Questions tagged [dirichlet-series]
For questions on Dirichlet series.
563
questions
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Approximate Riemann zeta function
Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.
In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.
My question is: Is there a Functional equation for ...
- 8,536
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votes
2
answers
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Convergence of sum in proof that $\Phi(s) - \frac{1}{s-1}$ extends to $\Re(s) > \frac{1}{2}$
Definitions: $\Phi(s) = \displaystyle\sum_{p} \frac{\log p}{p^s}$ where $p$ denotes a prime number.
$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^s}$ denotes the Riemann zeta function.
I ...
- 3,345
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How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form $$...
- 5,627
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votes
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An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
- 966
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answer
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Reference request: $L$-series and $\zeta$-functions
Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following:
...
- 3,783
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2
answers
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A question about an identity involving Dirichlet characters
Let $\chi$ be a Dirichlet character $\bmod q$. We have
$$\sum_{n=0}^{\infty} (-1)^{n-1} \chi(n) n^{-s}=\prod_p \left(1-(-1)^{p-1}\frac{\chi(p)}{p^s}\right)^{-1}=(1+\chi(2)2^{-s})^{-1}\prod_{p>2}\left(...
- 1,213
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2
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Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $...
- 5,627
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On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
