Effective estimates of exponential sums over primes

Abstract

The asymptotic behavior of the sum

$$S(x,\alpha ) = \sum\limits_{n \leqslant x} {\pi (n)e(n\alpha ),}$$

where α is real, e(α) = e^2πiα, and Λ is the von Mangoldt function, has been extensively studied by many authors. It plays a central role in Vinogradov’s solution of the 3-primes conjecture [10]. It is also a main tool in the study of the equidistribution of the sequence {pα, p prime} modulo 1.