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ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of the BombieriVinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.

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We prove that the range of exponents in Mockenhaupt's restriction theorem for Salem sets [12], with the endpoint estimate due to Bak and Seeger [1], is optimal.Mathematics Subject Classification: 28A78, 42A32, 42A38, 42A45Date: May 30, 2013 (revised).

Let Q be an infinite subset of Z, let Ψ : Z → [0, ∞) be positive on Q, and let θ ∈ R. DefineWe prove a lower bound on the Fourier dimension of E(Q, Ψ, θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of E(Q, Ψ, θ) in new cases. We also prove a higher-dimensional analog of our result.