Generalized Titchmarsh divisor problem

Karshiev, A. K.; Соколовский, А. В.

doi:10.1007/bf01094332

Cited by 2 publications

(4 citation statements)

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“…The remainder due to the substitution of (4) is (see Lemma 4) 2n (6) ,~ N2 -~ In ~ N E n(6)_<t --(1 + ln6) < N2-k lnl-P N.…”

Section: F~2(rrn) ( K "~ N(rm)122(rm) Lns ( K F~(rr)mentioning

confidence: 99%

“…V. Linnik [2] within the framework of the variance method, which he had developed. I-Ie found the asymptotics of the number of solutions to the first equation (1) n<N where in [5,8] n is equal to the product of two primes pl < N a and p2 < N 1-" , in [6] n is equal to the product of k primes with pi < N =~ , ax < a2 < --" < a~ < 1/7, and al -t-a2 ~--.. -{-ak = 1. Finally, in [7] n has at most six prime divisors greater than N 1/ssa .…”

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confidence: 99%

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The Titchmarsh problem with integers having a given number of prime divisors

Timofeev

Khripunova

1996

Math Notes

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ABSTRACT. The uymptotics for the number of representations of N as N ---* oo is expressed as the sum of a number having h prime divisors and a product of two natural numbers. The asymptotics is found for k _< (2 -e) In in N and where p is a prime and r(n) is the number of divisors of n. Therefore, the binary additive problemsare called Titchmarsh problem~.The first definitive solution of these problems was obtained in the sixties by Yu. V. Linnik [2] within the framework of the variance method, which he had developed. I-Ie found the asymptotics of the number of solutions to the first equation (1) n 0 and ln2 N = lnlnN. Note that research in this direction began in [9], where the asymptotics of the sum (2) was obtained for n having k prime divisors k < (2 -e)ln2 N and a = 1. Let ~(n) be the number of prime divisors of n, counting multiplicity. Denote N(x, = I{": "< x, = k} I,

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“…The remainder due to the substitution of (4) is (see Lemma 4) 2n (6) ,~ N2 -~ In ~ N E n(6)_<t --(1 + ln6) < N2-k lnl-P N.…”

Section: F~2(rrn) ( K "~ N(rm)122(rm) Lns ( K F~(rr)mentioning

confidence: 99%

mentioning

confidence: 99%

The Titchmarsh problem with integers having a given number of prime divisors

Timofeev

Khripunova

1996

Math Notes

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“…Indeed, the part of the sum containing such n is Here and subsequently, we assume that Q < v/-U/In A N. Let us write out and estimate the remainders R2 and R1 in (5). Let us begin with RI:…”

Section: N2mentioning

confidence: 99%

“…V. Linnik [2] within the framework of the variance method, which he had developed. Later, M. B. Bredikhin [3], M. S. Sarban [4l, A. K. Karshiev [5], Zh. A. Piyadina [6], and Ao fujii [7] have contributed to the generalization of the Titchmarsh problem.…”

Section: C= L+p(p-__l)mentioning

confidence: 99%

Additive problems for integers with a given number of prime divisors

Khripunova

1998

Math Notes

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AnSTRXCT. The asymptotics of sums of the form ~ r (Ibn -a[) (summation over n < N, w(n) = k) is studied, where to(n) is the number of distinct prime divisors of n, and ~-(n) is the number of all divisors.KEY WORDS: prime divisor, Titehmarsh problem, asymptotics of the sum of divisors.Let w(n) be the number of distinct prime divisors of the integer n, and let r(n) be the number of divisors of n. The goal of the present paper is to find the asymptotics of the sum r(Ib -al), where n is equal to the product of two primes satisfying certain conditions. For example, in [6] it was shown that n has at most six prime divisors greater than N l/as3 .

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Generalized Titchmarsh divisor problem

Cited by 2 publications

References 1 publication

The Titchmarsh problem with integers having a given number of prime divisors

The Titchmarsh problem with integers having a given number of prime divisors

Additive problems for integers with a given number of prime divisors

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