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Questions tagged [stieltjes-constants]

The Stieltjes constants appear in the Laurent series for the Riemann zeta function. They are a generalization of the Euler-Mascheroni constant.

The Stieltjes constants appear in the Laurent series for the Riemann zeta function. They are a generalization of the Euler-Mascheroni constant.

See the Wikipedia page for the Stieltjes constants for more details.

25 questions
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A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form of the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = \dfrac{5}{3}\zeta(3)-\dfrac{2}{3}\gamma^3-2\gamma \gamma_{1}-\gamma_{2} $$…
Olivier Oloa
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27
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8 answers

The sum of $(-1)^n \frac{\ln n}{n}$

I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$ This is a problem in Calculus by Simmons. It's in the end of chapter review and it's associated with the section about the alternating…
stuart
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Question on the paper Donal F. Connon, "Some integrals involving the Stieltjes constants"

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series expansion of the Hurwitz zeta function (formula $(2.3)$…
Vladimir Reshetnikov
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The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that the sum of $d(n)$ over all positive integers…
Anatoly
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4 answers

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion for this partial sum is $$ \displaystyle…
Anatoly
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12
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2 answers

A couple of definite integrals related to Stieltjes constants

In a (great) paper "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations" by Iaroslav V. Blagouchine, the following integral representation of the first Stieltjes…
Vladimir Reshetnikov
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3 answers

A closed form of the family of series $\sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}$ for $m\ge 1$

Introduction Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum $$s_m = \sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma…
Dr. Wolfgang Hintze
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Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) - (\ln k)^2/2$ as $k \to \infty$

I would like to obtain a closed form for the following limit: $$I_2=\lim_{k\to \infty} \left ( - (\ln k)^2/2 +\sum_{n=1}^{k} \psi(n) \, \psi'(n) \right)$$ Here $\psi(n)$ is digamma function. Using the method detailed in this answer, I was able to…
Slava Kashcheyevs
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What is the sign of the generalized Stieltjes constants $\gamma_{k}(a)$?

Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$: $$ \begin{align} \zeta(s) = \frac{1}{s-1}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}…
Olivier Oloa
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Regularization involving Stieltjes constants: $\displaystyle\sum_{k=1}^{\infty}\frac{\ln(k)^n}{k}\overset{\mathcal{R}}{=}\gamma_n$

Notation $\zeta(z)$ is the Riemann zeta function $\operatorname{Li}_{\nu}(z)$ is the polylogarithm function $\operatorname{Li}^{(n,0)}_{\nu}(z):=\frac{\partial^n}{\partial\nu^n}\operatorname{Li}_\nu(z)$ $\gamma_n$ is the $n$-th Stieltjes…
Math Attack
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Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$

It seems that the above identity is true. Can this be proven? Or are there references treating sums like the right hand side? The above constants, $\gamma_{n}$, are the Stieltjes constants. Thanks.
Neves
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Where does Laurent series for zeta function converge?

I am wondering in what region of the complex plane the Laurent series $$ \zeta(s)=\frac{1}{s-1} + \sum_{k=0}^{\infty} \frac{(-1)^k \gamma_k}{k!} (s-1)^k $$ converges. It is straight forward to derive this series formally from…
Dave
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2 answers

A finite series expression for infinite sums of powers of non-trivial zeros?

This WolframMathworld-page, mentions: $$Z(n) = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^n} + \frac{1}{(1-\rho_k)^n}\right) \quad n \in \mathbb{N}$$ where $\rho_k$ is the $k$-th non-trivial zero of the Riemann $\zeta$-function. The page also lists…
Agno
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Showing $\zeta(s)-{1\over s-1}$ is analytic

It is well known that Euler-Mascheroni constant has an alternative definition in terms of zeta function: $$ \gamma=\lim_{s\to1^+}f(s)\equiv\lim_{s\to1^+}\left[\zeta(s)-{1\over s-1}\right] $$ Using this definition, I would like to derive the Laurent…
TravorLZH
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Calculation of Integrals with reciproce Logarithm, Euler's constant $\gamma=0.577...$

Evaluate the improper integral $\int\limits_0^1\left(\frac1{\log x} + \frac1{1-x}\right)^2 dx = \log2\pi - \frac12 = 0.33787...$ With integration by parts we get from $\int\limits_0^1\left(\frac1{\log x} + \frac1{1-x}\right) dx = \gamma$ the similar…
skraemer
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