Exponential Sums Involving the k-th Largest Prime Factor Function

New Bounds for Gauss Sums Derived From k-th Powers, and for Heilbronn’s Exponential Sum

where p is prime, e(x) = exp(2πix), and ep(x) = e(x/p). In each case we shall assume that p | / a unless the contrary is explicitly stated. Gauss sums arise in investigations into Waring’s problem, and other additive problems involving k-th powers. Although they are amongst the simplest complete exponential sums, the question as to their true order of magnitude is far from being resolved. We re...

On sums involving reciprocals of the largest prime factor of an integer II

EXPONENTIAL SUM ESTIMATES OVER SUBGROUPS AND ALMOST SUBGROUPS OF Zq, WHERE q IS COMPOSITE WITH FEW PRIME FACTORS

In this paper we extend the exponential sum results from [B-K] and [B-G-K] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q, for any given δ > 0. The method consists in first establishing a ‘sum-product theorem’ for general subsets A of Z. If q...

Exponential Sums 1 Statement of Main Result

Let p be a prime number. We consider exponential sums which are expressions of the form S f = a mod p e 2πi f (a) p where f (x) ∈ Z [x] is a polynomial with integer coefficients. More generally, instead of the prime field Z /p we may work over any finite field K with q = p k elements. Let tr : K → Z /p denote the trace operator from K to its prime field. Then, an exponential sum over K is any e...

Exponential Sums, Cyclic Codes and Sequences: the Odd Characteristic Kasami Case

Let q = p with n = 2m and p be an odd prime. Let 0 ≤ k ≤ n − 1 and k 6= m. In this paper we determine the value distribution of following exponential(character) sums