Exponential sums with reducible polynomials

Exponential sums with reducible polynomials, Discrete Analysis 2019:15, 31 pp.

A sequence \((a_n)\) of real numbers in the interval \([0,1]\) is said to be equidistributed if for every subinterval \([a,b]\) of \([0,1]\), the proportion of the \(a_n\) that live in the interval \([a,b]\) tends to \(b-a\): that is, in a suitable sense, the distribution of the terms of the sequence converges to the uniform distribution on \([0,1]\). The concept of equidistribution is a central one in analytic number theory, and also in the subfield of combinatorics known as discrepancy theory.

A nice example of an equidistributed sequence is the Farey sequence, which is an enumeration of the rationals in \([0,1]\) that lists, for each positive integer \(n\) in turn, all the fractions \(a/n\) with \((a,n)=1\) in increasing order. The sequence begins
\[0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, \dots.\]
It was proved to be equidistributed in a 1949 paper of Eric Harold Neville.

In 1963, Christopher Hooley wrote a paper about sums of the form
\[\sum_{n\leq x}d(n^2+a)\]
where \(d(m)\) is the number of divisors of \(m\). He noted that if \(a=-k^2\) for a positive integer \(k\), so that \(n^2+a=(n+k)(n-k)\), then the character of the problem changed substantially and became a question that had essentially been solved by different techniques, so he concentrated on the case where the polynomial \(n^2+a\) is irreducible. As part of his investigation he found himself considering the sequence of rationals \(\nu/k\) with \(0<\nu<k\), arranged in increasing order for each \(k\) in turn, with the property that \(\nu^2+a\equiv 0\) (mod \(k\)), and proved that this sequence was equidistributed. In a later paper, he generalized this result to arbitrary irreducible polynomials: that is, he showed that if \(f\) is any such polynomial, then the sequence of rationals \(\nu/k\) such that \(f(\nu)\equiv 0\) (mod \(k\)) is equidistributed.

The main tool for proving equidistribution is exponential sums, and in particular a well-known criterion of Weyl, which roughly speaking says that a finite sequence is asymptotically equidistributed if it has no large Fourier coefficients. More precisely, if \((a_n)\) is a sequence that one would like to prove equidistributed, then Weyl’s criterion requires the sum \(x^{-1}\sum_{n\leq x}e(ma_n)\) to tend to zero for each fixed \(m\) as \(x\) tends to infinity. Here \(e(\alpha)\) is the usual shorthand for \(\exp(2\pi i\alpha)\).

This paper extends Hooley’s results to certain reducible polynomials. The authors obtain good estimates for the relevant exponential sums for polynomials that are products of linear factors, and also for the polynomial \(n(n^2+1)\). They identify main terms in each case, which is unusual for results about sums of this kind. As they state, it is probably possible to use their methods to extend the result about \(n(n^2+1)\) to any product of a linear and a quadratic polynomial, but there would be extra difficulties to contend with.