2
$\begingroup$

Given an Euler product of the form \begin{align} L(s) = \prod_{p}(1 - a_{p} p^{-s} - b_{p} p^{-2s})^{-1} \end{align} where $a_n$ and $b_n$ are not necessarily a multiplicative arithmetic functions of $n$, is there a prescription for computing the series coefficients, $L(s) = \sum_{n \geq 1} c_n \ n^{-s}$ given $a_p$ and $b_p$?

This example specializes to some familiar examples. For instance, if $b_{p} = 0$ for all primes, then $c_n = \prod_{p \mid n} a_{p}^{\text{ord}_{p}(n)}$. In particular, if $a_p = \chi(p)$, a Dirichlet character, then $L$ is the corresponding Dirichlet $L$-function with $c_n = \chi(n)$.

References are certainly welcome!

  • 1
    Notice that in your example where $b_p = 0$, the statement "then $c_n = a_n$" actually makes no sense, because all you are being given at first are the numbers $\{a_p,b_p\}$ as $p$ runs over primes. There is no such thing as $a_n$ initially.2012-03-14
  • 0
    Any book that discusses the $L$-series of elliptic curves or modular forms will show you a recursion when $b_p = -p$, and from that you can reverse engineer the work to figure out your answer for any $b_p$.2012-03-14
  • 1
    It makes more sense to just let $a_p$ be defined for prime $p$ only; anything more is superfluous, since only values at primes are relevant (and also the remark about complete multiplicativeness is false).2012-03-14
  • 0
    Edited. Thanks.2012-03-14

1 Answers 1