# Questions tagged [euler-product]

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34
questions

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### Are there extensions of Euler's infinite product for sine function? [migrated]

Euler product about sine function is $\frac{\sin(x)}{x} = \prod_{n=1}^\infty \left ( 1- \left(\frac{x}{n\pi}\right)^2 \right)$
I wonder if there is known results about slight modification of above ...

**8**

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**2**answers

884 views

### On modified Euler product

I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it.
Consider the modified Euler product as ...

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86 views

### Analytic continuation of the Euler product of odd primes to $s=1$?

From this question, the following expression emerges that is valid for $n \in \mathbb{N}, n \gt 1$:
$$\zeta(n) = \frac{\sigma^*_n - \sigma_n + 1}{1-2^{-n}} \tag{1}$$
or equivalently with $p$ = a prime:...

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327 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**

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**1**answer

613 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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131 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.

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107 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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**2**answers

253 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}......

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270 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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73 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,...

**11**

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**1**answer

216 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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65 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 ...

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votes

**1**answer

192 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 ...

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126 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
$$
F(s)=\prod_p\frac1{(1-p^{-s})^{n_p}}
$$
...

**2**

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**0**answers

131 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/{\...

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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 ...

**1**

vote

**1**answer

305 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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811 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 ...

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373 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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**1**answer

335 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)}{...

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155 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:
$$\...

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121 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{...

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131 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
$$
\...

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697 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

657 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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361 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 $p_n$ ...

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368 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),
...

**1**

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139 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}\...

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**1**answer

724 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:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

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552 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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128 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 ...

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votes

**1**answer

627 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 ...

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904 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)^{(...

**1**

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92 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 $\...