Angel V. Kumchev

Mean values of Dirichlet polynomials and applications to linear equations with prime variables

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to additive problems with prime variables. In particular, we are able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of the solutions of linear equations with prime variables.

On the Waring-Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

We investigate exceptional sets in the Waring-Goldbach problem. For example, in the cubic case, we show that all but O(N 79/84+ǫ ) integers subject to the necessary local conditions can be repre- sented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.

On the waring-goldbach problem for seventh powers

We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of 46 seventh powers of prime numbers.

On the Waring–Goldbach problem for eighth and higher powers

Recent progress on Vinogradovs mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function

On the distribution of prime numbers of the form [ n c ]

H(k)

A BINARY ADDITIVE EQUATION INVOLVING FRACTIONAL POWERS

in the Waring--Goldbach problem. We obtain new results for all exponents

ON SUMS OF THREE SQUARES

k\ge 8

The distribution of prime ideals of imaginary quadratic fields

, and in particular establish that

ON SUMS OF FOUR SQUARES OF PRIMES

H(k)\le (4k-2)\log k+k-7

A hybrid of two theorems of Piatetski-Shapiro

when