Bombieri's theorem in short intervals. II

Localised Bombieri–Vinogradov theorems in imaginary quadratic fields

“…In the rational case Timofeev has given a Bombieri-Vinogradov type theorem with a very general convolution, though not one with as long a summation of an unknown function such as h. Theorem 4 both generalises and extends a result due to Wu [31], though we cannot base our arguments, as he does, on [24].…”

“…His work [29] not only has stronger results and applies to more general functions than earlier papers but it also has aspects that enable the present generalisation. For instance, it makes no use of either the Pólya-Vinogradov Theorem or approximate functional equation for appropriate L-functions, both seen in [24]. The complication of detail of the proof in the present paper over [29] comes from the need (see Section 10) to introduce smooth functions.…”

“…Results for rational primes in short intervals have been given by Jutila [18], Huxley & Iwaniec [16], Ricci [27], Perelli, Pintz and Salerno [24,25], Zhan [32], and Timofeev [29].…”

On irreducible polynomials of small height over finite fields

“…We note that of course there is no particular significance in N -N ~/2, the only fact we really need is log N = O(log(N -M))). Actually it is an essentially relaxed version of those results that allow us to take Q = N ~ (with ~ = 1/40 and 7 = 1/30 in [2] and [8] respectively) and also allow us to take an arbitrarily large negative power of logarithm in the right-hand side of the bound.…”

Small gaps between products of two primes

“…The method of proof of both works [24] and [30] uses Heath-Brown's identity, therefore the analogue of (1.28) can be proved mutatis mutandis for E 2 -numbers as well. Accordingly, we will prove Theorem 4.…”