Questions tagged [euler-product]

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

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Dirichlet series of Artin $L$-function

I was trying to show how Artin $L$-function attached to irreducible Galois representation $\rho : G \rightarrow GL_2(\mathbb{C})$ for Galois group $G$ can be expressed as Dirichlet series. One paper I ...
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Dirichlet series and Euler product

For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation} My ...
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Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski

I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski. They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
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Derivatives of Euler products

Suppose I have an Euler product absolutely convergent for $\sigma >1$ $$\mathcal Z_\mathbf z(s)=\prod _p\left (1-\frac {1}{2p^{s}}\left (\frac {1}{p^z}+\frac {1}{p^{z'}}\right )\right );$$ note $$\...
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Riemann hypothesis like conjecture for a non-UFD?

I got inspired by quadratic rings and zeta functions. I know for instance that the norm $a^2 + 17 b^2$ for the ring $\Bbb Z[\sqrt{-17}]$ is multiplicative yet the ring $\Bbb Z[\sqrt{-17}]$ is not a ...
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Zeta function for Powerful Polynomials over Finite Field

I am currently working on a problem that requires me to get find a simpler expresion for: $$\sum_{f \in \mathcal{S}_h} \frac{1}{|f|^s} $$ Where $\mathcal{S}_h$ is the set of $h$-full polynomials (i.e. ...
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How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.

Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
matt stokes's user avatar
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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 $1$. ...
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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(...
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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 \...
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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. $
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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 ...
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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 + ...
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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 ...
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Validity of proof $\zeta(s) \neq 0$ for $\sigma\gt1$, where $\sigma$ is the real part of $s$

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 ...
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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 $$ \frac {e ^ {1/2} \Big (1 + e ^ {1/3} \big (1 + e ^ {1/5} (1 + e ^ {1/7} (\dots (1 + e ^ {1 / p (n)})} {p(n)} \sim c $$ ...
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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 ...
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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 ...
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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{\...
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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( \...
Penelope's user avatar
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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}...
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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, $...
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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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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 ...
Penelope's user avatar
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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)\...
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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 ...
Penelope's user avatar
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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 ...
ashbre87's user avatar
1 vote
1 answer
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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}...
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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 ...
Eric Snyder's user avatar
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6 votes
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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 ...
Unit's user avatar
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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 ...
24th_moonshine's user avatar
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305 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 ...
Rivers McForge's 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 ...
Penelope's user avatar
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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\,\,\...
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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/...
Penelope's user avatar
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1 vote
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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 ...
EllaFaulk's user avatar
1 vote
2 answers
146 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 ...
EllaFaulk's user avatar
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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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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.
Augusto Santi's user avatar
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1 answer
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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)\...
Agno's user avatar
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5 votes
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Maximal extension of domain of $\Psi(x)=\sum_{n=1}^\infty e^{\frac{\log(n)}{\log(x)}}$ using analytic continuation

Given $$ \Psi(x)=\sum_{n=1}^\infty e^{\frac{\ln(n)}{\ln(x)}}= \prod_{p~ \mathrm{prime}}\frac{1}{1-e^{\frac{\ln(p)}{\ln(x)}}}$$ What is the maximal extension of $\Psi$? I think that there is a ...
John Zimmerman's user avatar
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1 answer
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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 ...
zeraoulia rafik's user avatar
1 vote
1 answer
105 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 \...
UnusualMathem's user avatar
4 votes
1 answer
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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+\...
UnusualMathem's user avatar
2 votes
1 answer
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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_{\...
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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 ...
zeraoulia rafik's user avatar
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1 answer
117 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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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?
David's user avatar
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3 votes
2 answers
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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)}\...
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2 votes
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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}{...
angelene's user avatar