Vol. 16, No. 1, 1966

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ISSN: 0030-8730
Some averages of character sums

Herbert Walum

Vol. 16 (1966), No. 1, 189–192
Abstract

Let χ and ψ be nonprincipal characters mod p. Let f be a polynomial mod p and let a1,,ap be complex constants. We will assume aj = ak for j k(p), and thus have an defined for all n. Define

    ∑
S =    arχ(f(r))
r
(1)

and

       ∑
Jn(c) =    ψ(r)χ (rn − c),
r
(2)

where the variables of summation run through a complete system of residues mod p.

The averages in question are

     p−1
A1 = ∑  |Jn(a)|2
a=1
(3)

and

     ∑
A2 =    |S |2,
(4)

where the sum in (4) is over the coefficients mod p of certain fixed powers of the variables in f. Exact formulae for A1 will be obtained in all cases, and for A2 in an extensive class of cases.

Specifically, the following theorems are true.

Mathematical Subject Classification
Primary: 10.41
Milestones
Received: 21 November 1963
Revised: 16 June 1964
Published: 1 January 1966
Authors
Herbert Walum