On the Karatsuba divisor problem

On Dirichlet ' S Divisor Problem

tact, whereas those for a = 0.02 imply only 25% more frequent contacts than the all-ornone hypothesis but with rather high though rapidly diminishing attack rates. 10 Schuman and Doull, Amer. Jour. Pub. Health, 30, Supplement to March, 1940, p. 21, state: "From these estimates of carrier prevalence and froni the average annual increment in Shick-negatives, an estimate may be made of the number ...

The Circle and Divisor Problem

Generalizing the Titchmarsh Divisor Problem

Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem ∑ p≤x d(p− a) = cx + O ( x log log x log x ) where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: ∑ p≤x p≡a mod k d ( p− a k ) = Ckx + O ( x l...

Generalized divisor problem

In 1952 H.E. Richert by means of the theory of Exponents Pairs (developed by J.G. van der Korput and E. Phillips ) improved the above O-term ( see [8] or [4] pag. 221 ). In 1969 E. Krätzel studied the three-dimensional problem. Besides, M.Vogts (1981) and A. Ivić (1981) got some interesting results which generalize the work of P.G. Schmidt of 1968. In 1987 A.Ivić obtained Ω-results for ∫ T 1 ∆ ...

On the Riemann Zeta-function and the Divisor Problem Iii

Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of |ζ(1 2 + it)|. If E * (t) = E(t) − 2π∆ * (t/2π) with ∆ * (x) = −∆(x) + 2∆(2x) − 1 2 ∆(4x) and we set T 0 E * (t) dt = 3πT /4 + R(T), then we obtain R(T) = O ε (T 593/912+ε), T 0 R 4 (t) dt ≪ ε T 3+ε , and T 0 R 2 (t) dt = T 2 P 3 (log T) + O ε (T 11/6+ε), whe...