On periodic mod p sequences and G-functions (On a conjecture of Ruzsa).
On power-free numbers and polynomials. I.
On power.free numbers and polynomials. II.
On prime divisors of Mersenne numbers
On primitive divisors of Mersenne numbers
On primitive lattice points in planar domains
On sequences of ±1's defined by binary patterns [Book]
On short sums of certain multiplicative functions.
On some divisor problems
On some estimates involving the number of prime divisors of an integer
On some exponential sums involving divisor functions.
On some mean value results for the zeta-function in short intervals
Let denote the error term in the Dirichlet divisor problem, and let E(T) denote the error term in the asymptotic formula for the mean square of |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and , then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dtare...
On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...
On some sums involving the largest prime divisor of n
On some sums involving the largest prime divisor of n, II
On Summatory Functions of Additive Functions and Regular Variation
On Sums Involving Reciprocals of Certain Large Additive Functions
On sums involving reciprocals of certain large additive functions. II.
On sums involving reciprocals of the largest prime factor of an integer II
On sums of Ramanujan sums