Andrew Granville

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20242026

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math.NT2026

Mixed character sums modulo prime powers

Todd Cochrane, Andrew Granville

We obtain explicit estimates for the mixed character sum $S = S (χ, g, f, p^{m}) = \sum_{x = 1}^{p^{m}} χ (g (x)) e_{p^{m}} (f (x))$ , where $p^{m}$ is a prime power, $χ$ is a multiplicative character…

math.NT2026

Large values of exponential sums with multiplicative coefficients

Andrew Granville, Youness Lamzouri

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n \leq x} f (n) e (n α)$ where $f$ is a $1$ -bounded multiplicative function and $α \in R$ ,…

math.NT2026

Mixed incomplete character sums of rational functions with smooth moduli

Todd Cochrane, Andrew Granville, Junren Zheng

Let $χ = χ_{q}$ be a primitive character mod $q$ and fix $Δ > 0$ . In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M < n \leq M + N} χ (n)$ in intervals of length $N=q^Δ…

math.NT2025

Degrees of points on irreducible hypersurfaces

Lea Beneish, Andrew Granville

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $Q$ . We give a new explicit proof that t…

math.NT2024

Primes with small primitive roots

Kevin Ford, Mikhail R. Gabdullin, Andrew Granville

Let $δ (p)$ tend to zero arbitrarily slowly as $p \to \infty$ . We exhibit an explicit set $S$ of primes $p$ , defined in terms of simple functions of the prime factors of $p-…