# Questions tagged [euler-product]

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### Euler product over subsets of primes

It is well known that $$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$ Given an arbitrary prime $\,q\,$ is it true that $$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$ Thanks.
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### Does this product have analytic continuation?

The product $$F(s)=\prod_{p}\frac1{(1-p^{-s})^p},$$ converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any ...
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### How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? [duplicate]

In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ? Note01 : it ...
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### Euler product for sum of multiplicative function times log

(Cross-posted from StackExchange). Let $g$ be a multiplicative function which satisfies $0 \le g(p) \ll 1/p$ and $$\sum_{p\le x} g(p) = \log \log x + C + O((\log x)^{-10}).$$ Iwaniec and ...
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### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
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### Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product \begin{equation} \zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s), ...
I made a small tweak to the well-known Dirichlet L-function ($p$=prime): $$L(s, \chi_4) :=\prod_p \bigg(\frac {p^s}{p^s-\chi_4(p)} \bigg)=\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p \,\pi}{2}\... 1answer 627 views ### Does the Euler product for L(s,\chi_4) also converge in the right half of the critical strip? This question expands on this one from MSE. In the literature about Dirichlet L-series, I found that their Euler products:$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$... 2answers 543 views ### Do the roots of this equation involving two Euler products all reside on the critical line? This question loosely builds on the second part of this one. Take the Riemann \xi-function: \xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s). Numerical ... 0answers 123 views ### Existence of Euler product on critical line for L(\chi,s) L(\overline{\chi},1-s)? Generally there is no Euler product for Dirichlet L-functions L(\chi,s) in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?) But I would like to know if ... 1answer 584 views ### Euler product of Asai L-function? Let \pi be an automorphic form of GL(n)/\mathbb{Q} with standard L-function$$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$where \{\alpha_{p,i}:i=1,\dots,n\} are the ... 3answers 849 views ### Does this 'alternating' Euler product converge for all \Re(s) > 0? Does the following 'alternating' Euler product, with p_n the n-th prime number, converge for \Re(s)>0 ?$$\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 \$\...