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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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....
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0answers
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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{...
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1answer
67 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 ...
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1answer
42 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 ...
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1answer
25 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 ...
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0answers
41 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}$$\...
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2answers
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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}+\...
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0answers
52 views

Is there an Euler product for the Prime Zeta function?

The Riemann Zeta function, denoted by $\zeta$ is defined for all positive integers $n>1$ by $$ \zeta(n)=\sum_{k=1}^\infty k^{-n}. $$ It is also defined by its Euler product, running over all ...
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0answers
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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 ...
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1answer
126 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. ...
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1answer
39 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 ...
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1answer
79 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-\...
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1answer
47 views

Why is $\prod\limits_{p\le Y}\left(1+\frac{1+2e}{p}\right)\le(\log Y)^{10}$

Why is $\prod\limits_{p\le Y}\left(1+\frac{1+2e}{p}\right)\le(\log Y)^{10}$ Can I bring $(1+2e)$ to the exponent and write $\prod\limits_{p\le Y}\left(1+\frac{1}{p}\right)^{1+2e}$ which is lesser ...
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2answers
122 views

Why $\sum\frac{\mu(h)\mu(k)}{hk}\gcd(h,k)=\prod\limits_{p\le x}\left(1-\frac1p\right)$, where the sum enumerates the pairs $(h,k)$ of primes below $x$

Why is the following equality true ? $$\sum\limits_{h,k\atop p|hk\implies p\le X}\frac{\mu(h)\mu(k)}{h\cdot k}(h,k)=\prod\limits_{p\le X}\left(1-\frac1p\right)$$ The notation $p|hk\implies p\le X$ ...
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0answers
32 views

Approximation for $\prod\limits_{r<p\le P}\left( 1-\frac rp\right)^{-1}$

Is there an approximation for $\prod\limits_{r<p\le P}\left( 1-\frac rp\right)^{-1}$ when $r$ is fix and greater than $1$ and $p$ is prime For example if $r=1$ then the above is approximately $\...
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0answers
26 views

Why is the following estimation true?

Why is the following estimation true ? $\displaystyle\sum\limits_{h_1,k_1\atop p|h_1k_1\implies p\le Y}\frac{e^{\Omega(h_1)}}{lcm(h_1,k_1)}\ll\prod\limits_{p\le Y}\left(1+\frac{1+2e}{p}\right)$ ...
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0answers
37 views

Which sequences of integers admit an Euler Product?

Suppose that $A = \{a_n\}$ is an increasing sequence of positive integers and let $\zeta_A(s)=\sum_n \frac{1}{{a_n}^s}$. For which sequences does $\zeta_A (s)=\prod_{p_{n_k}}(1-{p_{n_k}}^{-s})^{-1}$...
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0answers
39 views

Closed forms at integer values for Euler products with Dirichlet $\chi_{4}(p^s)$. Could these be extended towards non-integer values?

I was experimenting with the following products over all primes: $$\prod_p \left(\frac{p^s}{{p^s-\sin\left(\dfrac{p^s \,\pi}{2}\right)}} \right)\,\cdot\,\prod_p \left(\frac{p^s+\sin\left(\dfrac{p^s \...
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1answer
91 views

Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $

I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product: $$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 = q \prod_{n=1}^\infty (1 - q^n)...
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0answers
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Does the Euler product stand for $a(n)=rad(n)$?

Does the Euler product stand for $a(n)=rad(n)$? Or more generally, for multiplicative functions which are not completely multiplicative? Where rad is the product of a number's distinct prime factors....
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2answers
213 views

How can the Zeta function be zero?

How can the Zeta function be zero? If the zeta function is the Euler product: $$\zeta(s)=\prod_p \frac{1}{1-p^{-s}}$$ Then being a product my first thought was that it could only be zero if one or ...
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1answer
79 views

Euler Product involving Möbius and Euler Functions

I am attempting to follow this line in a proof: $\frac{n}{\phi(n)}=\prod_{p|n}(1-\frac{1}{p})^{-1}=\prod_{p|n}(1+\frac{1}{p-1})=\sum_{d\delta=n}\frac{\mu^{2}(d)}{\phi(d)}.$ I follow the logic up to ...
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5answers
343 views

Find all $x$ such that $x^{35} + 5x^{19} + 11x^3$ is divisible by 17.

