The Dispersion Method and Definite Binary Additive Problems

Bredikhin, B. M.

doi:10.1070/rm1965v020n02abeh001170

Cited by 11 publications

(9 citation statements)

References 15 publications

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“…In 1937 Vinogradov [13] and proved that I (3) (1) where A > 0 is an arbitrarily large constant and S (3) …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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The ternary Goldbach problem with arithmetic weights attached to two of the variables

Tolev

2010

Journal of Number Theory

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We consider the ternary Goldbach problem with two prime variables of the form k 2 + m 2 + 1 and find an asymptotic formula for the number of its solutions. Notations. ByGreek letters we denote real numbers and by small Latin letters -integers. However, the letter p, with or without subscripts, is reserved for primes. By ε we denote an arbitrarily small positive number, not the same in different appearances. N is a sufficiently large odd integer and L = log N. We denote by J the set of all subintervals of the interval [1, N] and if J 1 , J 2 ∈ J then J = J 1 , J 2 is the corresponding ordered pair. Respectively k = k 1 , k 2 is two-dimensional vector with integer components k 1 , k 2 and, in particular, 1 = 1, 1 . We write (m 1 , . . . ,m k ) for the greatest common factor of m 1 , . . . ,m k . As usual τ (k) is the number of positive divisors of k; r(k) is the number of solutions of the equation m 2 1 + m 2 2 = k in integers m j ; ϕ(k) is the Euler function; Ω(k) is the number of the prime factors of k, counted with the multiplicity; χ (k) is the non-principal character modulo 4 and L(s, χ ) is the corresponding Dirichlet's L-function. We mark by an end of a proof, or its absence.

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“…In 1937 Vinogradov [13] and proved that I (3) (1) where A > 0 is an arbitrarily large constant and S (3) …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Obviously, the inequality Ω(md) β implies the validity of at least one of the inequalities Ω(m) 1 2 β and Ω(d) 1 2 β. Hence…”

Section: Lemma 4 For Any Constantmentioning

confidence: 97%

The ternary Goldbach problem with arithmetic weights attached to two of the variables

Tolev

2010

Journal of Number Theory

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“…Thus, it follows from formulas (2.18) and (2.19) and from the main 1emma that Furthermore, we find the asymptotics of A(n), which already is a ternary problem that can be solved by the Hooly lemma concerning the distribution of quasiprlme numbers in arithmetic progressions. Let us apply the sharpened version of this lemma (Lemma 3 of the paper [6] The proof of Theorem 2 is completed by the calculation of the inner sums in (2.25) by means of Lemma 2.3.2 of the paper [7] (we apply a modified version of this lemma that can be proved similarly). Finally we have where R is a fixed genus defined by the given form ~(u, v).…”

Section: Ipt R2 ~7]--v2r2 O'zt=ai( ~2--1pl )mentioning

confidence: 99%

An analog of the generalized Hardy-Littlewood problem with almost prime numbers

Piyadina¹

1996

Math Notes

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“…Taking c x = 2 in the Lemma, and G > B (2), we see that f Actually, it will be clear from the sequel that the weaker condition f(p")<l suffices for the purpose of estimating effectively the sums JSJ and <S 2 . Hence, if also / is nonnegative, the method will be seen to lead, under these alternative conditions on/, to good lower bounds for 8.…”

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confidence: 97%