Motivated by several constructions of permutation polynomials done by several authors (most notably by Zieve), we propose a unified treatment for a large set of classes of permutation polynomials of F q . Our approach yields a recipe for constructing several new and old classes of permutation polynomials of F q .

ABSTRACT. Let φ : [0, ∞) → R and let y 0 be a non-negative constant. Let (λ n ) n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let (r n ) n∈N be a complex sequence, and c a real number. Assume that φ is square-integrable on [0, y 0 ] and for y ≥ y 0 , φ can be expressed as φ(y) = c + ℜ λn≤X r n e iλny + E(y, X),We prove that, under certain assumptions on the exponents λ n and the coefficients r n , φ(y) is a B 2 -almost periodic function and thus possesses a limiting distribution. Also if {λ n } n∈N is linearly independent over Q, we explicitly calculate the Fourier transform of the limiting distribution measure. Moreover, we prove general versions of the above results for vector-valued functions. Finally, we illustrate some applications of our general theorems by applying them to several classical error terms which occur in prime number theory. Examples include the error term in the prime number theorem for an automorphic L-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

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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.

We count permutation polynomials of F q which are sums of m + 1 ( 2) monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.