# 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 ...
1 vote
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### Dirichlet series and Euler product

For a multiplicative function $f$, show that we have $$\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).$$ My ...
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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. ...
1 vote
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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 ...
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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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### 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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### 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 ...
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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 ...
1 vote
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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/...
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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 ...
1 vote
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### 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 ...
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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.
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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?
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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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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}{...