Radu-Octavian Vilceanu: An application of Dirichlet L-series to the computation of certain integrals, p.159-173

Abstract:

In this article we compute integrals of the form

$$\int_{0}^{1}\frac{\sum_{n=1}^{{q}-1}\chi_{-{q}}\left( n\right) x^{n-1}% }{1-x^{{q}}}\ln\left( \ln\frac{1}{x}\right) dx,$$

where $\chi_{{q}}$ is the odd Dirichlet character (mod $q$). The argument is based on the theory of L-series due to Dirichlet. Our formulas were initially found via the PSLQ algorithm of H.R.P. Ferguson.

Key Words: Dirichlet character, L-series, PSLQ algorithm.

2000 Mathematics Subject Classification: Primary: 28-04,
Secondary: 11M35.

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