Certain logarithmic integrals, including solution of Monthly problem #tbd, zeta values, and expressions for the Stieltjes constants

Abstract

We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values of these zeta functions at integer argument. Other summations with the $p_n$ coefficients are presented.

32 pages, no figures

References in corpus (3)

• Series representations for the Stieltjes constants
• A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications
• Sums of alternating products of Riemann zeta values and solution of a Monthly problem

Cited by in corpus (5)

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• On an integral involving the digamma function
• Functional equations for the Stieltjes constants
• A note on some constants related to the zeta-function and their relationship with the Gregory coefficients
• Note on the Stieltjes constants: series with Stirling numbers of the first kind