Questions tagged [euler-product]

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Abscissa of convergence for a very specific Dirichlet series / Euler product

I am interested in the convergence of the following Euler product: $$ \prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}. $$ The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
1 vote
0 answers
175 views

Convergence of zeta Euler product with additional term

Let's consider the following Euler product ($s=\sigma+it)$: $$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$ So for $\sigma>1$, it is clear the product converges and we have: $$...
  • 1,091
2 votes
1 answer
232 views

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 ...
  • 549
1 vote
1 answer
119 views

Sum of an arithmetic sequence involving Euler factors

I am trying to find an asymptotic formula for the following sum as $T \to \infty$. $$ \sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)...
  • 549
6 votes
2 answers
1k views

On modified Euler product

Consider the modified Euler product as follows: $$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$ Here $c$ is a constant My questions are Is there a compact representation for this ...
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2 votes
2 answers
489 views

Convergence of Euler product and Dirichlet series in the same half-plane?

I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there. Suppose we have an Euler product over the primes $$F(s) = \prod_{...
18 votes
1 answer
638 views

Could computing the next prime in a finite Euler product be made rigorous?

It is well known that: $$\zeta(s):=\prod_{n=1}^{\infty} \frac{1}{1-p_n^{-s}} \qquad \Re(s) \gt 1$$ with $p_n =$ the $n$-th prime. It also known that: $$\zeta(2n):= \frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{...
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1 vote
0 answers
139 views

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.
1 vote
0 answers
121 views

What is behind the constant in the functional equation for the Hasse-Weil zeta function?

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ of dimension $n$. The Weil conjectures assert that the zeta function $Z(X_0,t)$ satisfies the functional equation $$Z(X_0,t) = \pm q^{\...
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-3 votes
2 answers
256 views

The inequality $\Pi (1-\frac{1}{a_i})^{x_i} \le \Pi (1-\frac{1}{b_j})^{y_j} $ hold? [closed]

Question: Are the properties as follows holds? Version 1: the answer by Bjørn Kjos-Hanssen Let $P$ be a positive integers. We written: $P=$ $a_1^{x_1}a_2^{x_2}...a_n^{x_n}$ $=b_1^{y_1}b_2^{y_2}......
3 votes
1 answer
274 views

Reference request for Euler products in positive characteristic

Let $K$ be a global field with ring of integers $O$, and let $f$ some integer valued function whose domain is the set of ideals of $O$ (e.g. $f(I)=|O:I|$). Extending the ordinary definition from the ...
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2 votes
0 answers
75 views

Accelerating convergence of a product by multiplying by zeta values: history?

Let $R(s_1,\dotsc,s_n) = \prod_p r(p^{-s_1},\dotsc,p^{-s_n})$, where $r$ is a rational function on $n$ variables. Say we want to compute the value of $R(s_1,\dotsc,s_n)$ for some choice of $s_1,\dotsc,...
  • 515
11 votes
1 answer
227 views

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 $T$ ...
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1 vote
0 answers
68 views

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 ...
0 votes
1 answer
242 views

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
0 answers
132 views

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 $$ F(s)=\prod_p\frac1{(1-p^{-s})^{n_p}} $$ ...
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2 votes
0 answers
138 views

Interchanging limit and infinite product in Euler product for Dedekind function s=1

For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation $$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
11 votes
2 answers
1k views

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 ...
user avatar
1 vote
1 answer
346 views

What is the Euler product for double summations?

I know that the Euler product of a summation of multiplicative function is given by $$\sum_nf(n)=\prod_p (1+f(p)+f(p^2)+....),$$ and if we have the Möbius function then it will be $$\sum_n\mu (n)...
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4 votes
2 answers
907 views

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 ...
  • 1,091
3 votes
0 answers
382 views

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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5 votes
1 answer
376 views

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)}{...
1 vote
0 answers
162 views

Convergence of the Mellin transform of $\zeta(s)\, \Gamma(s)$ for line integrals with real part $\le 1$

This question is inspired by this one, however I believe is quite different. The Mellin transform for: $$\displaystyle \Gamma(s)\,\zeta(s)= \int_0^\infty x^{s-1} \frac{1}{e^x-1}\,dx$$ equals: $$\...
  • 3,989
1 vote
0 answers
130 views

Behavior of partial Euler product in the critical strip (with Dirichlet Character)

Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) : $$P(\chi,N)=\prod_{i=1}^{N} \frac{...
  • 1,091
3 votes
0 answers
135 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ \...
7 votes
2 answers
721 views

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 ...
3 votes
1 answer
667 views

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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6 votes
2 answers
377 views

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 $p_n$ ...
  • 3,989
8 votes
0 answers
382 views

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), ...
  • 3,649
1 vote
0 answers
141 views

An apparent closed form for a slightly tweaked Dirichlet L-function. Could it be proven? [closed]

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}\...
  • 3,989
5 votes
1 answer
831 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: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
  • 3,989
4 votes
2 answers
569 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 ...
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1 vote
0 answers
144 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 ...
  • 1,091
0 votes
1 answer
679 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 ...
  • 3,674
6 votes
3 answers
955 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)^{(...
  • 3,989
1 vote
0 answers
97 views

Question about the zeros of the sum/difference of two finite Euler products

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 $\...
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