# Questions tagged [zeta-functions]

Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.

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### Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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### Connecting two pictures of the Zeta function

Lets consider two views of zeta functions of curves. For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...
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### Deformations of the Riemann zeta function

Consider the Dirichlet series (for fixed $0 < a \leq 1$): $$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$ which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...
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### Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
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### Zeros of partial sums of the Ramanujan's zeta function

In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
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### Prove duality of multiple zeta values by Extended Double Shuffle Relations

It is easy to prove the duality theorem of multiple zeta values (MZVs) by the integral representation of MZVs. However, how does one prove MZV's duality theorem solely by Finite Double Shuffle ...
For $\alpha\in\mathbb{C}$, I defined the "complex-weighted" Hurwitz zeta function \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)}\int_0^{\infty} \frac{e^{-wt}}{(1-e^{-t})^{\...