Questions tagged [euler-product]
The euler-product tag has no summary.
41 questions
2
votes
0
answers
244
views
Consequences of irrationality of Euler-Mascheroni constant
It is well known that, due to Euler product formula for L-functions, one can derive the infinitude of primes from the proofs of the irrationality of pi or of Apery constant. These are in the radius of ...
- 1,125
2
votes
1
answer
262
views
How to efficiently calculate higher logarithmic derivatives of the Riemann zeta function?
I know how to calculate higher derivatives of the Riemann zeta function $$\left(\frac{d}{ds}\right)^i\zeta(s)=\left(\frac{d}{ds}\right)^i\sum_{n=1}^{\infty}\frac1{n^s}=\sum_{n=1}^{\infty}\frac{(-\log ...
- 298
0
votes
1
answer
348
views
Euler product with asymptotic
In Polymath8b project there is that equation,
Which I do not understand the steps. I tried to fix a j and factorise,
$$\displaystyle S_j=\sum_{d_j,e_j}\frac{\mu(d_j)\mu(e_j)}{[d_j,e_j]{d_j}^s{e_j}^t} ...
- 45
4
votes
0
answers
423
views
A question on vanishing of Godement–Jacquet-like zeta integral
$\DeclareMathOperator\GL{GL}$The classical Godement–Jacquet zeta integral is of this form:
$f$ is a matrix coefficient of a cuspidal automorphic representation of $\GL_n(\mathbb{A}_\mathbb{Q})$, and $\...
- 111
-2
votes
1
answer
218
views
Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L_4^*(s) = \...
- 3,198
1
vote
2
answers
564
views
Abscissa of convergence for a very specific Dirichlet series / Euler product
I am interested in the convergence of the following Euler product:
$$
\prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}.
$$
The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
- 3,198
1
vote
0
answers
256
views
Convergence of zeta Euler product with additional term
Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...
- 1,189
2
votes
1
answer
439
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 ...
- 597
1
vote
1
answer
213
views
Sum of an arithmetic sequence involving Euler factors
I am trying to find an asymptotic formula for the following sum as $T \to \infty$.
$$ \sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)...
- 597
6
votes
2
answers
1k
views
On modified Euler product
Consider the modified Euler product as follows:
$$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$
Here $c$ is a constant
My questions are
Is there a compact representation for this ...
- 159
2
votes
2
answers
918
views
Convergence of Euler product and Dirichlet series in the same half-plane?
I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there.
Suppose we have an Euler product over the primes
$$F(s) = \prod_{...
- 263
18
votes
1
answer
759
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:
$$\zeta(2n):= \frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{...
- 4,211
1
vote
0
answers
195
views
Euler product over subsets of primes
It is well known that
$$\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\;\;\;?$$
Thanks.
- 1,018
1
vote
0
answers
183
views
What is behind the constant in the functional equation for the Hasse-Weil zeta function?
Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ of dimension $n$. The Weil conjectures assert that the zeta function $Z(X_0,t)$ satisfies the functional equation
$$Z(X_0,t) = \pm q^{\...
- 4,272
-3
votes
2
answers
319
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}......
- 2,452