# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## The Minkowski question mark function: explicit series for the dyadic period function and momentsHTML articles powered by AMS MathViewer

by Giedrius Alkauskas
Math. Comp. 79 (2010), 383-418 Request permission

Corrigendum: Math. Comp. 80 (2011), 2445-2454.

## Abstract:

Previously, several natural integral transforms of the Minkowski question mark function \$F(x)\$ were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about \$F(x)\$. One of them, the dyadic period function \$G(z)\$, was defined as a Stieltjes transform. In this paper we introduce a family of “distributions” \$F_{\textsf {p}}(x)\$ for \$\Re \sf p\geq 1\$, such that \$F_{1}(x)\$ is the question mark function and \$F_{2}(x)\$ is a discrete distribution with support on \$x=1\$. We prove that the generating function of moments of \$F_\textsf {p}(x)\$ satisfies the three-term functional equation. This has an independent interest, though our main concern is the information it provides about \$F(x)\$. This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.
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