Find all $x$ such that $x^{35} + 5x^{19} + 11x^3$ is divisible by 17. I think we can use the fact that we can mod everything by 17 and want 0. But how exactly should we go about doing this?
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152 views

Convergence of Euler Product for Leibniz Pi Formula

In class we were shown a derivation of Leibniz's formula for pi: $$\frac{\pi}{4}=\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}$$ We can rewrite this formula using the following function on $\mathbb{N}$: $$\...
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0answers
148 views

Zeros of Riemann Zeta Function- Euler Product and Functional Equation

The problem statement, all variables and given/known data Question Use the functional equation to show that for : a) $k \in Z^+ $ that $ \zeta (-2k)=0$ b) Use the functional equation and the euler ...
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1answer
168 views

Riemann Zeta - Euler product convergence?

I'm fascinated by the Riemann Zeta hypothesis - I haven't yet taken a course in complex analysis but I'm curious what this sentence means on Wikipedia: The convergence of the Euler product shows ...
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1answer
53 views

Feller-Torier constant equals this sum involving the prime zeta function?

According to the Wikipedia and Mathworld pages, the Feller-Tornier constant can be defined by this sum involving the prime zeta function $$C_{\text{FT}}=\frac{1}{2}\left(1+\exp\left[-\sum_{n=1}^\infty ...
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Slightly unusual proof for divergence of the series of reciprocals of primes (need verification)

We know that : $ \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p} \frac{1}{1 - p^{-s}} $ From now on, assume the sum over $ n $ means $1$ to infinity. And assume that the sum or product over p means over ...
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2answers
155 views

Infinite Product of 1-1/n^4 [duplicate]

When I look at wolframalpha, I get $$\prod_{n=2}^\infty \left(1-\frac{1}{n^4}\right) = \frac{\sinh(4\pi)}{4\pi}.$$ My only guess where this comes from would be the euler's sine product formula$$\sin(...
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0answers
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Small question about convergence for Euler product of zeta-function

Can anyone offer some clarification on the following equivalence? Perhaps as happens too often, I was told the statement follows from 'basic analysis', and while this is probably true, I can't seem to ...
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0answers
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Show that coefficients of $\eta(z)^2 \eta(11z)^2$ are multiplicative

Can someone help me find the Euler product associated to this newform? I have a product of eta functions: $$ \eta(z)^2 \eta(11z)^2 = q \prod_{n=1}^\infty (1-q^n)^2 (1 - q^{11n})^2 = q - 2q^2 - q^3 +...
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0answers
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Dirichlet L-function has no zeroes in Re(s) > c implies Euler product converges in Re(s) > c?

If a Dirichlet $L$-function has no zeroes in $\Re(s) \gt c$, does its Euler product necessarily converge in $\Re(s) \gt c$? So I know the proof that (conditional) convergence of the Euler product $\...
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0answers
73 views

Multiplicative coefficients of Selberg functions

Let $F(s)$ be a Selberg function, $F(s)=\sum_{n=1}^\infty\frac{a(n)}{n^{s}}$ and $\log F(s)=\sum_{n=2}^\infty\frac{b(n)\Lambda(n)}{\log n}\frac{1}{n^s}$ where $b(n)$ is zero unless $n$ is a positive ...
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337 views

Evaluating $\prod_p(1\pm4/p^2)$ in Closed Form

Do either of the two infinite products $~\displaystyle\prod_{p~\in~\mathbb P}\bigg(1+\frac{2^2}{p^2}\bigg)~$ and $~\displaystyle\prod_{p~\in~\mathbb P}^{p~>~2}\bigg(1-\frac{2^2}{p^2}\bigg)~$ ...
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1answer
94 views

Evaluating $\prod\limits_{n=1}^{\infty}(1+e^{-x2^n})$

In this page, it is stated that $$ \prod\limits_{n=1}^{\infty}\left(1+e^{-x2^n}\right) = \frac{1}{2}(1+\coth(x)) $$ How can one show this? In the webpage it is under the title "Euler's product", which ...
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Values where infinite products of primes and composites are equal in arithmetic progressions under the Dirichlet theorem.

