left( i{ {\rm e}^{2\,i\pi \,x}} \right) \right) ^{2}{\it csgn} \left( i{ {\rm e}^{i\pi \,x}} \right) +{\it csgn} \left( i{{\rm e}^{2\,i\pi \,x} } \right) \left( {\it csgn} \left( i{{\rm e}^{i\pi \,x}} \right) \right) ^{2}- \left( {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x }}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( {\frac {i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) - \left( {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}} -1}} \right) \right) ^{2}{\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) - \left( {\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x} }}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}{\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) - \left( {\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2}- \left( {\it csgn} \left( {\frac {2\,i}{{{\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{2 }+ \left( {\it csgn} \left( i{{\rm e}^{2\,i\pi \,x}} \right) \right) ^{3}+ \left( {\it csgn} \left( {\frac {i{{\rm e}^{2\,i\pi \,x}}}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{3}+ \left( {\it csgn} \left( {\frac {2\,i{{\rm e}^{2\,i\pi \,x}}}{{{\rm e}^{2\,i\pi \,x}}-1 }} \right) \right) ^{3}+ \left( {\it csgn} \left( {\frac {2\,i}{{ {\rm e}^{2\,i\pi \,x}}-1}} \right) \right) ^{3} \right) {s}^{-1} \left( s-1 \right) ^{-1} \left( {x}^{s+1} \right) ^{-1}$$

Computer friendly:

"-I/Pi/s*x/(x^(s+1))ln(exp(IPi*x))+1/4*x*(2-2*s-scsgn(Iexp(2*IPix)/(e\ xp(2*IPix)-1))csgn(Iexp(2*IPix))*csgn(I/(exp(2*IPix)-1))-csgn(2\ *I/(exp(2*IPix)-1))*csgn(I/(exp(2*IPix)-1))-s*csgn(2*I*exp(2*IPix\ )/(exp(2*IPix)-1))csgn(Iexp(2*IPix)/(exp(2*IPix)-1))-s*csgn(2*I\ /(exp(2*IPix)-1))^2*csgn(I/(exp(2*IPix)-1))+csgn(I*exp(2*IPix)/(e\ xp(2*IPix)-1))csgn(Iexp(2*IPix))*csgn(I/(exp(2*IPix)-1))+2*scs\ gn(Iexp(2*IPix))^2*csgn(Iexp(IPix))-scsgn(I*exp(2*IPix))csgn(\ Iexp(IPix))^2+scsgn(Iexp(2*IPix)/(exp(2*IPix)-1))^2*csgn(I/(ex\ p(2*IPix)-1))+scsgn(Iexp(2*IPix)/(exp(2*IPix)-1))^2*csgn(I*exp(\ 2*IPix))+s*csgn(2*I*exp(2*IPix)/(exp(2*IPix)-1))^2*csgn(I*exp(2*I\ Pix)/(exp(2*IPix)-1))-csgn(2*I/(exp(2*IPix)-1))^2+s*csgn(2*I/(exp\ (2*IPix)-1))^2+csgn(2*I*exp(2*IPix)/(exp(2*IPix)-1))csgn(Iexp(2\ IPi*x)/(exp(2*IPix)-1))+s*csgn(2*I*exp(2*IPix)/(exp(2*IPix)-1))\ ^2+csgn(2*I/(exp(2*IPix)-1))^2*csgn(I/(exp(2*IPix)-1))-scsgn(Iexp\ (2*IPix))^3-scsgn(Iexp(2*IPix)/(exp(2*IPix)-1))^3-s*csgn(2*I*ex\ p(2*IPix)/(exp(2*IPix)-1))^3-s*csgn(2*I/(exp(2*IPix)-1))^3-2*csgn\ (I*exp(2*IPix))^2*csgn(Iexp(IPix))+csgn(Iexp(2*IPix))csgn(Iex\ p(IPix))^2-csgn(I*exp(2*IPix)/(exp(2*IPix)-1))^2*csgn(I/(exp(2*I*\ Pix)-1))-csgn(Iexp(2*IPix)/(exp(2*IPix)-1))^2*csgn(I*exp(2*IPix\ ))-csgn(2*I*exp(2*IPix)/(exp(2*IPix)-1))^2+csgn(I*exp(2*IPix))^3+\ csgn(I*exp(2*IPix)/(exp(2*IPix)-1))^3+csgn(2*I*exp(2*IPix)/(exp(2\ IPi*x)-1))^3+csgn(2*I/(exp(2*IPix)-1))^3+s*csgn(2*I/(exp(2*IPix)-\ 1))*csgn(I/(exp(2*IPix)-1))-csgn(2*I*exp(2*IPix)/(exp(2*IPix)-1))\ ^2*csgn(I*exp(2*IPix)/(exp(2*IPix)-1))+4*x)/s/(-1+s)/(x^(s+1))"

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  • $\begingroup$ Mathematica will produce LaTeX output for you. Just wrap TeXForm[ ] around your expression. $\endgroup$ – Stopple Aug 24 '13 at 14:00
  • $\begingroup$ @Stopple I made it computer friendly on purpose, maybe will add it in latex. $\endgroup$ – joro Aug 24 '13 at 14:22
  • $\begingroup$ The latex doesn't provide much info, just added it. $\endgroup$ – joro Aug 24 '13 at 14:26
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    $\begingroup$ formally, $\int (\pi x^{s+1})^{-1}{\rm arctan}[{\rm cot}(\pi x)]dx = [x^{s}\pi s(s-1)]^{-1}\{[\pi x-(s-1){\rm arctan}[{\rm cot}(\pi x)]\}$, with a limit $x\rightarrow 1$ of $(3-s)[2s(s-1)]^{-1}$, without any complications. $\endgroup$ – Carlo Beenakker Aug 24 '13 at 17:12

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