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 $\Delta ^j(n+\alpha )$.
Citation
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
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