[Math] Are L-functions uniquely determined by their values at negative integers

analytic-number-theorydirichlet-seriesl-functions

Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that

  • the corresponding L-function
    $$L_{\{a_n\}}(s):=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$$
    converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$

  • $L_{\{a_n\}}(s)$ has analytic continuation to a meromorphic function on the whole complex plane

  • $L_{\{a_n\}}(n)=0$ for all negative integers $n$
  • not all of $a_n$ are zero?

Added : It was suggested in the answers that I should have used the term "Dirichlet series of integer sequences" instead of "L-function" as it lacks Euler product. I apologize for the confusion 🙂

Best Answer

Are L-functions uniquely determined by their values at negative integers?

No. The rescaled Riemann zeta function $$ \zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} = \sum_{n=1}^\infty \frac{a_n}{n^s}, $$ corresponding to the coefficient sequence $$ a_n = \begin{cases} 1 & \textrm{if $n$ is a square}, \\ 0 & \textrm{otherwise}, \end{cases} $$ is an example of an $L$-function that has a meromorphic continuation to all of $\mathbb{C}$ and vanishes at the negative integers.

Note that the formulation of your question seems to mix up the notion of an $L$-function with the more general notion of a Dirichlet series.

There are also some interesting Dirichlet series that are not $L$-functions but still satisfy the properties you are asking about (meromorphic continuation and zeros at the negative integers). One such function is the so-called Witten zeta function of the group $SU(3)$, as I proved in "On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function" (see theorem 1.3 on page 5). The coefficient sequence for that function is \begin{align} a_n &= \#\{ j,k\ge 1 : n = jk(j+k) \} \\ &= \textrm{the number of inequivalent irreducible} \\ & \quad \textrm{ representations of $SU(3)$ of dimension $n/2$.} \end{align}

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