2 TeXed up

I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

Z_+(s)

$$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)p)$$

and

Z_-(s)

$$Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)p)$$

where S(n,m;p) $S(n,m;p)$ is the Kloosterman sum. This sum converges when s $s$ has a large real part and can be analytically continued to define Z_{+/-}(s) $Z_{+/-}(s)$ on the whole complex s $s$ plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether Z_{+/-}(s) $Z_{+/-}(s)$ has poles for any integer values of s, $s$, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions but I have not found quite what I'm looking for. Most treatments focus on the general properties of the Kloosterman Zeta function, which are quite complicated.
My suspicion is that because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.

1

# Poles of Kloosterman Zeta Function

I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)

and

Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)

where S(n,m;p) is the Kloosterman sum. This sum converges when s has a large real part and can be analytically continued to define Z_{+/-}(s) on the whole complex s plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether Z_{+/-}(s) has poles for any integer values of s, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions but I have not found quite what I'm looking for. Most treatments focus on the general properties of the Kloosterman Zeta function, which are quite complicated.
My suspicion is that because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.