For $\Re(s) > 1$, the following product is valid (see also this question): $$\displaystyle \prod _{n=2}^{\infty } \left( \dfrac{1}{1-\frac{1}{n^{s}}} \right)=\prod _{primes}^{\infty } \left( \...
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1answer
199 views

Intuitive proof for the Euler products formula

I have memorized the Euler product formula but don't actually understand the proof of it given in the Wikipedia and in several books. The formula I am referring to is $$\varphi(n) = n\prod_{p\mid n}\...
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3answers
259 views

Prove that the product $\prod_{k= 2}^n(1-k^{-3})$ is always greater than $1/2$.

Prove that $$\prod_{k=2}^{n}\left(1-\frac{1}{k^{3}}\right)\geq \frac{1}{2},\,\forall n\geq2.$$ Intuitively it looks very true and this looks very similar to the reciprocal of the Euler product. I ...
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2answers
433 views

Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...
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1answer
56 views

Find the value of the Infinite product in terms of k which is a positive integer

$$\prod_{n=k+1}^{+\infty}\left(1-\frac{k^2}{n^2}\right)$$ The only help we have been able to find is that of Euler, anything would be amazing!
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1answer
484 views

density of squarefree numbers in $\mathbb{Z}$ that are 1 mod 4

It's common exercise to show the "density" of square-free numbers in $\mathbb{Z}$ is $\frac{6}{\pi^2}$ which we could say is $6 \times \frac{1}{3^2} = \frac{2}{3}$ or possibly $ 6 \times (\frac{7}{22})...
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1answer
192 views

Euler product for sum of multiplicative function times log

Let $g$ be a multiplicative function. Iwaniec and Fouvry claim the following identity on p. 273, identity (7.19). Why is this Euler product identity true? $$-\sum_n \mu(n)g(n)\log n = \prod_{p} (1-g(...
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0answers
47 views

Can't find the error with the following sum involving multiplicative functions

I apologize for this question which might be trivial but I'm stuck with this issue and I'm probably doing some mistake over and over again. Here's the framework. Let $f$ and $h$ be two multiplicative ...
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1answer
81 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer $...
3
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0answers
100 views

Why do you need to prove the error term goes to zero for the complete derivation of the Euler Product Formula?

I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I ...
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0answers
67 views

Conditions of Euler Product

We know that if the infinite sum of a multiplicative function is absolute convergent, then the sum can be expressed as infinite product and the infinite product is absolutely convergent. Does there ...
8
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1answer
561 views

Surprising behavior of Leibniz formula for Pi (as Euler product)

I wrote a program to compute successive approximations of Pi using the following Euler product: π/4 = (3/4)*(5/4)*(7/8)*(11/12)*(13/12)... in which the ...
0
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1answer
101 views

Estimating size of partial euler product

What estimates are there for product over primes $p \leq x$ $\prod_{p \leq x}(1-\frac{1}{p^{r}})$ given $r$ is positive integer. Something better than $\prod_{p \leq x}(1-\frac{1}{p^{r}}) \leq 1-\...
5
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1answer
453 views

Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?

The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as: $$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$ It has the following Euler product (I used that Dirichlet character $\chi_{4}(...
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1answer
91 views

Euler product of $\sum (2^k n + 1)^{-s}$

do we know for a given $k > 2$ the Euler product of $\ \displaystyle\sum_{n=0}^\infty (2^k n + 1)^{\textstyle-s} \ $ ? I saw that every prime numbers will appear in it, as well as some non-prime ...