Questions tagged [euler-product]
For questions on Euler products, an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
130
questions
-1
votes
0
answers
64
views
Partial Euler product for $\zeta$-function
I want to study the function $\zeta_x (\sigma)=\prod_{p\leq x}\left(1-\dfrac{1}{p^{\sigma}}\right)^{-1}$, where $p$ is a prime number. And find the following limit or estimate its value:
$$
\lim_{x\to ...
- 445
0
votes
0
answers
44
views
Euler product formula and zeta function, Stein's proof in Fourier Analysis book
I have been reading through Stein's Fourier Analysis book and in Chapter 8 I noticed something that I cannot quite get my head around.
Theorem 1.10: For every $s >1$, we have $$ \zeta(s) = \prod_{...
- 33
3
votes
1
answer
57
views
Counting numbers up to $n$ whose prime factorizations have exactly $k$ prime factors with exponent $1$
Question. Let $N_k(n)$ count how many numbers $1\le x\le n$ for which $x$ has exactly $k$ unitary prime divisors, or equivalently $x$'s prime factorization has exactly $k$ primes with exponent $1$. ...
- 15.5k
0
votes
1
answer
38
views
Does the Euler product for the Rankin-Selberg convolution of two eigenforms require the Ramanujan conjecture?
Suppose $f$ and $g$ are weight $k$ eigenforms for the Hecke operators, normalized, with Fourier coefficients $a_{n}$ and $b_{n}$ respectively. I wanted to see if I could derive a Euler product for
$$L(...
- 645
3
votes
1
answer
88
views
How to prove that $\sqrt{3}\pi/6=\prod_{p \equiv 1 \pmod{6}} \frac{p}{p-1}\prod_{p \equiv 5 \pmod{6}} \frac{p}{p+1}$ with $p \in \mathbb{P}$?
I would like to prove the formula $$\frac{\sqrt{3}\pi}{6}=\left(\prod_{\substack{p \equiv 1 \pmod{6} \\ p \in \mathbb{P}}} \frac{p}{p-1}\right) \cdot \left(\prod_{\substack{p \equiv 5 \pmod{6} \\ p \...
- 904
0
votes
0
answers
54
views
A variant of the Euler product formula?
Does the following limit make sense? $\prod_{2 \leq p \leq n} (1-\frac{1}{p})\ln n \rightarrow 1,$ as $n \rightarrow \infty. $
- 1,755
2
votes
1
answer
80
views
Are there extensions of Euler's infinite product for sine function?
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 ...
Kimdongyeob
2
votes
0
answers
33
views
A way of splitting Euler products
I bumped into a claim I am not understanding completely about Euler products. Is it true that
\begin{align}
&\prod_p (1 + a(p) p^{-s} + a(p^2) p^{-2s} + \cdots) \\
&\hspace{2cm}= \prod_p (1 + ...
- 973
0
votes
0
answers
78
views
Trying to prove that $\varphi(s)$ is related to prime numbers
My goal is to show that $\varphi(s)$ is related to prime numbers.
Define $$ \varphi(s)=\sum_{n=1}^\infty \big(e^{-n^{-s}}-1\big). $$
Note that $e^x-1=\sum_{k\geq 1} x^k/k!$ for all $x$. One can use ...
- 861
3
votes
1
answer
133
views
Validity of proof $\zeta(s) \neq 0$ for $\sigma >1$.
The Riemann zeta function is can be expressed as an infinite series, as well as an infinite Euler product over primes $p$.
$$ \zeta(s) = \sum_n 1/n^s = \prod_p(1-1/p^s)^{-1} $$
Here $s=\sigma+it$ and ...
- 1,772
7
votes
1
answer
172
views
A prime numbers property: $4/\pi=e^{1/2}\Big(1+e^{1/3}\big(1+e^{1/5}(\dots)\big)\Big)/p_n, \ \ n\to\infty$
It is possible to show that the limit of the following fraction is a constant $$ \frac {e ^ {1/2} \Big (1 + e ^ {1/3} \big (1 + e ^ {1/5} (1 + e ^ {1/7} (\dots (1 + e ^ {1 / p (n)})} {p(n)} \sim c $$
...
