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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.
11M06 \(\zeta (s)\) and \(L(s, \chi)\)