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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Question about a limit of the euler-product of the riemann zeta function
The limit $$\lim_{s\to 1} (s-1)\zeta(s)=1=\lim_{s\to 1}(s-1)\prod_{p\in\mathbb P} (1-p^{-s})^{-1}$$
is well-known.
Consider that there are infinitely many distinct subsets $\mathbb P_{k}\subsetneq\...
- 67
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Meromorphic continuation of Euler product
Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$
Longer version: I realise I have some misconceptions about ...
- 1,662
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Euler product proof by the fundamental theorem of arithmetic
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.”...
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Dirichlet series and Euler product
For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation}
My ...
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Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski
I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski.
They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
- 1,145
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Derivatives of Euler products
Suppose I have an Euler product absolutely convergent for $\sigma >1$ $$\mathcal Z_\mathbf z(s)=\prod _p\left (1-\frac {1}{2p^{s}}\left (\frac {1}{p^z}+\frac {1}{p^{z'}}\right )\right );$$ note $$\...
- 1,662
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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 ...
- 16.4k
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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. ...
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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 $1$. ...
- 152k
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Does the Euler product for the Rankin-Selberg convolution of two eigenforms require the Ramanujan conjecture?
Suppose $f$ and $g$ are weight $k$ eigenforms for the Hecke operators, normalized, with Fourier coefficients $a_{n}$ and $b_{n}$ respectively. I wanted to see if I could derive a Euler product for
$$L(...
user427380
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How to prove that $\sqrt{3}\pi/6=\prod_{p \equiv 1 \pmod{6}} \frac{p}{p-1}\prod_{p \equiv 5 \pmod{6}} \frac{p}{p+1}$ with $p \in \mathbb{P}$?
I would like to prove the formula $$\frac{\sqrt{3}\pi}{6}=\left(\prod_{\substack{p \equiv 1 \pmod{6} \\ p \in \mathbb{P}}} \frac{p}{p-1}\right) \cdot \left(\prod_{\substack{p \equiv 5 \pmod{6} \\ p \...
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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. $
- 2,040
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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 ...
Kimdongyeob
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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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