The Average Number of Divisors of the Output of a Quadratic Polynomial

Abstract: Let $d(n)$ count the number of divisors of $n$. In 1963, Hooley studied partial sums $n < X$ of $d(n^2+h)$ and showed that the result was asymptotic to $c X \log X + c' X + O(X^{8/9})$ as $X$ tends to infinity (assuming $h$ not a negative square). In other words, the irreducible polynomial $Q(x) = x^2 + h$ has outputs with, on average, $\sim \log x$ many divisors. Hooley's error bound was improved by Bykoskii in 1987 to $O(X^{2/3})$ using the spectral theory of automorphic forms. This talk describes a new proof of Bykovskii's result in a new framework, now using Dirichlet series and automorphic forms of half-integral weight. This new framework has limitations but is also quite flexible. To demonstrate this, we develop in tandem counts for the average number of divisors of $Q(x,y) = x^2+y^2+h$ for $x^2+y^2+h < X$.

number theory

Audience: researchers in the topic

London number theory seminar

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