Questions tagged [euler-product]
For questions on Euler products, an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
118
questions
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1answer
83 views
On modified Euler product: [closed]
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 ...
1
vote
1answer
61 views
Where does $\frac{3}{2}$ in $1<s\cdot\zeta(s+1) < \frac{3}{2}$ come from?
In his short note on Euler and the prime harmonic series, Professor of Mathematics Paul Pollack, makes the following intermediate statement about the Riemann zeta function $\zeta(s)$ for $s>1$, ...
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2answers
77 views
Circular argument for infinitude of primes from Euler product?
Many textbooks will explain how the Euler product formula can be used as a basis for proving the infinite of primes:
$$\sum_{n}\frac{1}{n^{s}}=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}$$
My concern ...
1
vote
1answer
89 views
Counting function and its Artin L-function
I am working with the following function: given a polynomial, $P(x)$, with non-negative integer coefficients and passing through origin, we can define
$$f(d)=\{1\leq a\leq d \ | \ P(a)\equiv 0 \mod(d)\...
8
votes
5answers
533 views
Convergence of infinite products flaw in this derivation of Euler product formula?
I am trying to write a derivation of the Euler product formula in a way that is accessible to younger students and those not mathematically trained at university.
Because of this I am starting from ...
0
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1answer
44 views
I want to find maxima for a particular Euler product
I would like to obtain large numbers for a particular Euler product given by:
$\zeta(s) = {\displaystyle \prod_{p} \left( 1-p^{-s} \right)^{-1} }$
for $s = 1 + ia$
$i$ being the imaginary unit.
In ...
0
votes
0answers
27 views
Do we always take the imaginary part when solving for sine and the real part when solving for cosine?
I’m currently working on differential equations with particular solutions by guessing a complex solution that includes both a real and an imaginary part. For example, for the following Problem, I have ...
1
vote
1answer
66 views
Cotangent in integral form
Could you help me to establish the following relation, without going through the Eulerian development
$\displaystyle \pi \operatorname{cotan}(\pi z) = \frac{1}{z} + \sum_{n=1}^{+\infty} \left( \frac{1}...
2
votes
1answer
95 views
Euler products, Merten's theorems, and an unexpected result
I'm going to start by saying I'm mostly out of my depth here. I'm an amateur recreational mathematician. But I've been looking at the Twin Prime Conjecture lately, because it is so fascinating. Easy ...
5
votes
1answer
52 views
Abelian group zeta function
Let $s \in \mathbb{C}$. What's known about
$$\zeta_{\mathrm{ab}}(s) := \sum_G \frac{1}{o(G)^s} \tag{1}$$
where the sum is over all finite abelian groups $G$ up to isomorphism?
By the primary ...
0
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0answers
38 views
Analytic continuation of $\Psi(s)= \sum_{n=1}^\infty e^{-n^s+n^{-s}} $
Prescribe a function$$\Psi(s)= \sum_{n=1}^\infty e^{-n^s+n^{-s}} $$
$\Psi$ converges for $\Re(s)>0.$
What is the analytic continuation of $\Psi?$
I think $$ \bigg(\prod_{p\in\Bbb P} \frac{1}{1+e^{...
0
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0answers
35 views
Difference of sum of reciprocal of primes in two different congruence classes
Let $P$ denote the set of primes and let $A$ and $B$ be distinct non-zero congruence classes modulo a prime $q$. What do we know about the infinite sum:
$$S = \sum_{p \in P \, \cap A} \frac{1}{p} - \...
1
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0answers
30 views
What is the value of this prime product formula
What is the value of this prime product formula:
$$\prod_{p=1,2\pmod5}\left(1-\frac{1}{p}\right)^{-1}\prod_{q=3,4\pmod5}\left(1+\frac{1}{q}\right)^{-1}$$
I tried to translate into sums over certain ...
5
votes
0answers
197 views
Convergence of Euler product implies convergence of Dirichlet series?
