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

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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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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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Who extended the Euler Product Formula to all real $s>1$?

I believe Euler discovered this identity but only wrote them for particular values of $s$, then Chebychev extended to real $s>1$. However, I read in the book Riemann's Zeta Function, H.M. Edwards ...
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53 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 ...
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57 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} ...
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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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52 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 ...
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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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43 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 ...
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29 views

Clarify mertens' theorem?

If merten's theorem states this http://mathworld.wolfram.com/MertensTheorem.html (equation on the second line) specifically, then what is the error described as for finite n?
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41 views

Is this a finite product describing the partial harmonic series sums?

http://mathworld.wolfram.com/EulerProduct.html In the second last line, it gives a product P(n). Is this supposed to be describing the finite terms of the harmonic series sum? I don't see how it ...
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272 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 ...
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1answer
35 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 ...
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222 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 ...
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69 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 ...
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1answer
58 views

Euler product question

im doing a small research and i want to know if $$P = 2 \times 3 \times 5 \times 7 \times 11 \times \cdots \times P_n$$ can be obtained using the euler product (since $s = 1$ on euler product gives ...
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Effective bounds on L(1,Chi) for Chi a Dirichlet Character

I have $\chi$, a Dirichlet Character $\bmod n$, and I have established that $L(1,\chi) \geq C / \log \log n$ for some constant C under the generalized Riemann Assumption. I'll call this proposition ...
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55 views

Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do ...
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91 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
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65 views

A question about the convergence of partial products of zeta of one.

Recently I've been toying around with the Totient function and the Prime Number Theorem and came up with the odd result that the following limit $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
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3answers
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Landau's proof that $\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}$

In the Handbuch, Landau proves that for all $s>1$ the following equality holds $$\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}.$$ I'm having trouble with the following part of his proof: Landau says that ...
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1answer
42 views

Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to Of course these formulas give you the same value when they reach infinity, but my question is that say $s=1$. What would be the ...
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161 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
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113 views

Euler product of Dirichlet series

Let $f$ be an arithmetic function such $f(n_1n_2)=f(n_1)f(n_2)$ for all $n_1,n_2 \in \mathbb{N}$ with $\gcd(n_1,n_2)=1$. Suppose we know that the Dirichlet series $$F(s) = \sum_{n=1}^{\infty}f ...
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98 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
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102 views

Euler product for Riemann zeta and analytic continuation

the Euler product for the Riemann zeta $$ \zeta (s)= \prod _{p}\left( \sum_{n=0}^{\infty}p^{-ns}\right) $$ this is only valid for $ \Re(s) >1 $ however we could use the Borel transform so $$ ...
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Rate of growth of an Euler Product

Merten's Theorem gives $$ \prod_{p < x} \left( 1 - \frac{1}{p} \right) \sim k(\log x)^{-1} $$ I also know that $$ \prod_{p < x} \left(1 - \frac{n}{p} \right) \leq \prod_{p < x} \left(1 - ...
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Conjectures about zeta functions and poles

Let $p^*_n$ be the $n$ th element of a subset of primes such that $p^*_{n+1}>p^*_n$ and $p^*_n < O((n+2) ln((n+2))^3)$. Define $f(z)$ as the analytic continuation of $\prod_{n>0} ...
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Reduce formula using Euler's?

I am performing a self-study, and I am lost as to a derivation that has taken place. I basically started with this equation: $$ \Upsilon(\phi) = e^{-j\frac{N-1}{2}\phi} \ \Big[ \frac{1 - e^{j N ...
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Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
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Merthen's third theorem and uncertainty of prime hits

Conjecture(1) Merten's third theorem says: $$\lim_{L\to\infty}\ln L\prod_{p\le L}\left(1-\frac1p\right)=e^{-\gamma}$$ we have a wild discussion here around the table whether it is possible to ...
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The partial sum and partial product of $\zeta$function

Taking the partial sum of the $\zeta $ function: $$\zeta^H(s,k)=\sum_{n=0}^k \frac{1}{n^s}$$ and the partial product f the $\zeta $ function: $$\zeta^P(s,j)=\prod_{i=0}^j \frac{1}{1-p_i^{-s}}$$ I ...
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Representing $\prod_{k=2}^{\infty}\left(1+\frac{1}{k^{2}-1}\right)$ as an Euler's product.

The above formula corresponds to the following identity $$ 2=\prod_{k=2}^{\infty}\left(1+\frac{1}{k^{2}-1}\right) $$ I wonder if this can be represented as an Euler's product. Could anyone find ...
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638 views

The meaning of the Euler Formula for Zeta

Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes? I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
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489 views

Euler's proof for the infinitude of the primes

I am trying to recast the proof of Euler for the infinitude of the primes in modern mathematical language, but am not sure how it is to be done. The statement is that: $$\prod_{p\in P} ...
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The sum of the reciprocals of fourth powers [duplicate]

This problem is an extension of the well known basel problem and involves finding the sum of 1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity Euler managed to prove ...
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Proving $ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$

Suppose $m,n \in \Bbb N$, $k$ is product of all prime number such that divide $m,n$ How to prove that: $$ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$$
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Why do authors claim that Euler gave no proof to his “$\sin(\pi x)= \pi x\prod\limits_{k=1}^{\infty}\left(1-\frac{x^2}{k^2} \right )$” when…

When he proved the relation between $\pi \cot(\pi x)$ and the harmonic series in "Introductio in analysin infinitorum" which states that $$\pi \cot(\pi x)=\sum_{k \to \infty}^{\infty} ...
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66 views

Convergence of 'imaginary' Euler product

Take the following 'tweaked' Euler product ($i = \text{the imaginary unit})$: $$Eul_i(s) := \prod_{p\in\mathbb{P}} = \dfrac{1}{1-\dfrac{1}{(ip)^s}}$$ It is not difficult to see that for $s=2k+1, ...
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Using Eulers equation to find General Solutions

I have the problem $y''+4y = 0$ where $y(0) = 1$ , and $y'(0) = 0$ I have to find a particular solution, making a guess it is in the form $y(x) = e^{rx}$ I have the solution here, but I cannot quite ...
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Euler product on the critical line

can the Euler product be defined on the critical line $ 1/2+it $ ?? i mean the Euler product for the Riemann Zeta function $$ \zeta(s)= \prod _{n}(1-p^{-s})^{-1} $$ for example from the ...
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254 views

Proof of discovering two large prime numbers in polynomial time

$N=p*q$ is a product of two distinct primes. Show that if $\phi(N)$ and 2N are known, then it is possible to compute p and q in polynomial time. so, I know that $\phi(N)=(p-1)(q-1)$ Given this, if ...
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Multiplicative Dirichlet Series

How do I go about proving that where $f(k)$ is multiplicative that: $$\sum_{k\geq 1}{\frac{f({k})}{k^{s}}}=\prod_{p}\sum_{k\geq 0}{\frac{f(p^{k})}{p^{ks}}}$$ I first tried to use the fundamental ...
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187 views

Euler product of Dirichlet Series

For $n$ a positive integer, let $f(n)$ be the squarefree part of $n$. Find the Euler product for $\mathfrak D_{f}(s)$ where $\mathfrak D_{f}(s)$ is the Dirichlet Series of $f$.
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Evaluating a series with the Möbius function and greatest common divisor.

Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series ...