On some modifications of two theorems of Erdős
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 some tauberian theorems related to the prime number theorem
On Squarefree Numbers in Arithmetic Progressions.
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 Involving Reciprocials Of Certain Arithmetical Functions
On Sums of Large Differences between Consecutive Primes.
On sums of Ramanujan sums
On sums of seven cubes of almost primes
On Sums of Two Coprime k-th Powers.
On sums of two cubes: an Ω₊-estimate for the error term
The arithmetic function counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of leads in a natural way to a certain error term which is known to be in mean-square. In this article it is proved that as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently...
On sums of two kth powers: a mean-square asymptotics over short intervals
On sums of two kth powers: an asymptotic formula for the mean square of the error term
On sums over primes
On the additive complements of the primes and sets of similar growth