Questions tagged [euler-product]

For questions on Euler products, an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

Filter by
Sorted by
Tagged with
-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 ...
user avatar
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_{...
user avatar
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 $1$. ...
user avatar
  • 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(...
user avatar
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 \...
user avatar
  • 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. $
user avatar
  • 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 ...
user avatar
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 + ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
  • 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 $$ \frac {e ^ {1/2} \Big (1 + e ^ {1/3} \big (1 + e ^ {1/5} (1 + e ^ {1/7} (\dots (1 + e ^ {1 / p (n)})} {p(n)} \sim c $$ ...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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{\...
user avatar
  • 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( \...
user avatar
  • 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}...
user avatar
  • 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, $...
user avatar
  • 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 $s>1$, ...
user avatar
  • 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 ...
user avatar
  • 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)\...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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}...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 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\,\,\...
user avatar
  • 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/...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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.
user avatar
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)\...
user avatar
  • 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 ...
user avatar
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 \...
user avatar
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+\...
user avatar
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_{\...
user avatar
  • 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 ...
user avatar
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}...
user avatar
  • 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?
user avatar
  • 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)}\...
user avatar
  • 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}{...
user avatar
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}^\...
user avatar
  • 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, ...
user avatar
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}{\...
user avatar
  • 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^{...
user avatar
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 ...
user avatar
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{...
user avatar
  • 2,791