(Crossposted to Math Overflow) Suppose we have an Euler product over the primes
$$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$
where each $a_p \in \mathbb{C}$. The Euler product is ...
16
votes
3answers
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Proof of Infinitude of Primes by Euler's Product Formula is Circular?
Many guides will refer to Euler's product formula as simple way to prove that the number of primes is infinite.
$$\sum_n\frac{1}{n} = \prod_p \frac{1}{1-\frac{1}{p}}$$
The argument is that if the ...
0
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0answers
26 views
Finite Jacobi triple product
With regard to the post Is this correct for this expression?
Could we obtain an expression for the double finite product below?
$$
\prod_{i=1}^n\prod_{j=1}^k(1-x_{ij}),
$$
Where $0<x_{ij}<1\,\,\...
0
votes
0answers
73 views
Inequality between Finite Sum of Reciprocals and Finite Euler product formula using Square-free Factor
The Wikipedia page on the divergence of the sum of reciprocals of primes has a proof of the log log divergence rate.
https://en.wikipedia.org/wiki/...
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3answers
66 views
How would you find the number of solutions to $m_1+2m_2+3m_3=n$?
Assuming $m_i$ are nonnegative integers
I understand that we need to use $C(n+k-1,n-1)$ here but I am not sure how the coefficients of $m_i$ affect the equation?
For example to find the number of ...
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vote
2answers
85 views
Explain why the coefficient of $x^n$ in the Euler product is equal to the number of partitions of n?
The Euler product is
$\prod_{k=1}^{\infty}\frac{1}{1-x^k}=\prod_{k=1}^{\infty}(1 + x^k + x^{2k} + x^{3k} + ...)$
Why does the coefficient $x^n$ equal the number of partitions of n from the Euler ...
0
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1answer
89 views
Explain which partitions of $n$ does the coefficient of $x^n$ count in the following…
$\frac{1}{1-x}\frac{1}{1-x^3}(1+x^2+x^4)$.
I've expanded out the 2 fractions with Taylor series then found that it equals $1+\sum_{n=1}^{\infty}nx^n$ but I'm not sure if this is right/ what I can say ...
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0answers
45 views
Euler product over Landau primes
Is there anyone who can suggest me some papers on the following constant (Euler product over Landau primes)?
$$\prod_{p\,=\,n^2+1}\,(1-p^{-1})=0.35588892...$$
Many thanks.
1
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1answer
43 views
The argument of a finite Euler product at a non-trivial zero of $\zeta(s)$.
With $p_n$ is the $n$-th prime number, we know that:
$$\arg\left(\zeta(s)\right)=\arg\prod_{n=1}^{\infty} \frac{1}{1-\frac{1}{p_n^s}}=\sum_{n=1}^{\infty} \arg\left(\frac{1}{1-\frac{1}{p_n^s}}\right)\...
0
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0answers
41 views
Is the following true $\log \big(\zeta(s)\big)=s \times\int_0^\infty J(x)x^{-s-1}dx=-\sum_p \log\bigg(1-p^{-s}\bigg)$?
Define$$ f(s)=\log\bigg(\sum_{n=1}^\infty n^{-(e^s)}\bigg)$$
where,
$$\exp\bigg(f\big(\log(s)\big)\bigg)=\sum_{n=1}^\infty n^{-s}=\zeta(s). $$
With $f(s),$ I substituted the euler product formula ...
0
votes
1answer
62 views
Does $\zeta(s)=\prod \frac{1}{1-p^{-s}}$ converge for $ \Re(s) >1$ for $p= iq $ (Gaussian prime)?what about $\zeta(2),\zeta(4),\cdots$?
The Riemann zeta function is defined by the Euler product as :$\zeta(s)=\prod \frac{1}{1-p^{-s}}$ it is converge for $ \Re(s) >1$ , now if we plug instead of ordinary prime $p$ the Gaussian prime ...
