Asymptotic estimates of sums involving the Moebius function. II

Author:
Krishnaswami Alladi

Journal:
Trans. Amer. Math. Soc. **272** (1982), 87-105

MSC:
Primary 10H15; Secondary 10H25

DOI:
https://doi.org/10.1090/S0002-9947-1982-0656482-7

MathSciNet review:
656482

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a positive integer and the Moebius function. If , let denote its largest prime factor and put . We study the asymptotic behavior of the sum as and discuss a few applications.

**[1]**Krishnaswami Alladi,*Duality between prime factors and an application to the prime number theorem for arithmetic progressions*, J. Number Theory**9**(1977), no. 4, 436–451. MR**0476666**, https://doi.org/10.1016/0022-314X(77)90005-1**[2]**Krishnaswami Alladi,*Asymptotic estimates of sums involving the Moebius function*, J. Number Theory**14**(1982), no. 1, 86–98. MR**644903**, https://doi.org/10.1016/0022-314X(82)90060-9**[3]**Tom M. Apostol,*Introduction to analytic number theory*, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR**0434929****[4]**N. G. de Bruijn,*On the number of uncancelled elements in the Sieve of Eratosthenes*, Indag. Math.**12**(1950), 247-256.**[5]**-,*On the number of integers**and free of prime factors*, Indag. Math.**13**(1951), 50-60.**[6]**N. G. de Bruijn,*The asymptotic behaviour of a function occurring in the theory of primes*, J. Indian Math. Soc. (N.S.)**15**(1951), 25–32. MR**0043838****[7]**K. Chandrasekharan,*Arithmetical functions*, Die Grundlehren der mathematischen Wissenschaften, Band 167, Springer-Verlag, New York-Berlin, 1970. MR**0277490****[8]**H. Halberstam and H.-E. Richert,*Sieve methods*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR**0424730****[9]**B. V. Levin and A. S. Fainleib,*Applications of some integral equations to problems of number theory*, Russian Math. Surveys**22**(1967), 119-204.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0656482-7

Keywords:
Moebius function,
asymptotic estimate,
largest and smallest prime factors

Article copyright:
© Copyright 1982
American Mathematical Society