# Questions tagged [euler-product]

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### Pólya–Vinogradov like inequality for a character sum with Euler factors

Let $M$ be a large positive integer, $d$ an odd positive integer and $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{R}$. For a non-principal character $\chi_d = \chi$ with modulus $d$, I ...
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### Partial product of Euler factors

Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let $$\zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}},$$ where $\mathrm{Re}(s)>1$. Is there any $T$ such that $T$ ... 1 vote
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### Some theoretical question on Euler product

It is very rare that the Euler product $\lim_{X \to \infty}\prod_{p \leq X}(1 + a(p)p^{-s})$ conditionally converges for $\sigma > A$ with some $0 < A \leq 1$ when $|a(n)| = 1$. Suppose ...
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### Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

Is anything useful known about the function defined by $f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ?$ Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...
1 vote
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### Euler products without meromorphic continuation

For every prime number $p$ fix a natural number $n_p$ with $$C\frac{p+1}2\le n_p\le C(p+1)$$ for some constant $C>0$, independent of $p$. The product $$F(s)=\prod_p\frac1{(1-p^{-s})^{n_p}}$$ ... 142 views

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### Riemann Hypothesis and Euler product

It is conjecture that under certain conditions a L-function satisfies RH. Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional ...
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### Generalization of Trigonometric, Hyperbolic, $\Gamma$ and $\zeta$ Functions

It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$ $a\not\in\mathbf M,~$ and $~\Re(s)>1,~$...
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### If a Dirichlet series converges Conditionally, how can I apply Euler product?

In 1737, Euler discovered that if $f(n)$ is multiplicative and $\sum f(n)/n^{s}$ converges absolutely for ${\rm Re}(s) > \sigma_a$ then we have \begin{equation} \sum_{n=1}^{\infty} \frac{f(n)}{...
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### Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE. In the literature about Dirichlet $L$-series, I found that their Euler products: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
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### Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
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### Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?) But I would like to know if ...
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### Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function $$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$ where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
### Does this 'alternating' Euler product converge for all $\Re(s) > 0$?
Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ? \displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} \right)^{(...
The conjecture Are all zeros of $\zeta(0+s) \pm \zeta(0-s)$ except a finite few on the line $\Re(s)=0$? was shown to be unconditionally true. The proof can even be extended towards the domain \$\...