On integers with a special divisibility property
In this note, we study those positive integers which are divisible by , where is the Carmichael function.
On irregularities in the graph of generalized divisor functions
On isolated, respectively consecutive large values of arithmetic functions
On Mirsky's generalisation of a problem of Evelyn-Linfoot and Page in additive theory of numbers.
On oscillations in the additive divisor problem, 1
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