On the local behavior of
Author:
Adolf Hildebrand
Journal:
Trans. Amer. Math. Soc. 297 (1986), 729751
MSC:
Primary 11N25
MathSciNet review:
854096
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Abstract 
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Abstract: denotes the number of positive integers and free of prime factors . In the range , can be well approximated by a "smooth" function, but for , this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that behaves "locally" more regular by giving a sharp estimate for , valid in the range , .
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 K. Alladi, The TuránKubilius inequality for integers without large prime factors, J. Reine Angew. Math. 335 (1982), 180196. MR 667466 (84a:10045)
 [2]
 N. G. de Bruijn, On some Volterra equations of which all solutions are convergent, Indag. Math. 12 (1950), 257265.
 [3]
 , On the number of positive integers and free of prime factors , Indag. Math. 12 (1951), 5060.
 [4]
 K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22 (1930), 114.
 [5]
 D. R. HeathBrown, Finding primes by sieve methods, Proc. Internat. Congr. Math. (Warsaw 1983), Vol. 1, pp. 487492. MR 804704 (87a:11083)
 [6]
 D. Hensley, A property of the counting function of integers with no large prime factors, J. Number Theory 22 (1986), 4674. MR 821136 (87f:11065)
 [7]
 A. Hildebrand, On the number of positive integers and free of prime factors , J. Number Theory 22 (1986), 289307. MR 831874 (87d:11066)
 [8]
 K. Norton, Numbers with small prime factors, and the least kth power nonresidue, Mem. Amer. Math. Soc. No. 106 (1971). MR 0286739 (44:3948)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608540960
PII:
S 00029947(1986)08540960
Keywords:
Integers free of large primes factors,
asymptotic estimates
Article copyright:
© Copyright 1986
American Mathematical Society
