# Tagged Questions

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### Derivative of the Selberg $\zeta$-function

I want to compute the derivative of the Selberg $\zeta$-function: $$\mathcal{Z}(s)=\prod_{\gamma \; \text{primitive}} \prod_{n=0}^\infty (1-e^{-l(\gamma)(n+s)}); \qquad \Re(s)>1.$$ Where ...
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### Approximation of Products of Truncated Prime $\zeta$ Functions

The problem arose, while I was looking at products of power prime zeta functions $$P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks},$$ with $k\in \mathbb{N}$ and $s=it$ with real $t$. By using ...
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### Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...
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### When functions, analytically continued, carry over certain properties

Let $\Omega$ be a sufficiently smooth planar region in $\mathbb{R}^2$ with spectrum $\Gamma$ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary \$ \partial ...
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### Square-free zeta function zeros

It is a well known fact that the geometric series $$1+x+x^2+x^3+\ldots$$ has the following form $$\frac{1}{1-x}$$ Another possible representation is $$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$ ...