Bombieri's mean value theorem

Simple Zeros of the Riemann Zeta-Function

“…The main term arises from q 1; the case 1 < q < h is treated using Siegel's Theorem, and the case h < q < y depends on an identity for A n and the large sieve. This is analogous to the proofs of Bombieri's Theorem given by Gallagher [7] and Vaughan [17].…”

New equidistribution estimates of Zhang type

“…In proving these estimates, we will rely on the following general "bilinear" form of the Bombieri-Vinogradov theorem (the principle of which is due to Gallagher [22] and Motohashi [42]). Theorem 2.9 (Bombieri-Vinogradov theorem).…”

On power-free numbers and polynomials. I.

“…Then, bringing in real numbers λ ά with the feature that λ ι = l and l d = 0 for i/>z, we use the non-negative function ( £ λ α ) 2 The discussion leading to (21) ensures that the implicit condition l' d >0 is retained. 4 ) It would be premature to weaken the conditions of summation in the first sum below without using the modulus sign, since we cannot yet be entirely sure that Ti (k) is positive.…”