Periodica Mathematica Hungarica

, Volume 10, Issue 4, pp 261–271 | Cite as

On the order of the error function of the square-full integers

  • D. Suryanarayana
Article
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Revised:

Abstract

LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that
$$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$
where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that\(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.:
$$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$
(1)
$$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$
(2)
for any integer ν≥1.

In fact, we prove some general results and deduce the above from them.

On the basis of (1) and (2) above, we conjecture that\(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-\nulldelimiterspace} {12}} + \varepsilon } )\) under the assumption of R.H.

AMS (MOS) subject classifications (1970)

Primary 10H15 Secondary 10H25 

Key words and phrases

Square-full integers Riemann Zeta function Riemann hypothesis 
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References

  1. [1]
    R. Ayoub,An introduction to the analytic theory of numbers, Amer. Math. Soc. Providence, 1963. (Reprinted 1974)MR 28 # 3954Google Scholar
  2. [2]
    P. T. Bateman, Square-full integers (Solution to Problem 4459),Amer. Math. Monthly 61 (1954), 477–479.Google Scholar
  3. [3]
    P. T. Bateman andE. Grosswald, On a theorem or Erdős and Szekeres,Illinois J. Math. 2 (1958), 88–98.MR 20 # 2305Google Scholar
  4. [4]
    H. M. Edwards,Riemann's Zeta function, Academic Press, New York, 1974.Zbl 315, 10035Google Scholar
  5. [5]
    P. Erdős andG. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem,Acta Sci. Math. (Szeged) 7 (1934), 95–102.Zbl. 10, 294Google Scholar
  6. [6]
    S. W. Golomb, Powerful numbers,Amer. Math. Monthly,77 (1970), 848–855.MR 42 # 1780Google Scholar
  7. [7]
    G. H. Hardy,A course of pure mathematics, tenth edition, Cambridge University Press, 1952. (Reprinted 1967)MR 14-145Google Scholar
  8. [8]
    G. H. Hardy andE. M. Wright,An introduction to the theory of numbers, fourth edition, Oxford University Press, 1960. (Reprinted 1965)MR 16-673Google Scholar
  9. [9]
    E. Landau,Handbuch der Lehre von der Verteilung der Primzahlen, third edition, Chelsea, New York, 1974. (Originally published in 1909)Google Scholar
  10. [10]
    H. Rademacher,Topics in analytic number theory, Springer, Berlin, 1973.MR 51 # 358Google Scholar
  11. [11]
    A. Sklar, Square-full integers (Solution to Problem 4459),Amer. Math. Monthly 60 (1953), 55.Google Scholar
  12. [12]
    D. Suryanarayana andR. Sita Rama Chandra Rao, The distribution of squarefull integers,Ark. Mat. 11 (1973), 195–201.MR 49 # 8948Google Scholar
  13. [13]
    E. C. Titchmarsh,The theory of the Riemann Zeta function. Oxford University Press, 1951. (Reprinted 1967)MR 13-741Google Scholar

Copyright information

© Akadémiai Kiadó 1979

Authors and Affiliations

  • D. Suryanarayana
    • 1
  1. 1.Department of MathematicsAndhra UniversityWaltairIndia

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