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


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.


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

Mathematics Subject Classification (2000)

11M06 11N37 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Heath-Brown, D.R.: The twelfth power moment of the Riemann zeta-function. Q. J. Math. (Oxford) 29, 443–462 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ivić, A.: The Riemann Zeta-Function. Wiley, New York, (1985). 2nd edn. Dover, New York (2003) zbMATHGoogle 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) zbMATHGoogle 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) zbMATHCrossRefGoogle Scholar
  9. 9.
    Ivić, A.: On the Riemann zeta function and the divisor problem III. Ann. Univ. Bp. Sect. Comput. 29, 3–23 (2008) zbMATHGoogle Scholar
  10. 10.
    Ivić, A.: On the Riemann zeta function and the divisor problem IV. Uniform Distribution Theory 1, 125–135 (2006) zbMATHMathSciNetGoogle 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) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Jutila, M.: A Method in the Theory of Exponential Sums. Tata Lectures on Math., vol. 80. Springer, Berlin (1987) zbMATHGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jutila, M.: Riemann’s zeta-function and the divisor problem II. Ark. Mat. 31, 61–70 (1993) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefGoogle Scholar
  22. 22.
    Ramachandra, K., Sankaranarayanan, A.: On an asymptotic formula of Srinivasa Ramanujan. Acta Arith. 109, 349–357 (2003) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rankin, R.A.: Modular Forms and Functions. Cambridge Univ. Press, Cambridge (1977) zbMATHGoogle Scholar
  25. 25.
    Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591, 1–20 (2006) zbMATHMathSciNetGoogle 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) zbMATHGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zhai, W.: On higher-power moments of Δ(x) II. Acta Arith. 114, 35–54 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zhai, W.: On higher-power moments of Δ(x) III. Acta Arith. 118, 263–281 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Zhai, W.: On higher-power moments of E(t). Acta Arith. 115, 329–348 (2004) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Katedra Matematike RGF-aUniversitet u BeograduBelgradeSerbia

Personalised recommendations