A SURVEY ON PURE AND MIXED EXPONENTIAL SUMS MODULO PRIME POWERS TODD COCHRANE AND ZHIYONG ZHENG 1. Introduction In this paper we give a survey of recent work by the authors and others on pure and mixed exponential sums of the type p m  x=1 ep m(f (x)), p m  x=1 χ(g(x))ep m(f (x)), where p m is a prime power, ep m(·) is the additive character ep m(x)= e 2πix/p m and χ is a multiplicative character (mod p m ). The goals of this paper are threefold; first, to point out the similarity between exponential sums over finite fields and exponential sums over residue class rings (mod p m ) with m ≥ 2; second, to show how mixed exponential sums can be reduced to pure exponential sums when m ≥ 2 and third, to make a thorough review of the formulae and upper bounds that are available for such sums. Included are some new observations and consequences of the methods we have developed as well as a number of open questions, some very deep and some readily accessible, inviting the reader to a further investigation of these sums. 2. Pure Exponential Sums We start with a discussion of pure exponential sums of the type (2.1) S(f,q)= q  x=1 eq (f (x)) where f (x) is a polynomial over Z, q is any positive integer and eq (f (x)) = e 2πif (x)/q . These sums enjoy the multiplicative property S(f,q)= k  i=1 S(λi f,p m i i ) where q =  k i=1 p m i i and ∑ k i=1 λi q/p m i i = 1, reducing their evaluation to the case of prime power moduli. For the case of prime moduli we have the fundamental result of Weil [92] that for any polynomial f (x) of degree d not divisible by p, there exists a set of d − 1 complex numbers ω1,...ω d−1 , each of modulus √ p such that (2.2) S(f,p)= ω1 + ω2 + ··· + ω d−1 . Date : September 8, 2009. 1991 Mathematics Subject Classification. 11L07;11L03. Key words and phrases. exponential sums. The first author wishes to thank Tsinghua University, Beijing, P.R.C., and the National Center for Theoretical Sciences, Hsinchu, Taiwan, for hosting his visits and supporting this work. The second author was supported by the N.S.F. of the P.R.C. for distinguished young scholars. 1