On oscillations in the additive divisor problem, 1
On periodic mod p sequences and G-functions (On a conjecture of Ruzsa).
On polynomials that are sums of two cubes.
It is proved that, if F(x) be a cubic polynomial with integral coefficients having the property that F(n) is equal to a sum of two positive integral cubes for all sufficiently large integers n, then F(x) is identically the sum of two cubes of linear polynomials with integer coefficients that are positive for sufficiently large x. A similar result is proved in the case where F(n) is merely assumed to be a sum of two integral cubes of either sign. It is deduced that analogous propositions are true...
On Popov's explicit formula and the Davenport expansion
We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function with the periodic Bernoulli polynomial weight and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order or gives the well-known explicit formula for respectively the partial...
On positive integers with a certain nondivisibility property.
On power-free numbers and polynomials. I.
On power.free numbers and polynomials. II.
On prime divisors of Mersenne numbers
On prime divisors of remarkable sequences.
On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c)...
On prime factors of sums of integers I
On prime numbers in an arithmetic progression with a prime-power difference
On prime primitive roots
On prime values of reducible quadratic polynomials
It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer such that the polynomial represents at least r distinct primes.
On primitive divisors of Mersenne numbers
On primitive lattice points in planar domains
On Representing Integers as Products of the p + 1.
On rough and smooth neighbors.
We study the behavior of the arithmetic functions defined byF(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1)where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.
On sequences of ±1's defined by binary patterns [Book]
On sets characterizing number-theoretical functions (II) (The set of "prime plus one"'s is a set of quasi-uniqueness)