- 1,507
0
votes
0
answers
91
views
Validity of taking log of divergent sums in proof $\sum 1/p \rightarrow \infty$
This short proof that $\sum 1/p \rightarrow \infty$ from this source, pdf the author takes logs of divergent series.
I have seen many comments saying that such proofs are invalid because you can't ...
- 1,772
2
votes
2
answers
89
views
Does this sum of two Euler products converge for $s=1$?
The following sum of two Euler products is always real for real $s$:
$$A(s):=\prod_{p=prime} (1 - (i\,p)^{-s})^{-1}+\prod_{p=prime} (1 - (-i\,p)^{-s})^{-1}$$
ADDED:
Some (trivial) closed forms that I ...
- 2,539
2
votes
0
answers
55
views
Using Viete's Product show that $\frac{3}{\pi}= \frac{\sqrt{2+\sqrt{3}}}{2} * \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2} * ...$
Viete's product is
$$
\frac{2}{\pi}=\frac{\sqrt{2}}{2} * \frac{\sqrt{2+\sqrt{2}}}{2} * \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}*...
$$
Show that
$$
\frac{3}{\pi} = \frac{\sqrt{2+\sqrt{3}}}{2} * \frac{\...
- 27
1
vote
1
answer
79
views
validity of proof $\pi$ is irrational from Euler's product formula
Many articles and blog posts (and some textbooks) mention in passing that Euler's product formula can be used to prove that $\pi$ is irrational.
$$ \zeta(s) = \sum_n \frac{1}{n^s} = \prod_n \left( \...
- 1,772
2
votes
2
answers
112
views
Why is Euler's statement $\exp{(A + \frac{1}{2}B + \frac{1}{3}C+ \ldots)} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ true?
Euler, in his paper Variae observationes circa series infinitas [src], makes the following statements in his Theorem 19.
$$ \exp{(A + \frac{1}{2}B + \frac{1}{3}C+ \frac{1}{4}D + \ldots)} = 1 + \frac{1}...
- 1,772
0
votes
1
answer
94
views
Why is Euler product destroyed?
Consider the lower branch of the hyperbola $f(x)=\frac{1}{x}.$ We can add up the images on the curve corresponding to pre-images equal to $-{n}$ for $n=1,2,3,\cdot\cdot\cdot.$
Upon doing this we get, $...
- 861
1
vote
1
answer
80
views
Where does $\frac{3}{2}$ in $1<s\cdot\zeta(s+1) < \frac{3}{2}$ come from?
In his short note on Euler and the prime harmonic series, Professor of Mathematics Paul Pollack, makes the following intermediate statement about the Riemann zeta function $\zeta(s)$ for $s>1$, ...
- 1,772
1
vote
2
answers
109
views
Circular argument for infinitude of primes from Euler product?
Many textbooks will explain how the Euler product formula can be used as a basis for proving the infinite of primes:
$$\sum_{n}\frac{1}{n^{s}}=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}$$
My concern ...
- 1,772
1
vote
1
answer
100
views
Counting function and its Artin L-function
I am working with the following function: given a polynomial, $P(x)$, with non-negative integer coefficients and passing through origin, we can define
$$f(d)=\{1\leq a\leq d \ | \ P(a)\equiv 0 \mod(d)\...
- 141
8
votes
5
answers
648
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 ...
- 1,772
0
votes
1
answer
49
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 ...
- 1
1
vote
1
answer
81
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}...
- 157
4
votes
2
answers
162
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 ...
- 1,385
6
votes
1
answer
76
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 ...
- 7,391
1
vote
0
answers
31
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 ...
- 629
5
votes
0
answers
234
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 ...
- 4,534
15
votes
3
answers
3k
views
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 ...
- 1,772
0
votes
0
answers
33
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\,\,\...
- 139
0
votes
0
answers
84
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/...
- 1,772
1
vote
3
answers
69
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 ...
