Jun Furuya and Yoshio Tanigawa

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 20, 2014, Number 2, Pages 44—51

**Download full paper: PDF, 196 Kb**

## Details

### Authors and affiliations

Jun Furuya

^{1}* Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine
Handayama 1-20-1, Higashi-ku, Hamamatsu city, Shizuoka, 431-3192, Japan*

^{2}

*Department of Integrated Arts and Science, Okinawa National College of Technology*

Nago, Okinawa, 905-2192, Japan

Nago, Okinawa, 905-2192, Japan

Yoshio Tanigawa

*Graduate School of Mathematics, Nagoya University
Nagoya, 464-8602, Japan*

### Abstract

In this paper, we show the relation between the shifted sum of a number-theoretic error term and its continuous mean (integral). We shall obtain a certain expression of the shifted sum as a linear combination of the continuous mean with the Bernoulli polynomials as their coefficients. As an application of our theorem, we give better approximations of the continuous mean by a shifted sum.

### Keywords

- The circle problem
- Mean value of error terms
- Shifted sum
- Bernoulli polynomial

### AMS Classification

- 11N37

### References

- Cao, X., J. Furuya, Y. Tanigawa, W. Zhai. On the differences between two kinds of mean value formulas of number-theoretic error terms. Int. J. Number Theory, DOI: 10.1142/S1793042114500195
- Cao, X., J. Furuya, Y. Tanigawa,W. Zhai. On the mean of the shifted error term in the theory of the Dirichlet divisor problem, to appear in “Rocky Mountain J. Math.”.
- Furuya, J. On the average orders of the error term in the circle problem. Publ. Math. Debrecen, Vol. 67, 2005, 381–400.
- Furuya, J. On the average orders of the error term in the Dirichlet divisor problem. J. Number Theory, Vol. 115, 2005, 1–26.
- Furuya, J., Y. Tanigawa. On integrals and Dirichlet series obtained from the error term in the circle problem, to appear in “Funct. Approx. Comment. Math.”.
- Furuya, J., Y. Tanigawa, W. Zhai. Dirichlet series obtained from the error term in the Dirichlet divisor problem. Monatsh. Math., Vol. 160, 2010, 385–402.
- Graham, S. W., G. Kolesnik, Van der Corput’s Method of Exponential Sums, London Math. Soc. Lect. Note Series, Vol. 126, Cambridge University Press, 1991.
- Huxley, M. N. Exponential sums and lattice points III. Proc. London Math. Soc., Vol. 87, 2003, 591–609.
- Ivić, A. The Riemann Zeta-Function, John Wiley & Sons, New York, 1985 (2nd ed., Dover, Mineola, NY, 2003).
- Ivić, A., P. Sargos. On the higher moments of the error term in the divisor problem. Illinois J. Math., Vol. 51, 2007, 353–377.
- Krätzel, E. Lattice Points, Kluwer Academic Publishers, Dordrecht, 1988.
- Lau, Y. K., K. M. Tsang. Mean square of the remainder term in the Dirichlet divisor problem. J. Théorie Nombres Bordeaux, Vol. 7, 1995, 75–92.
- Lau, Y. K., K. M. Tsang. On the mean square formula of the error term in the Dirichlet divisor problem. Math. Proc. Cambridge Phil. Soc., Vol. 146, 2009, 277–287.
- Zhai, W. On higher-power moments of Δ(x) II. Acta Arith., Vol. 114, 2004, 35–54.
- Zhai, W. On higher-power moments of Δ(x) III. Acta Arith., Vol. 118, 2005, 263–281.
- Zhang, D., W. Zhai, On the fifth-power moment of Δ(x). Int. J. Number Theory, Vol. 7, 2011, 71–86.

## Related papers

## Cite this paper

Furuya, J., & Tanigawa, Y. (2014). Mean values of the error term with shifted arguments in the circle problem. Notes on Number Theory and Discrete Mathematics, 20(2), 44-51.