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

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3
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1answer
33 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 $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
2
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3answers
68 views

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 ...
0
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0answers
19 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 ...
1
vote
1answer
75 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 ...
2
votes
1answer
55 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 ...
1
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0answers
40 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 $$ ...
1
vote
2answers
55 views

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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0answers
30 views

Question about the zeros and poles of the PrimeZeta function.

The Euler product over all primes, $$\displaystyle \zeta(s) := \prod_{p\in\mathbb{P}} \dfrac{1}{1-\dfrac{1}{(p)^s}}$$ is only valid for $\Re(s) >1$. However, when taking the log on both sides ...
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0answers
71 views

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} ...
4
votes
2answers
82 views

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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0answers
106 views

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!} + ...
2
votes
0answers
54 views

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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0answers
91 views

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 ...
5
votes
1answer
100 views

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 ...
4
votes
1answer
337 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 ...
7
votes
1answer
165 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} ...
1
vote
0answers
38 views

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 ...
5
votes
3answers
523 views

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: $$ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$$
4
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1answer
333 views

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} ...
0
votes
1answer
57 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, ...
0
votes
2answers
100 views

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 ...
2
votes
1answer
69 views

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 ...
1
vote
1answer
108 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 ...
1
vote
1answer
77 views

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 ...
2
votes
1answer
138 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 $f$.
6
votes
1answer
372 views

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 ...