- 21
1
vote
2
answers
118
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 ...
- 21
0
votes
1
answer
98
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 ...
user791837
0
votes
0
answers
60
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.
- 1,806
1
vote
1
answer
49
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:
$$\arg\left(\zeta(s)\right)=\arg\prod_{n=1}^{\infty} \frac{1}{1-\frac{1}{p_n^s}}=\sum_{n=1}^{\infty} \arg\left(\frac{1}{1-\frac{1}{p_n^s}}\right)\...
- 2,539
0
votes
1
answer
90
views
Does $\zeta(s)=\prod \frac{1}{1-p^{-s}}$ converge for $ \Re(s) >1$ for $p= iq $ (Gaussian prime)?what about $\zeta(2),\zeta(4),\cdots$?
The Riemann zeta function is defined by the Euler product as :$\zeta(s)=\prod \frac{1}{1-p^{-s}}$ it is converge for $ \Re(s) >1$ , now if we plug instead of ordinary prime $p$ the Gaussian prime ...
- 6,578
1
vote
1
answer
81
views
Proof of an identity concerning the prime $\zeta$ function
I have to prove the following identity:
let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then
\begin{equation}
P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)).
\end{equation}
I proved that
\...
- 187
4
votes
1
answer
205
views
Estimate for $\sum_{n\leq x}2^{\Omega(n)}$
I need some help to find a mistake in my proof.
I have to prove that $\sum_{n\leq x}2^{\Omega(n)}\sim cx\log^2x$ for $x\rightarrow+\infty$, where $\Omega(p_1^{k_1}\cdot\ldots\cdot p_j^{k_j})=k_1+\...
- 187
2
votes
1
answer
169
views
Euler product exists for the Dedekind zeta function
The Dedekind zeta function of a number field $K$, denoted by $\zeta_K(s)$, is defined for all complex numbers $s$ with $\Re(s) > 1$ by the Dirichlet series
\begin{equation*}
\zeta_K(s) = \sum_{\...
- 1,015
2
votes
0
answers
82
views
Does this sum $(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{...^{(1-\frac{1}{p^2})}}}$ also related to Riemann zeta function?
I'm interesting for iterated exponention sum of the form $(z_1)^{z_2)^{...^{z_k}}}$ such that $z_1$ and $z_2$, $z_k$ are differents real exponents , This kind of sum was studied by many Authors such ...
- 6,578
0
votes
1
answer
88
views
Zeta function to Euler product
I heard that the progress zeta function to Euler product is like below
$$\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}...
- 23
0
votes
0
answers
27
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?
- 23
3
votes
2
answers
123
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 $$\frac{\zeta(3)\zeta(4)\zeta(5)}{\zeta(8)\zeta(10)}\...
- 6,055
2
votes
1
answer
163
views
Number of twin primes
I am reading article about pseudorandomness of primes and there is a part that I don't understand.
Based on Cramer model(without even numbers) the number of twin primes in $[1,..,N]$
is ~ $2\frac{N}{...
- 43
4
votes
1
answer
124
views
Trouble Proving $\sum_{n=1}^\infty \frac{\mathrm{d}(n)^2}{n^s}=\frac{\zeta(s)^4}{\zeta(2s)}$
I am running into considerable trouble trying to prove the identity in the question. I figure the solution will come from Euler-products, so here was my attempt. I want to show that
$$
\sum_{n=1}^\...
- 1,814
1
vote
1
answer
395
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, ...
user727355
2
votes
2
answers
246
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.
$$\pi = \frac{2}{\...
- 1,409
1
vote
1
answer
82
views
Euler product for L-series of a modular form
I might not be seeing the wood for all the trees in what follows.
If $f(z)=\sum_{n}a_{n}z^{-n}$ is a Hecke eigenform, the coefficients $a_{n}$ satisfy the relation
$$a_{p^{j}}=a_{p^{j-1}}a_{p}-pa_{p^{...
- 1,432
0
votes
0
answers
66
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 ...
- 1,942
0
votes
1
answer
63
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{...
- 2,791