Abstract:We improve the range of p (Z d )-boundedness of the integral k-spherical maximal functions introduced by Magyar. The previously best known bounds for the full k-spherical maximal function require the dimension d to grow at least cubically with the degree k. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we improve upon bounds in the authors' previous work [1] on the ergodic Waring-Goldbach problem, which is the analogous problem of p (Z d )-boundedness of the k-spherical maximal functions whose coordinates are restricted to prime values rather than integer values.

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In this paper we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. Using new exponential sums in tandem with a sieve method we are able to provide stronger "minor arc" estimates than previous authors, thereby improving the saving obtained in the exponent by a factor 8/7.

Let c q (n) denote the Ramanujan sum modulo q, and let x and y be large reals, with x = o(y). We obtain asymptotic formulas for the sums n≤y q≤x c q (n) k (k = 1, 2).

We obtain new results concerning the simultaneous distribution of prime numbers in arithmetic progressions and in short intervals. For example, we show that there is an absolute constant δ > 0 such that 'almost all' arithmetic progressions a mod q with q ≤ x δ and (a, q) = 1 contain prime numbers from the interval (x − x 0.53 , x].