Skip to main content
Log in

On the Mean Square of the Error Terms Corresponding to Exponential Sums Involving the Ideal Counting Function

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this paper we consider the asymptotic behaviour of the error term Q(x;h/q), which is defined by (1.2). In particular, we derive a certain lower bound estimate of this function when h, k and q are fixed integers, and study the non-trivial upper bound estimates of the mean value of Q(x;h/q).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. H. Cramér, Über zwei Sätze von Herrn G.H. Hardy, Math. Zeit. 15 (1922), 200–210.

  2. J. Furuya, “Mean square of an error term related to a certain exponential sum involving the divisor function,” in Number Theory and its Applications (S. Kanemitsu-K. Gyõry, Eds.), Devel. in Math. vol 2, Kluwer 1999, pp. 111–127.

  3. J. Furuya, “On exponential sums involving the ideal counting function in quadratic number fields,” Monatsh. Math. 137 (2002), 129–156.

    Google Scholar 

  4. J. Furuya, “Mean square of error terms related to exponential sums involving some arithmetical functions,” Thesis. Nagoya University, 2001.

  5. A. Ivić, The Riemann Zeta-Function, John Wiley & Sons, New York, 1985.

    Google Scholar 

  6. Y.K. Lau and K.M. Tsang, “Mean square of the remainder term in the Dirichlet divisor problem,” J. Théor. Nombres Bordeaux 7 (1995), 75–92.

    Google Scholar 

  7. T. Meurman, “On the mean square of the Riemann zeta-function,” Quart. J. Math. 38(2) (1987), 337–343.

    Google Scholar 

  8. W. Müller, “On the asymptotic behaviour of the ideal counting function in quadratic number fields,” Monatsh. Math. 108 (1989), 301–323.

    Google Scholar 

  9. E. Preissmann, “Sur la moyenne quadratique du terme de reste du probléme du cercle,” C.R. Acad. Sci. Paris Sér. I 306 (1988), 151–154.

    Google Scholar 

  10. R.A. Smith, “On r (n)r (n +a),” Proc. Nat. Inst. Sci. India. Part A 34 (1968), 132–137.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Furuya, J. On the Mean Square of the Error Terms Corresponding to Exponential Sums Involving the Ideal Counting Function. The Ramanujan Journal 8, 177–198 (2004). https://doi.org/10.1023/B:RAMA.0000040480.23461.32

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:RAMA.0000040480.23461.32

Navigation