Articles of stieltjes constants

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 compute simpler, related series: $$\lim_{k\to \infty} \sum_{n=1}^{k} \left (\psi'(n) -1/n \right) =1 $$ $$\sum_{n=1}^{\infty} \psi”(n) […]

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 from $n=1$ to $x$, when $x$ tends to infinity, is asymptotic to […]

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. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = \dfrac{5}{3}\zeta(3)-\dfrac{2}{3}\gamma^3-2\gamma \gamma_{1}-\gamma_{2} $$ where \begin{align} & H_{n}: =\sum_{k=1}^{n}\frac{1}{k} \\ &\gamma: =\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{1}{k}-\ln n\right) \\ & \gamma_{1}:=\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{\ln k}{k}-\frac{1}{2}\ln^2 n\right)\\& \gamma_{2}: =\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{\ln^2 k}{k}-\frac{1}{3}\ln^3 n\right), \end{align} $\gamma_1, \gamma_2$ being Stieltjes constants. […]

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 series test. There’s a hint: refer to an equation from a previous section on the integral test. […]

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 constant $\gamma_1$ is given (on page 539): $$\gamma_1=-\left[\gamma-\frac{\ln2}2\right]\ln2+i\int_0^\infty\frac{dx}{e^{\pi x}+1}\left\{\frac{\ln(1-ix)}{1-ix}-\frac{\ln(1+ix)}{1+ix}\right\}.\tag1$$ It’s possible to get rid of imaginary numbers in this formula, […]

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler’s constant has the following representations (Euler-Mascheroni constant expression, further simplification,, Question on Macys formula for Euler-Mascheroni Constant $\gamma$) $$ \gamma= \lim_{n \to \infty} {\left(2H_n-H_{n^2} \right)}=\sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=(n-1)^2+1}^{n^2} \frac{1}{j}\right) $$ $$ \gamma= \lim_{n \to \infty} {\left(2H_n-H_{n(n+1)} \right)}=\sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right) $$ $$ \gamma= \lim_{n \to \infty} {\left(2H_n-\frac{1}{6}H_{n^2+n-1}-\frac{5}{6}H_{n^2+n}\right)}$$$$=\frac{7}{12}+\sum_{n=1}^{\infty}\left(\frac{2}{n+1}-\frac{1}{3n(n+1)(n+2)}-\sum_{k=n(n+1)+1}^{(n+1)(n+2)}\frac{1}{k}\right) $$ Hardy (1912) and Kluyver (1927) derived formulas for […]