No abstract

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written aswhere p j are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.

We study the exponential sum involving multiplicative function f under milder conditions on the range of f , which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of additively twisted coefficients of automorphic L-function on GL m (m 4), uniformly in the additive character.

In 1995, Sankaranarayanan studied a divisor problem related to the Epstein zeta-function. By the theory of modular forms and the Riemann zeta-function, we are able to improve his result for a number of cases.

It is proved that each sufficiently large integer N ≡ 5 (mod24) can be written aswhere pj are primes. This result, which is obtained by an iterative method and a hybrid estimate for Dirichlet polynomial, improves the previous results in this direction.