Expressing a series of complex exponentials as an asymptotic expansion

Question

Lately I've been working on a certain problem. To solve it, I wish to find out if I can find a 'nicer' representation of the following series:

$$f(t)=\sum_{n=-\infty}^{\infty}\frac{\Gamma(s_n)t^{-s_n}\zeta_R(2s_n)l^{h-1}(1-l^{1-ws_n})}{\pi^{2s_n}\cdot wln(l)\cdot l^{h-ws_n}}$$ Where $s_n=\frac{h}{w}+\frac{2\pi in}{wln(l)}$, and $\zeta_R$ is the Riemann zeta function (all parameters $w,h,l$ are given positive real numbers). More generally, I'm interested in a general method for finding a 'nice' expression for: $$f(t)=\sum_{n=-\infty}^{\infty}a_n t^{-s_n}=\frac{1}{t^{h/w}}\sum_{n=-\infty}^{\infty}a_n t^{-2\pi in/wln(l)}$$ By 'a nice expression', I ideally mean a way to express $f(t)$ as an asymptotic exapnsion for $t\rightarrow0^+$ - if it is at all possible. If there isn't a good way to do this, I'm open to hearing other suggestions for ways to view this series "more simply" as $t\rightarrow0^+$. I'm purposely keeping this a bit vague, since this is an open question and so I'm open to all sorts of ideas. I'm mainly interested in finding out if there is some asymptotic expansion for this function which contains a constant term independent of $t$ (this is actually my main question).

The motivation for my question is the asymptotic expansion for the trace of the heat kernel on manifolds, which is usually something of the form:

$$K(t,x,x)\sim \frac{1}{(4\pi t)^{n/2}} \sum_{k=0}^\infty a_kt^k$$

I'm working on constructing a similar expansion for "different" spaces other than manifolds, and I eventually got to the expression above. But my current formula has various problems:

1. The function $t^{ik}$ is not actually a function, due to having different branches. I thought of simplifying the formula above by using a Laurent series, but since I have a problem with branches, I don't think I can do this around $t=0$.Maybe since I assume $t$ to be real and positive I can go around this problem, but I'm not sure how.

2. The power term $t^{-h/w}$ does not have an integer power, so I probably won't get a 'nice' expression like in the usual heat kernel on manifolds. But I hope to get something at least remotely nice.

If anyone has any suggestions on how to approach this problem, I'd be happy to hear. Any ideas on how to simplify expressions of the form above (or even the specific $f$ I gave) in order to get an asymptotic expansion similar to the one for the trace of the heat kernel?

Thanks in advance.

Answer 1

Why do you keep all those parameters. You are looking at $$f(x)=\sum_n \Gamma(a+ibn) \zeta(2(a+ibn))e^{2\pi in x}=\sum_n G(bn) e^{2i\pi n x}$$ Then $G(t)=\Gamma(a+it)\zeta(2(a+it))$ is the Fourier transform of $g(t)=e^{u a} \sum_{m\ge 1} e^{-m^2 e^u}$ and the Fourier series gives that $$\sum_n G(bn) e^{2i\pi nx}=\frac1b \sum_k g((k-x)/b)$$ (up to many typos)