How Can Contour Integration Yield Specific Values for Variable Conditions?

[zoorna]

greetings . this is my first post here . i am preparing myself for a complex analysis course that i will be taking next semester . i came across this problem , which is probably a very simple one , but i don't know how to go about it , so bare with me

we have the contour integration
I(x)=\frac{1}{2\pi i} \int_{\sigma-iT}^{\sigma+iT}\left(\frac{x}{n}\right)^{s}ds

where \Re(s) ,\sigma > 1

x is a variable , and n is a constant .

i need the integration to yield a constant if x=n, and zero otherwise ??

my guess is that the function \frac{x}{n} should be somehow modified to yield the desired result - constant for x=n , zero other wise - . or is it the contour that should be changed ??

your help is appreciated .

 

[zoorna]

i'll try to make the question a bit clearer . we have the dirichlet series :

I(s)= \sum_{n=1}^{\infty}\frac{\alpha(n)}{n^{s}} , \Re(s)>1

where \alpha(n) is some arithmetic function of n .

now i am trying to use mellin transform, or any kind of transform akin to that of fourier's, to extract \alpha(n) . i was hoping for a kernel - function of s - that is orthogonal to all terms except for the one containing the integer i want to extract \alpha(n) for . meaning , i am trying to find a function f(x,s) such that :

<br /> &lt; f(x,s),n^{-s}&gt;=\left\{\begin{matrix}<br /> k &amp; , &amp; x=n &amp; \\ <br /> 0 &amp; , &amp; o.w &amp; <br /> \end{matrix}\right.<br />

k is a constant

i hope this makes it clearer .

 

[zoorna]

i think i found it . using Perron's formula :

\alpha(n)=\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}I(s)\frac{\left(n+1/2\right)^{s}-\left(n-1/2 \right )^{s}}{s}ds