On sign-changes in the remainder-term of the prime-number formula, III
On some estimates involving the number of prime divisors of an integer
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 tauberian theorems related to the prime number theorem
On Sums Involving Reciprocals of Certain Large Additive Functions
On sums involving reciprocals of certain large additive functions. II.
On Sums Involving Reciprocials Of Certain Arithmetical Functions
On Sums of Large Differences between Consecutive Primes.
On the Brun-Titchmarsh theorem.
On the Brun-Titchmarsh theorem
The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size when logx/logq ≥ 8 and is bounded....
On the comparative theory of primes
On the composite Lehmer numbers with prime indices, III
On the congruence , where q is a prime and f is a k-normal polynomial
On the coprimality of certain multiplicative functions
On the coprimality of values of Euler's function at consecutive integers.
On the density of certain sets in arithmetical semigroups
On the density of some sets of primes connected with cyclotomic polynomials
On the density of some sets of primes, I
On the density of some sets of primes, II