The divisor function $d_3(n)$ in arithmetic progressions

D. Heath-Brown

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Displaying similar documents to “The divisor function $d_{3} (n)$ in arithmetic progressions”

The ternary Goldbach problem in arithmetic progressions

Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N, r) = b \in ℕ ³ : 1 \leq b_{j} \leq r, (b_{j}, r) = 1 a n d b ₁ + b ₂ + b ₃ \equiv N (m o d r) .$ It is known that $(N, r) = r ² \prod_{p | r} p | N ((p - 1) (p - 2) / p ²) \prod_{p | r} p ∤ N ((p ² - 3 p + 3) / p ²)$ . Let ε > 0 be arbitrary and $R = N^{1 / 8 - ε}$ . We prove that for all positive integers r ≤ R, with at most $O (R l o g^{- A} N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_{j} \equiv b_{j} (m o d r),$ j = 1,2,3,⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

On the differences of primes in arithmetical progressions

Martin Huxley (1969)

Acta Arithmetica

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Upper bounds for k-th coset representatives modulo n

Karl Norton (1969)

Acta Arithmetica

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Distributions of the values of some arithmetical functions

Paul Erdös, Andrzej Schinzel (1961)

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On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin (2001)

Journal de théorie des nombres de Bordeaux

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Let $q, q_{1}, \dots, q_{s} \geq 2$ be integers, and let $α_{1}, α_{2}, \dots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${(x n)}_{n = 1}^{M N}$ in $s$ -dimensional unit cube $(s, M, N = 1, 2, \dots)$ . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (Korobov’s problem).

The divisor problem for arithmetic progressions with small modulus

Charles E. Chace (1992)

Acta Arithmetica

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A short intervals result for linear equations in two prime variables.

M. B. S. Laporta (1997)

Revista Matemática de la Universidad Complutense de Madrid

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Given A and B integers relatively prime, we prove that almost all integers n in an interval of the form [N, N+H], where N exp(1/3+e) ≤ H ≤ N can be written as a sum Ap1 + Bp2 = n, with p1 and p2 primes and e an arbitrary positive constant. This generalizes the results of Perelli et al. (1985) established in the classical case A=B=1 (Goldbach's problem).

A generalization of Bombieri's prime number theorem to algebraic number fields

Jürgen Hinz (1988)

Acta Arithmetica

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On mean values of L-functions and short character sums with real characters

Matti Jutila (1975)

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Displaying similar documents to “The divisor function d 3 ( n ) in arithmetic progressions”

Displaying similar documents to “The divisor function $d_{3} (n)$ in arithmetic progressions”