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2016 On the mean of the shifted error term in the theory of the Dirichlet divisor problem
Xiaodong Cao, Jun Furuya, Yoshio Tanigawa, Wenguang Zhai
Rocky Mountain J. Math. 46(1): 105-124 (2016). DOI: 10.1216/RMJ-2016-46-1-105

Abstract

Let $0\lt \alpha \lt 1$. We show that Bernoulli polynomials appear in the difference $\sum _{n \leq x}\Delta ^j(n+\alpha )-\int _1^x \Delta ^j(t)\,dt$ for $j=1, \ldots , 4$. As a corollary of this fact, we get better approximations of $\int _1^x \Delta ^j(t)\,dt$ by using zeros of Bernoulli polynomials. For $j=1,2$, we give some interpretation of this fact by means of Dirichlet series with the coefficients $\Delta ^j(n+\alpha )$.

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Xiaodong Cao. Jun Furuya. Yoshio Tanigawa. Wenguang Zhai. "On the mean of the shifted error term in the theory of the Dirichlet divisor problem." Rocky Mountain J. Math. 46 (1) 105 - 124, 2016. https://doi.org/10.1216/RMJ-2016-46-1-105

Information

Published: 2016
First available in Project Euclid: 23 May 2016

zbMATH: 1344.11058
MathSciNet: MR3506080
Digital Object Identifier: 10.1216/RMJ-2016-46-1-105

Subjects:
Primary: 11M41, 11N37

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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Vol.46 • No. 1 • 2016
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