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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

In mathematics, a Dirichlet series is any series of the form $$ \sum_{n=1}^{\infty} \frac{a_n}{n^s}, $$ where $s$ and $a_n$ are complex numbers and $n = 1, 2, 3, \dots$ . It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann $\zeta$ function is a Dirichlet series with $a_n=1$, as also are the Dirichlet $L$-functions.

It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet.

587 questions
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Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the Dirichlet Eta function. Could it be proved that…
Neves
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3 answers

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 real numbers), and assuming that $g(s)$ can be…
J. M. ain't a mathematician
  • 76,540
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6 answers

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - \ldots $$ Examining the pattern of the…
Gottfried Helms
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Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit $$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$ It is clear that the Dirichlet series $\sum_{n=1}^\infty\frac{\sin n}{n^s}$…
HGF
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4 answers

$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

In a paper about Prime Number Races, I found the following (page 14 and 19): This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln t}} - \# \{\text{primes}\le x\} } {\sqrt x/\ln x}…
draks ...
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Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches infinity. $\gamma$ is the Euler-Mascheroni constant,…
Mats Granvik
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Is series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?

I've done the following exercise: Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent? My approach: We're going to use the Dirichlet's criterion for convergence of series. Let $\displaystyle\…
Relure
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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 cosine infinite product $$\cos{(\pi x)}=\prod_{n\ge…
Zacky
  • 30,116
15
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1 answer

An entire function interpolating $\mu(n)$

This is in order to repair the pdf and answers of this user. $$f(x)=2\sum_{k\ge 0}\frac{x^{2k+1}}{\zeta(2k+2)}=2x\sum_{n\ge 1} \frac{\mu(n)/n^2}{1-x^2/n^2}, \qquad |x|<1$$ The RHS extends meromorphically to $\Bbb{C}$ and we have $-\mu(n) = \lim_{x…
reuns
  • 79,880
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2 answers

The Definite Integral Problem (with a twist)?

The Definite Integral Problem (with a twist) In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$. Here's something I asked myself related to the Riemann…
More Anonymous
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13
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3 answers

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ \beta(x)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{(2k-1)^{x}} $$ is…
Neves
  • 5,701
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2 answers

Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are equally spaced along lines of fixed real component.…
graveolensa
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1 answer

Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$

The paper mentioned a proposition: $$ \int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }. $$ Its equivalent…
Setness Ramesory
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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)\;\;\text{ converges for } \Re(s)>1 \\ &\color{red}{\text{ and…
reuns
  • 79,880
12
votes
1 answer

An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$

Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$ $$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{2332\,\pi ^6}{2679 075}\,4^{-3…
Vladimir Reshetnikov
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