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Science China Mathematics

, Volume 62, Issue 3, pp 447–468 | Cite as

The Selberg-Delange method in short intervals with some applications

  • Zhen Cui
  • Guangshi Lü
  • Jie WuEmail author
Articles
  • 57 Downloads

Abstract

In this paper, we establish a quite general mean value result of arithmetic functions over short intervals with the Selberg-Delange method and give some applications. In particular, we generalize Selberg’s result on the distribution of integers with a given number of prime factors and Deshouillers-Dress-Tenenbaum’s arcsin law on divisors to the short interval case.

Keywords

asymptotic results on arithmetic functions Selberg-Delange method arithmetic functions distribution of integers 

MSC(2010)

11N37 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671253, 11771252 and 11531008), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20120073110059), Program for Innovative Research Team in University of Ministry of Education of China (Grant No. IRT16R43) and Taishan Scholars Project, the Program PRC 1457-AuForDiP (CNRS-NSFC). Finally, the authors are grateful to Y-K Lau for his help during the preparation of this paper, and to the referee for a careful reading of our manuscript and helpful suggestions.

References

  1. 1.
    Cui Z, Wu J. The Selberg-Delange method in short intervals with an application. Acta Arith, 2014, 163: 247–260MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Delange H. Sur les formules dues à Atle Selberg. Bull Sci Math, 1959, 83: 101–111MathSciNetzbMATHGoogle Scholar
  3. 3.
    Delange H. Sur les formules de Atle Selberg. Acta Arith, 1971, 1: 105–146MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Deshouillers J-M, Dress F, Tenenbaum G. Lois de répartition des diviseurs, 1. Acta Arith, 1979, 23: 273–283CrossRefzbMATHGoogle Scholar
  5. 5.
    Garaev M Z, Luca F, Nowak W G. The divisor problem for d 4(n) in short intervals. Arch Math, 2006, 86: 60–66CrossRefzbMATHGoogle Scholar
  6. 6.
    Hooley C. On intervals between numbers that are sums of two squares III. J Reine Angew Math, 1974, 267: 207–218MathSciNetzbMATHGoogle Scholar
  7. 7.
    Huxley M N. The difference between consecutive primes. Invent Math, 1972, 267: 164–170zbMATHGoogle Scholar
  8. 8.
    Ivić A. The Riemann Zeta-Function. New York-Chichester-Brisbane-Toronto-Singapore: John Wiley & Sons, 1985zbMATHGoogle Scholar
  9. 9.
    Kátai I. A remark on a paper of K. Ramachandra. In: Lecture Notes in Mathematics, vol. 1122. Berlin-Heidelberg: Springer, 1985, 147–152MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kátai I, Subbarao M V. Some remarks on a paper of Ramachandra. Lith Math J, 2003, 43: 410–418MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Landau E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974zbMATHGoogle Scholar
  12. 12.
    Montgomery H L. Topics in Multiplicative Number Theory. Berlin-New York: Springer, 1971CrossRefzbMATHGoogle Scholar
  13. 13.
    Motohashi Y. On the sum of the Möbius function in a short segment. Proc Japan Acad Ser A Math Sci, 1976, 52: 477–479zbMATHGoogle Scholar
  14. 14.
    Ramachandra K. Some problems of analytic number theory. Acta Arith, 1976, 31: 313–324MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Richert H. E. Zur abschatzung der Riemannschen zetafunktion in der nähe der vertikalen б = 1. Math Ann, 1967, 169: 97–101MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sathe L G. On a problem of Hardy and Ramanujan on the distribution of integers having a given number of prime factors. J Indian Math Soc (NS), 1953, 17: 63–141zbMATHGoogle Scholar
  17. 17.
    Sathe L G. On a problem of Hardy and Ramanujan on the distribution of integers having a given number of prime factors. J Indian Math Soc (NS), 1954, 18: 27–81zbMATHGoogle Scholar
  18. 18.
    Sedunova A A. On the asymptotic formulae for some multiplicative functions in short intervals. Int J Number Theory, 2015, 11: 1571–1587MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Selberg A. Note on the paper by L. G. Sathe. J Indian Math Soc (NS), 1954, 18: 83–87zbMATHGoogle Scholar
  20. 20.
    Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory. Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge: Cambridge University Press, 1995Google Scholar
  21. 21.
    Titchmarsh E C. The Theory of Function, 2nd ed. Oxford: Oxford University Press, 1952Google Scholar
  22. 22.
    Titchmarsh E C. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford: Clarendon Press, 1986zbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.School of MathematicsShandong UniversityJinanChina
  3. 3.School of Mathematics and StatisticsYangtze Normal UniversityChongqingChina
  4. 4.Institut Élie Cartan de LorraineUniversité de LorraineVandœuvre-lès-NancyFrance

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