Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of L-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

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We give a new bound for the large sieve inequality with power moduli q k that is uniform in k. The proof uses a new theorem due to T. Wooley from his work on efficient congruencing.

We show that for every xed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and suciently large, and let Q 1 = Q 2 := n 1/2 (log n) −ϑ and Q 3 := (log n) θ . Then for all q 3 Q 3 , all reduced residues a 3 mod q 3 , almost all q 2 Q 2 , all admissible residues a 2 mod q 2 , almost all q1 Q1 and all admissible residues a1 mod q1, there exists a representation n = p1 + p2 + p3 with primes pi ≡ ai (qi), i = 1, 2, 3.

We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem.We present two selected applications: First, we prove a new kth derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree k ≥ 4.1991 Mathematics Subject Classification. Primary 11L15, Secondary 11J54.