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.
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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(...
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Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$ [closed]
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.
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Abel Summation for d(n)/n
If $d(n) = \sum_{d|n}1$ is the divisor function, it is a fact that $d$ has summatory behaviour
\begin{equation}
\sum_{n \leq x} d(n) = x \log x + (2 \gamma - 1)x + O(\sqrt{x}).
\end{equation}
I'm ...
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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 ...
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$\gamma_0$ in terms of $\gamma_{1-Inf}$
I’ve numerically observed that $\sum_{n=1}^{\infty}\gamma_n/n!$ = $1/2 - \gamma_0$ , where $\gamma$ are the Stieltjes constants.
Is there a recurrence explanation for this or a known proof?
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How does one arrive at the basic limit formulation for the Stieltjes constants?
The starting definition that I am using is:
$$\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty\gamma_n\cdot\frac{(-1)^n}{n!}(s-1)$$
If I naively differentiate, I find:
$$\zeta'(s)=\sum_{m=1}^\infty-\frac{\ln(...
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Proof of an integral representation of Stieltjes constants
Pre-requisite definition
Stieltjes constants can be thought of as a generalization of Euler-Mascheroni constants defined as,
\begin{align}
\gamma_n &= \lim_{m \rightarrow \infty} \left( \sum_{...
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A finite series expression for infinite sums of powers of non-trivial zeros?
This WolframMathworld-page, mentions:
$$Z(n) = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^n} + \frac{1}{(1-\rho_k)^n}\right) \quad n \in \mathbb{N}$$
where $\rho_k$ is the $k$-th non-trivial zero of ...
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Is $\zeta(x)>\frac{1}{x-1}$ when $1<x<2$?
I had originally found that $\lfloor\zeta(\zeta(n))\rfloor$, where $\zeta(n)$ is the Riemann Zeta Function, seemed to be relatively close to $\left\lfloor\frac{1}{\zeta(n)-1}\right\rceil$ for $n \in \...
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Showing $\zeta(s)-{1\over s-1}$ is analytic
It is well known that Euler-Mascheroni constant has an alternative definition in terms of zeta function:
$$
\gamma=\lim_{s\to1^+}f(s)\equiv\lim_{s\to1^+}\left[\zeta(s)-{1\over s-1}\right]
$$
Using ...
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Calculation of Integrals with reciproce Logarithm, Euler's constant $\gamma=0.577...$
Evaluate the improper integral $\int\limits_0^1\left(\frac1{\log x} + \frac1{1-x}\right)^2 dx = \log2\pi - \frac12 = 0.33787...$
With integration by parts we get from
$\int\limits_0^1\left(\frac1{\...
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Variant of Stieltjes constants
New Identities
For any positive integers $k\geq1$ and $j\geq2$, let $x_j=\frac{j+\sqrt{j^2-4}}{2}$. Let us define $(\text{A}_k)_{k\geq1}$ by the following constants "which are variants of Stieltjes ...
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Decreasing Combinations of Stieltjes Coefficients
I have noticed that if we take the Laurent expansion of the Riemann zeta function about $s=1$ $$
\zeta(s) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n (s-1)^n
$$
which defines $\...
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A closed form of the family of series $\sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}$ for $m\ge 1$
Introduction
Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum
$$s_m = \sum _{k=1}^{\infty } \frac{\...
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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 ...
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