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

2012/01/16 by Mark W. Coffey, Coffey, Mark W. · 2 citations
Mathematics · #Mathematical functions and polynomials #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1201.3393

Abstract

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

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