1
vote
1answer
53 views
Proof of an identity concerning the prime $\zeta$ function
I have to prove the following identity:
let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then
\begin{equation}
P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)).
\end{equation}
I proved that
\...
3
votes
1answer
118 views
Estimate for $\sum_{n\leq x}2^{\Omega(n)}$
I need some help to find a mistake in my proof.
I have to prove that $\sum_{n\leq x}2^{\Omega(n)}\sim cx\log^2x$ for $x\rightarrow+\infty$, where $\Omega(p_1^{k_1}\cdot\ldots\cdot p_j^{k_j})=k_1+\...
2
votes
1answer
93 views
Euler product exists for the Dedekind zeta function
The Dedekind zeta function of a number field $K$, denoted by $\zeta_K(s)$, is defined for all complex numbers $s$ with $\Re(s) > 1$ by the Dirichlet series
\begin{equation*}
\zeta_K(s) = \sum_{\...
2
votes
0answers
77 views
Does this sum $(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{…^{(1-\frac{1}{p^2})}}}$ also related to Riemann zeta function?
I'm interesting for iterated exponention sum of the form $(z_1)^{z_2)^{...^{z_k}}}$ such that $z_1$ and $z_2$, $z_k$ are differents real exponents , This kind of sum was studied by many Authors such ...
0
votes
1answer
67 views
Zeta function to Euler product
I heard that the progress zeta function to Euler product is like below
$$\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+...$$
$$\frac{1}{2^s}\zeta(s) = \frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}...
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0answers
21 views
Domain of Euler product
I heard that Euler product is defined where s>1(real number).
$$\zeta(s) =\prod_{p=prime} (1 - p^{-s})^{-1}$$
but, Why? Why it couldn't be defined where s is complex number?
2
votes
1answer
68 views
Efficiently summing reciprocal of polynomial over primes.
For fun, I have been trying to calculate the sum of the reciprocals of the cube-full numbers. I have managed to show that the limit is equal to $$\frac{\zeta(3)\zeta(4)\zeta(5)}{\zeta(8)\zeta(10)}\...
2
votes
1answer
128 views
Number of twin primes
I am reading article about pseudorandomness of primes and there is a part that I don't understand.
Based on Cramer model(without even numbers) the number of twin primes in $[1,..,N]$
is ~ $2\frac{N}{...
3
votes
1answer
90 views
Trouble Proving $\sum_{n=1}^\infty \frac{\mathrm{d}(n)^2}{n^s}=\frac{\zeta(s)^4}{\zeta(2s)}$
I am running into considerable trouble trying to prove the identity in the question. I figure the solution will come from Euler-products, so here was my attempt. I want to show that
$$
\sum_{n=1}^\...
1
vote
1answer
236 views
What is the Euler product?
I have two or three questions directly related to this wikipedia article:
wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
I think it is exactly about my main Q (see my other Q, ...
2
votes
2answers
190 views
Is there a proof for Euler's “other” formula for pi?
Some context: The webpage Pi Formulas on Wolfram.com has a list of some well-known formulas for $\pi$. However, there is a formula, as stated below, which I can't find a proof for.
$$\pi = \frac{2}{\...
1
vote
1answer
55 views
Euler product for L-series of a modular form
I might not be seeing the wood for all the trees in what follows.
If $f(z)=\sum_{n}a_{n}z^{-n}$ is a Hecke eigenform, the coefficients $a_{n}$ satisfy the relation
$$a_{p^{j}}=a_{p^{j-1}}a_{p}-pa_{p^{...
0
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0answers
49 views
How to show Euler’s Product? [duplicate]
Calling $P$ the set of prime numbers, Euler proved that $$\sum_{n=1}^{\infty}\frac 1{n^s}=\zeta (s)=\prod_{p\in P} \frac 1{1-\frac 1{p^s}}$$
At University my professor proved that in this way:
First ...
