On The Simultaneous Distribution Of The Fractional Parts Of Different Powers Of Prime Numbers

In this paper we study the distribution modulo 1 of the sequence of vectors(p^α1 ... , p^αk), where k≥2 is a fixed integer andα₁,…,αk are fixed real numbers lying in the interval(s, s + 1) and p runs over the set of prime numbers.In Chapter one, we consider the simultaneous distribution of the fractional parts of different powers of prime numbers with s = 1. It suffices to bound the following Weyl exponential sumWhen 1＜αk＜…＜α₁＜2, we mainly use van der Corput method to estimate the above Weyl exponential sum, obtained theorem 1 and theorem 2 in this article.In Chapter two, we consider the simultaneous distribution of the fractional parts of different powers of prime numbers with s sufficiently large. It suffices to bound the following Weyl exponential sumWhen s＜α_k＜…＜α₁＜s + 1 (s is a sufficiently large integer), We mainly use Vinogradov method to estimate the above Weyl exponential sum, obtained theorem 3 in this article.Keywords: exponential sum; simultaneous distribution; van der Corput method; Vinogradov method.