On some complex explicit formulae connected with the Mobius function. II
On some complex explicit formulae connected with the Mobius function. I
On Some Connections Between Zeta-Zeros and Square-Free Integers.
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 Extremal Problems of Landau
2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.The prime number theorem with error term presents itself as &pi'(x) = ∫2x [dt/ logt] + O ( x e- K logL x). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial 2/3 (1+cos x)2 = 1+4/3 cosx +1/3 cos2x. Landau considered more general positive definite nonnegative...
On some inequalities concerning .
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 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