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A logarithmic analog of the Leibniz series, and some integrals associated with the Riemann zeta-function. (Russian. English summary) Zbl 1028.11049
The Leibniz series is given by $$\sum^\infty_{j=2}(-1)^j\frac{\ln(j)}{j}$$ and its value $$C \ln 2-\frac{\ln^2(2)}{2}$$, $$C$$ being Euler’s constant, is evaluated by using the integral $J=\int\limits_0^{\infty} \frac{\ln (x) dx}{1+\text{exp} (x)}= - \frac{\ln^2 (2)}{2}.$ The proof is based on the decomposition of the Riemann zeta-function $$\zeta(s)$$ into powers of $$s$$ and the assumption that the values $$\zeta(0)$$ and $$\zeta'(0)$$ are known. The main purpose of the paper under review is to provide a direct method to calculate the integral $$J$$. In particular, the introduced method is not based on properties of the zeta-function and allows for the calculation of several related integrals.
##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
logarithmic Leibniz series; Riemann zeta-function