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

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16 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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0answers
72 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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1answer
46 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
46 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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0answers
25 views

Deeply confused while trying to understand the derivation of infinite product representation of gamma function

I am trying to understand how in the world Euler figured out the infinite product representation of Gamma function. $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$ Of ...
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0answers
27 views

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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1answer
39 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 ...
2
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1answer
49 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, ...
3
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1answer
59 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
92 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 ...
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30 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 ...
3
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1answer
110 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 ...
3
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1answer
82 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 ...
3
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1answer
84 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 ...
2
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1answer
77 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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2answers
70 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
87 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
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2answers
99 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
120 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!} + ...
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0answers
55 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
138 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 ...
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1answer
116 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 ...
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1answer
464 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
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1answer
302 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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0answers
40 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
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3answers
837 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: $$ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$$
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1answer
380 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} ...
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1answer
62 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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2answers
108 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 ...
3
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1answer
81 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 ...
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1answer
172 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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1answer
88 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 ...
3
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1answer
165 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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1answer
425 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 ...