Questions tagged [dirichlet-series]
For questions on Dirichlet series.
184
questions with no upvoted or accepted answers
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Odd values for Dirichlet beta function
I would like to find a proof for the generating formula for odd values of Dirichlet beta function, namely: $$\beta(2k+1)=\frac{(-1)^kE_{2k}\pi^{2k+1}}{4^{k+1}(2k)!}$$
My try was to start with the ...
- 28.1k
12
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0
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473
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How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
- 78.4k
9
votes
0
answers
1k
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How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\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
8
votes
0
answers
683
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Eisenstein series twisted by a Dirichlet character
On page 17 of https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf, we see a remark where the author mentioned that If $\chi$ is a non-trivial Dirichlet character ...
- 458
8
votes
0
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323
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The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$
With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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7
votes
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Dirichlet series, abscissa of absolute convergence $\neq$ abscissa of uniform convergence
It is well known that Dirichlet series, series of the form $$\sum_{n=1}^{\infty}\dfrac{a_n}{n^s},$$
where $\{a_n\}$ is a complex sequence and s is a complex variable, converge in half planes. The ...
- 659
6
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194
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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 $$\sum_{n\le x}\...
- 1,145
6
votes
0
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198
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What do we know about the analytic continuations of Dirichlet series?
Let $s=\sigma+it$ be a complex number and define the function:
$$F(s)=\sum_{k=2}^{\infty}\frac{p_\pi(k)}{k^s}$$
Where $p_\pi(k)$ is the number of unordered factorizations of $k$, corresponding to OEIS ...
6
votes
0
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86
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Question on the distribution of the values of $f(x)=\sum\limits_{n=1}^x a(n)$ where $a(n)=\sum\limits_{d|n}\mu(d)\ \mu\left(\frac{n}{d}\right)$
Consider the function $a(n)$ defined in formula (1) below and it's summatory function $f(x)$ defined in formula (2) below where $f(x)$ is related to the Riemann zeta function $\zeta(s)$ as illustrated ...
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6
votes
0
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202
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Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$
This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$.
(1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$
(2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
- 7,631
5
votes
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101
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Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
5
votes
0
answers
327
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Convergence of Euler product implies convergence of Dirichlet series?
(Crossposted to Math Overflow) Suppose we have an Euler product over the primes
$$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$
where each $a_p \in \mathbb{C}$. The Euler product is ...
- 5,830
5
votes
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570
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Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form
Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
4
votes
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Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,810
4
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What is the explicit formula for $\Phi(x)=\sum\limits_{n=1}^x\phi(n)$?
I ran across the following claimed explicit formula for $\Phi(x)$.
(1) $\quad\Phi(x)=\sum\limits_{n=1}^x\phi(n)$
(2) $\quad \frac{\zeta(s-1)}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\phi(n)}{n^s}$
(...
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