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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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The Minkowski question mark function: explicit series for the dyadic period function and moments
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by Giedrius Alkauskas PDF
Math. Comp. 79 (2010), 383-418 Request permission

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


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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Additional Information
  • Giedrius Alkauskas
  • Affiliation: The Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, Lithuania and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • Email:
  • Received by editor(s): September 15, 2008
  • Received by editor(s) in revised form: January 17, 2009
  • Published electronically: May 12, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 79 (2010), 383-418
  • MSC (2000): Primary 11A55, 26A30, 32A05; Secondary 40A15, 37F50, 11F37
  • DOI:
  • MathSciNet review: 2552232