Register to reply

A very intriguing integral

by riemannian
Tags: integral, intriguing
Share this thread:
Mar1-12, 11:29 AM
P: 5
greetings . the following integral appears in some references on analytic number theory . i am really intrigued by it . and would love to understand it .
[tex] \int_{1}^{\infty}\frac{\left \{x \right \}}{x}\left(\frac{1}{x^{s}-1}\right)dx[/tex]

[itex]\Re(s)>1 [/itex] , [itex]\left \{x \right \} [/itex] is the fractional , sawtooth function .

i have tried the Fourier expansion of the sawtooth function :

[tex] \int_{1}^{\infty}\frac{\left \{x \right \}}{x}\left(\frac{1}{x^{s}-1}\right)dx = \int_{1}^{\infty}\frac{1}{x(x^{s}-1)}\left(\frac{1}{2}-\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2\pi i nx)}{n} \right )dx =\int_{1}^{\infty}\frac{1}{x(x^{s}-1)}\left(\frac{1}{2}+\frac{1}{2\pi i}\ln \left(\frac{1-q^{2}}{1-q^{-2}} \right)\right )dx[/tex]

where [itex] q [/itex] is the nome :
[tex] q=e^{i \pi x}[/tex]

but that brought me no where near a solution !! any suggestions on how to do the integral ??
Phys.Org News Partner Science news on
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
Mar1-12, 01:54 PM
P: 5
after some manipulation , the integral reduces to :

[tex] \frac{1}{2\pi i }\int_{1}^{\infty}\frac{\left(\pi i +\ln(-e^{2\pi i x}) \right )}{x(x^{s}-1)}dx[/tex]

Register to reply

Related Discussions
Intriguing thought experiment Classical Physics 1
Intriguing Question Special & General Relativity 5
Quite Intriguing. What I'm learning. General Discussion 2
Very intriguing stuff... General Discussion 14
Intriguing Statistical/Probability Question Set Theory, Logic, Probability, Statistics 0