5,111 research outputs found
On the Waring--Goldbach problem for eighth and higher powers
Recent progress on Vinogradov's 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 in the Waring--Goldbach problem.
We obtain new results for all exponents , and in particular establish
that when is large, giving the first improvement
on the classical result of Hua from the 1940s
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
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