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.
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Cui Z, Wu J. The Selberg-Delange method in short intervals with an application. Acta Arith, 2014, 163: 247–260
Delange H. Sur les formules dues à Atle Selberg. Bull Sci Math, 1959, 83: 101–111
Delange H. Sur les formules de Atle Selberg. Acta Arith, 1971, 1: 105–146
Deshouillers J-M, Dress F, Tenenbaum G. Lois de répartition des diviseurs, 1. Acta Arith, 1979, 23: 273–283
Garaev M Z, Luca F, Nowak W G. The divisor problem for d 4(n) in short intervals. Arch Math, 2006, 86: 60–66
Hooley C. On intervals between numbers that are sums of two squares III. J Reine Angew Math, 1974, 267: 207–218
Huxley M N. The difference between consecutive primes. Invent Math, 1972, 267: 164–170
Ivić A. The Riemann Zeta-Function. New York-Chichester-Brisbane-Toronto-Singapore: John Wiley & Sons, 1985
Kátai I. A remark on a paper of K. Ramachandra. In: Lecture Notes in Mathematics, vol. 1122. Berlin-Heidelberg: Springer, 1985, 147–152
Kátai I, Subbarao M V. Some remarks on a paper of Ramachandra. Lith Math J, 2003, 43: 410–418
Landau E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974
Montgomery H L. Topics in Multiplicative Number Theory. Berlin-New York: Springer, 1971
Motohashi Y. On the sum of the Möbius function in a short segment. Proc Japan Acad Ser A Math Sci, 1976, 52: 477–479
Ramachandra K. Some problems of analytic number theory. Acta Arith, 1976, 31: 313–324
Richert H. E. Zur abschatzung der Riemannschen zetafunktion in der nähe der vertikalen б = 1. Math Ann, 1967, 169: 97–101
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: 63–141
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: 27–81
Sedunova A A. On the asymptotic formulae for some multiplicative functions in short intervals. Int J Number Theory, 2015, 11: 1571–1587
Selberg A. Note on the paper by L. G. Sathe. J Indian Math Soc (NS), 1954, 18: 83–87
Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory. Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge: Cambridge University Press, 1995
Titchmarsh E C. The Theory of Function, 2nd ed. Oxford: Oxford University Press, 1952
Titchmarsh E C. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford: Clarendon Press, 1986
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.
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Cui, Z., Lü, G. & Wu, J. The Selberg-Delange method in short intervals with some applications. Sci. China Math. 62, 447–468 (2019). https://doi.org/10.1007/s11425-017-9172-7
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DOI: https://doi.org/10.1007/s11425-017-9172-7
Keywords
- asymptotic results on arithmetic functions
- Selberg-Delange method
- arithmetic functions
- distribution of integers