Functiones et Approximatio Commentarii Mathematici

On integrals and Dirichlet series obtained from the error term in the circle problem

Jun Furuya and Yoshio Tanigawa

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Abstract

In this paper, we shall investigate several properties of integrals defined by $\int_1^{\infty}t^{-\theta}P(t)\log^jtdt$ with a complex variable $\theta$ and a non-negative integer $j$, where $P(x)$ is the error term in the circle problem of Gauss. We shall also study the analytic continuation of several types of the Dirichlet series related with the circle problem, and study a proof of the functional equation of the Dedekind zeta-function associated with the Gaussian number field ${\mathbb{Q}}(\sqrt{-1})$.

Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 2 (2014), 303-333.

Dates
First available in Project Euclid: 26 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1417010856

Digital Object Identifier
doi:10.7169/facm/2014.51.2.5

Mathematical Reviews number (MathSciNet)
MR3282630

Zentralblatt MATH identifier
1358.11106

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions

Keywords
analytic continuation the circle problem the Dedekind zeta-function periodic Bernoulli functions

Citation

Furuya, Jun; Tanigawa, Yoshio. On integrals and Dirichlet series obtained from the error term in the circle problem. Funct. Approx. Comment. Math. 51 (2014), no. 2, 303--333. doi:10.7169/facm/2014.51.2.5. https://projecteuclid.org/euclid.facm/1417010856


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