Distributional properties of the largest prime factor

Banks, William D.; Harman, Glyn; Shparlinski, Igor E.

doi:10.1307/mmj/1133894172

Cited by 15 publications

(14 citation statements)

References 10 publications

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“…We now recall the asymptotic formula 1 * ' "~ =0.6243... x t-1 logn 2<n<x 6 for the average logarithmic size of the largest prime factor (see [27, Exercise 3, Chapter III.5]). Assuming that the shifts qj -a, where q 0 = q, and qj = P{qj-\ -a), j = 1,2,..., behave as 'typical' integers, then it is reasonable to expect that the bound k a {q) <£ log log q holds for almost all primes q.…”

Section: Discussionmentioning

confidence: 99%

“…Introduction 1.1. Background Let V be the set of prime numbers, and for every integer n > 1, let P(n) € V be the largest prime factor of n. The function P : {2,3,...} ^ V arises naturally in many number theoretic situations and has been the subject of numerous investigations; see, for example, [5,6,8,14,16,17,18,20,24,28] and the references contained therein.…”

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confidence: 99%

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On values taken by the largest prime factor of shifted primes

Banks¹,

Shparlinski²

2007

J. Aust. Math. Soc.

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Let V denote the set of prime numbers, and let P (n) denote the largest prime factor of an integer n > 1. We show that, for every real number 32/17 < r) < (4 + 3\/2)/4, there exists a constant c(r\) > 1 such that for every integer a ^ 0, the set \p e V : p = P(q -a) for some prime q with p'has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 < t) < (4 + 3\/2)/4, one can take c(r)) = 1 + e for any fixed e > 0. In particular, our results imply that for every real number 0.486 < # < 0.531, the relation P(q -a) x q" holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q H> P(q -a) for a > 0, and show that for infinitely many primes

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

On values taken by the largest prime factor of shifted primes

Banks¹,

Shparlinski²

2007

J. Aust. Math. Soc.

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“…Estimate exponential and character sums with the largest squarefree divisor of n and with the squarefree part of n, that is, Comments: Bounds of character sums with these functions are given by H. Liu and W. Zhang [92], but the method of [92] does not apply to exponential sums. Exponential sums with P (n) have been estimated in [7]; exponential and character sums with the Euler function are estimated in [2, 9, 13].…”

Section: Arithmetic Functions Problem 30 Estimate Exponential Sums Wmentioning

confidence: 99%

Open Problems on Exponential and Character Sums

Shparlinski

2009

Number Theory

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“…Exponential sum over an arithmetic function is a topic in the number theory (see [2,3]). In particularly, Kerr [5] considered the rational exponential sums over the divisor function…”

Section: Introductionmentioning

confidence: 99%