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The Ramanujan Journal

, Volume 19, Issue 2, pp 207–224 | Cite as

On the divisor function and the Riemann zeta-function in short intervals

  • Aleksandar Ivić
Article

Abstract

We obtain, for T ε U=U(T)≤T 1/2−ε , asymptotic formulas for
$$\int_{T}^{2T}\Bigl(E(t+U)-E(t)\Bigr)^{2}{\mathrm{d}}{t},\qquad \int_{T}^{2T}\Bigl(\Delta (t+U)-\Delta (t)\Bigr)^{2}{\mathrm{d}}{t},$$
where Δ(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for \(|\zeta(\frac{1}{2}+\mathit{it})|\) . Upper bounds of the form O ε (T 1+ε U 2) for the above integrals with biquadrates instead of square are shown to hold for T 3/8U=U(T) T 1/2. The connection between the moments of E(t+U)−E(t) and \(|\zeta(\frac{1}{2}+\mathit{it})|\) is also given. Generalizations to some other number-theoretic error terms are discussed.

Keywords

Riemann zeta-function Divisor functions Power moments in short intervals Upper bounds 

Mathematics Subject Classification (2000)

11M06 11N37 

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References

  1. 1.
    Atkinson, F.V.: The mean value of the Riemann zeta-function. Acta Math. 81, 353–376 (1949) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bugeaud, Y., Ivić, A.: Sums of the error term function in the mean square for ζ(s). Mon.hefte Math. (2008, in press). arXiv:0707.4275
  3. 3.
    Coppola, G., Salerno, S.: On the symmetry of the divisor function in almost all short intervals. Acta Arith. 113, 189–201 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Heath-Brown, D.R.: The twelfth power moment of the Riemann zeta-function. Q. J. Math. (Oxford) 29, 443–462 (1978) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ivić, A.: The Riemann Zeta-Function. Wiley, New York, (1985). 2nd edn. Dover, New York (2003) MATHGoogle Scholar
  6. 6.
    Ivić, A.: The Mean Values of the Riemann Zeta-Function. Tata Inst. of Fundamental Research, Bombay, LNs, vol. 82. Springer, Berlin (1991) MATHGoogle Scholar
  7. 7.
    Ivić, A.: On the Riemann zeta function and the divisor problem. Centr. Eur. J. Math. 2(4), 1–15 (2004) Google Scholar
  8. 8.
    Ivić, A.: On the Riemann zeta function and the divisor problem II. Centr. Eur. J. Math. 3(2), 203–214 (2005) MATHCrossRefGoogle Scholar
  9. 9.
    Ivić, A.: On the Riemann zeta function and the divisor problem III. Ann. Univ. Bp. Sect. Comput. 29, 3–23 (2008) MATHGoogle Scholar
  10. 10.
    Ivić, A.: On the Riemann zeta function and the divisor problem IV. Uniform Distribution Theory 1, 125–135 (2006) MATHMathSciNetGoogle Scholar
  11. 11.
    Ivić, A.: Some remarks on the moments of \(|\zeta (\frac{1}{2}+\mathit{it})|\) in short intervals. Acta Math. Hung. (2008, to appear). math.NT/0611427
  12. 12.
    Ivić, A.: On moments of \(|\zeta (\frac{1}{2}+\mathit{it})|\) in short intervals. The Riemann Zeta Function and Related Themes. Ramanujan Math. Soc. LNS 2. pp. 81–97 (2006). Papers in honour of Professor Ramachandra Google Scholar
  13. 13.
    Ivić, A.: On the mean square of the zeta-function and the divisor problem. Ann. Acad. Sci. Fenn. Math. 32, 1–9 (2007) Google Scholar
  14. 14.
    Ivić, A., Sargos, P.: On the higher moments of the error term in the divisor problem. Ill. J. Math. 81, 353–377 (2007) Google Scholar
  15. 15.
    Jutila, M.: On the divisor problem for short intervals. Ann. Univer. Turkuensis Ser. AI 186, 23–30 (1984) MATHMathSciNetGoogle Scholar
  16. 16.
    Jutila, M.: A Method in the Theory of Exponential Sums. Tata Lectures on Math., vol. 80. Springer, Berlin (1987) MATHGoogle Scholar
  17. 17.
    Jutila, M.: Mean value estimates for exponential sums. In: Number Theory, Ulm 1987. Lect. Notes in Math., vol. 1380, pp. 120–136. Springer, Berlin (1989) CrossRefGoogle Scholar
  18. 18.
    Jutila, M.: Riemann’s zeta-function and the divisor problem. Ark. Mat. 21, 75–96 (1983) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jutila, M.: Riemann’s zeta-function and the divisor problem II. Ark. Mat. 31, 61–70 (1993) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kaczorowski, A., Perelli, A.: The Selberg class: a survey. In: Györy, K., et al. (eds.) Number Theory in Progress. Proc. Conf. in Honour of A. Schinzel, pp. 953–992. de Gruyter, Berlin (1999) Google Scholar
  21. 21.
    Kühleitner, M., Nowak, W.G.: The average number of solutions of the Diophantine equation u 2+v 2=w 3 and related arithmetic functions. Acta Math. Hung. 104, 225–240 (2004) MATHCrossRefGoogle Scholar
  22. 22.
    Ramachandra, K., Sankaranarayanan, A.: On an asymptotic formula of Srinivasa Ramanujan. Acta Arith. 109, 349–357 (2003) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rankin, R.A.: Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions II. The order of the Fourier coefficients of integral modular forms. Proc. Camb. Philos. Soc. 35, 357–372 (1939) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rankin, R.A.: Modular Forms and Functions. Cambridge Univ. Press, Cambridge (1977) MATHGoogle Scholar
  25. 25.
    Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591, 1–20 (2006) MATHMathSciNetGoogle Scholar
  26. 26.
    Selberg, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvidensk. 43, 47–50 (1940) MathSciNetGoogle Scholar
  27. 27.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Univ. Press, Oxford (1986) MATHGoogle Scholar
  28. 28.
    Tsang, K.-M.: Higher power moments of Δ(x), E(t) and P(x). Proc. Lond. Math. Soc. 65(3), 65–84 (1992) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Zhai, W.: On higher-power moments of Δ(x). Acta Arith. 112, 367–395 (2004) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zhai, W.: On higher-power moments of Δ(x) II. Acta Arith. 114, 35–54 (2004) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zhai, W.: On higher-power moments of Δ(x) III. Acta Arith. 118, 263–281 (2005) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Zhai, W.: On higher-power moments of E(t). Acta Arith. 115, 329–348 (2004) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Katedra Matematike RGF-aUniversitet u BeograduBelgradeSerbia

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