Determine Fourier coefficients by the values of its L-Series.

Question

suppose I know the values of $\sum_{n=1}^\infty \frac{a_n}{n^k}$ for all $k=1,2,...$. Is there a way/tool to determine the coefficients $a_n$ from this (which might not be unique)?

I would appreciate any idea or keyword !

Answer 1

Suppose $$ L(s) = \sum_{n \geq 1} \frac{a(n)}{n^s}$$ is your Dirichlet series and that it converges in some half-plane. You know $L(k)$ for all positive integers $k$. Then $$ a(1) = \lim_{k \to \infty} L(k),$$ as all terms but the $1$st go to zero in the limit. This is how to play the game. Now that you know $a(1)$, you consider $$ a(2) = \lim_{k \to \infty} \left( L(k) - a(1) \right) 2^k = \lim_{k \to \infty} \left( a(1) - a(1) + \frac{a(2)}{2^k} + \sum_{n \geq 3} \frac{a(n)}{n^k} \right) 2^k.$$ You can then compute $a(3)$, and so on, recursively.

So yes, the Fourier coefficients are completely determined by the values of the $L$-series on the positive integers. Further, this shows that they are unique.