0
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1answer
46 views
Approximating Euler Product
Let $E(p)= \prod_{p}\frac{p+1}{p}$ for prime $p$.
Then $E(2)=\frac{2}{1}$, $E(3)=\frac{2}{1}*\frac{3}{2}$, $E(5)=\frac{2}{1}*\frac{3}{2}*\frac{5}{4}$, $E(7)=\frac{2}{1}*\frac{3}{2}*\frac{5}{4}*\frac{...
0
votes
1answer
92 views
Minimum and maximum of a partial Euler product?
Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be
$$
\beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s};
$$
which for ...
11
votes
2answers
218 views
What can be said about $\prod_{s=2}^{\infty} \zeta(s) $?
Another problem from quora.
What can be said about
$v
=\prod_{s=2}^{\infty} \zeta(s)
$?
Wolfy says that
$v
\approx 2.294856591673313794183
$.
The Inverse Symbolic Calculator
(http://wayback.cecm....
3
votes
0answers
51 views
An Euler product involving the prime counting function. Does its alternating version converge for $s=1$?
The classical Euler product $\displaystyle \prod_{p \in \mathbb{P}} \left( \frac{1}{1-p^{-s}} \right)$ could be rewritten as:
$$eu_0(s):=\prod_{n=2}^{\infty} \left( \frac{1-\frac{1}{(n+1)^s}}{1-\frac{...
2
votes
1answer
163 views
Partial Euler product
The Riemann Zeta function defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$ For $\Re(s)>1$ is convergent and admits the Euler product representation $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ For ...
1
vote
1answer
85 views
How does one compute the Euler product for the Dirichlet Beta function?
In this post, the author derives the Euler product for Dirichlet Beta function, defined as
$$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$
for $\Re(s)>0$ and obtains
$$\beta(s) = \prod_p ...
-1
votes
1answer
44 views
Permutation group of Satake parameters
Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $ L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $ G_{p}$ of the ...
1
vote
0answers
63 views
Need someone to show me how the Zeta function is equal to Euler's product formula?
$\zeta(x)$=$1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+\frac{1}{6^x}+\cdots$Euler's product formula states, for all $x$ greater than $1$ we have:$\zeta(x)$=$1\over 1-\frac{1}{2^x}$$\...
2
votes
2answers
71 views
Need help showing that $\zeta (x)$= ${1\over 1-2^{1-x}}$ $\eta (x)$
This question is I am working on is the extension of the domain of zeta function.$\zeta(x)$=$1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+\frac{1}{6^x}+ \cdots$$\eta(x)$=$1-\frac{1}{2^x}+\...
2
votes
0answers
127 views
Convergence of Euler Product on the line Re(s) = 1
Consider the Euler product for the Riemann Zeta function:
\begin{equation}
\zeta(s) =? \prod_p (1-p^{-s})^{-1}
\end{equation}
When I started studying this product, I read that for $Re(s) > 1$, the ...
5
votes
1answer
213 views
Not-too-slow computation of Euler products / singular series
I'd like to compute, to at least a few digits of accuracy, the constants that arise in Hardy-Littlewood conjecture F / Bateman-Horn conjecture, in particular for just a single quadratic polynomial.
...
1
vote
1answer
46 views
Could be Euler product for Riemann zeta function runs over pseudo-prime?
The Euler product over primes defined as :$$\zeta(s)=\prod_{p \ \text{prime}} \frac{1}{1-p^{-s}}\tag{01}$$ , My question Here is : is it possible to write this product $(01)$ for which run or ...
2
votes
1answer
111 views
Understanding a sum with logs and prime numbers.
I'm reading a proof and I don't understand this step:
$q$ are the primes
$$-\sum_q \ln \left(1-\frac{1}{q^s}\right) = \sum_q \frac{1}{q^s} + \mathcal{O}(1)$$
Working with
$$-\ln \left(1-\frac{1}{q^...
