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 :

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

k is a constant

i hope this makes it clearer .

 

[zoorna]

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

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