Ronald J. Evans

The determination of Gauss sums

The first equality in (1.1) is a simple consequence of the fact that the sequence {n}9 1 < n < p — 1, runs through the set of kth power residues (mod/?) exactly k times. The primary purpose of this paper is to survey the present knowledge on the values of the Gauss sums §(k) and G(x), and to convey some of the principal ideas used in their determinations. We also briefly discuss more general Gauss sums. We begin by making some elementary remarks about the values of Gauss sums. It is easily verified by direct multiplication that, for nonprincipal x>

Sums of gauss, jacobi, and jacobsthal☆

wherep is a prime withp = 1 (mod k). The evaluation of G, was first achieved by Gauss. The sums Gk for k = 3,4, 5, and 6 have also been studied. It is known that G, is a root of a certain irreducible cubic polynomial. Except for a sign ambiguity, the value of G4 is known. See Hasse’s text [24, pp. 478-4941 for a detailed treatment of G, and G, , and a brief account of G, . For an account of G, , see a paper of E. Lehmer [29]. In Section 3, we shall determine G, (up to two sign ambiguities). Using our formula for G, , the second author [18] has recently evaluated G,, (up to four sign ambiguities). We shall also evaluate G, , G,, , and Gz4 in terms of G, . For completeness, we include in Sections 3.1 and 3.2 short proofs of known results on G, and G 4 ; these results will be used frequently in the sequel. (We do not discuss G, , since elaborate computations are involved, and G, is not needed in the sequel.) While evaluations of G, are of interest in number theory, they also have

Multidimensional q -beta integrals

A multidimensional extension of a q-beta integral of Andrews and Askey is evaluated. As an application, a short new proof of an important q-Selberg integral formula is given.

On the mod p2 determination of ((p − 1)4(p − 1)2)

Abstract The Gross-Koblitz formula and a formula of Diamond are used to prove the congruence A= 1+ 2 p−1 −1 2 2a− p 2a ( mod p 2 ) (p a prime number ≡ 1 (mod 4), p = a2 + b2 (a, b ∈ Z , a ≡ 1 (mod 4))), proposed by F. Beukers which refines the well-known congruence A ≡ 2a (mod p) for the binomial coefficient A= p−1 2 p−1 4

Silverman's Game on Intervals

(1979). Silvermans Game on Intervals. The American Mathematical Monthly: Vol. 86, No. 4, pp. 277-281.

Pure Gauss sums over finite fields

New classes of pairs e,p are presented for which the Gauss sums corresponding to characters of order e over finite fields of characteristic p are pure, i.e., have a real power. Certain pure Gauss sums are explicitly evaluated. §

Hypergeometric _3F_2(1/4) evaluations over finite fields and Hecke eigenforms

Let H denote the hypergeometric 3 F 2 function over F p whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating H at the argument 1/4. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Onos result by evaluating an infinite family of 3 F 2 (1/4) over F q in terms of Jacobi sums. Here we present five new 3 F 2 (1/4) over F q (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between 3 F 2 (1/4) and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.

CONGRUENCES FOR SUMS OF POWERS OF KLOOSTERMAN SUMS

The nth power-moments Sn of classical Kloosterman sums (mod p) are known explicitly only for n ≤ 6. In 2002, we conjectured formulas for Sn (mod 4) for each n > 1, valid for all primes p > n. Here we prove these formulas, and give conjectural congruences for Sn modulo some higher powers of 2 for a few small values of n. For example, we conjecture that if p ≡ 17 (mod 120), so that p = 3s2 + 5t2, then

Generalized Lambert series

Let q = exp(2πiτ) with Im τ > 0, so 0 < |q| < 1. For any positive integer n, define