Lower Bounds on Odd Order Character Sums†

Abstract

Abstract.A classical result of Paley shows that there are infinitely many quadratic characters χ (mod q) whose character sums get as large as √ q log log q; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.

Large Values of for Th Order Characters and Applications to Character Sums

“…under the assumption of GRH. Shortly afterwards, Goldmakher and Lamzouri [10] proved that this bound is, in fact, optimal. More precisely, they showed that for any > 0 and any fixed odd integer g 3, there exist arbitrarily large q and primitive characters χ (mod q) of order g satisfying M(χ) g, √ q(log log q) (g/π ) sin (π/g)− .…”

“…Let z = (log Q) 10 . Then it follows, from Proposition 2.4, that, for all but at most Q 1/4 fundamental discriminants d with 0 < εδd < Q…”

“…After developing their ideas further, Goldmakher [9] showed that these bounds hold with the exponent instead of , and hence under the assumption of GRH. Shortly afterwards, Goldmakher and Lamzouri [10] proved that this bound is, in fact, optimal. More precisely, they showed that for any and any fixed odd integer , there exist arbitrarily large and primitive characters of order satisfying …”

On character sums and exponential sums over generalized arithmetic progressions

“…Moreover, it would also be nice to be able to construct a family of characters χ (mod q) so that S(χ, r) is large, as is done by Paley [13] in the case r = 1. See also [8] and [10] for an improvement of Paley's result.…”

Large even order character sums

“…No other unconditional lower bounds were proved until quite recently, when the authors [3] established that there are arbitrarily large q and primitive characters χ (mod q) of fixed odd order g such that…”