Hilbert's Inequality

Short intervals asymptotic formulae for binary problems with primes and powers, II: density 1

“…By symmetry we can integrate over [0, ξ]. We use Corollary 2 of Montgomery and Vaughan [6] (see also the remark after their statement) with T = ξ, a r = exp(−r ℓ /N) and λ r = 2πr ℓ thus getting …”

On the fractional parts of $x/n$ and related sequences. I

“…The following theorem (together with the observations made in Lemmas 2, 3,4) shows that the (R, Xj asymptotic distribution function modulo 1 of x\n can exist under very general conditions provided that y is not too small compared with x.…”

“…4 at every a at which F is continuous, the counting function (1.6) A([a,p),/c,(^)) = Card {n: 1 ^ n ^ /c, a ^ {x^} < (3} being here defined for all real numbers a and (3. The conditions (1.4) mean that F is continuous at 0 and 1, and imply that F is constant on the intervals (-oo, 0] and [1, oo).…”

The Mean Square of the Logarithm of the Zeta-Function

“…As we shall see, the proof of Theorem 1 follows from an application of Hilbert's inequality (in the version of Montgomery-Vaughan [12]) and the convergence of the…”

“…In order to evaluate the integral of j log ' itj 2 , we use the following lemma, due to Montgomery and Vaughan (see [11, (28) p. 140] and [12]). Lemma 1.…”