The Diaconu–Goldfeld–Hoffstein secondary-term conjecture for quadratic Dirichlet moments

Let $r\ge 4$ and let $N\ge 1$ be an integer. Choose a real number $\Theta$ satisfying

$$(N+1)^{-1}<\Theta<N^{-1}.$$

For the quadratic Dirichlet moment $M_r(D)$, let $Q_{n,r}(x)$ denote polynomials.

Diaconu–Goldfeld–Hoffstein conjecture. As $D\to\infty$,

$$M_r(D)=\sum_{n=1}^{N}D^{\frac12+\frac{1}{2n}}Q_{n,r}(\log D)+O\!\left(D^{(1+\Theta)/2}\right),$$

for some polynomials $Q_{n,r}(x)$.

The conjecture refines the usual main-term asymptotic by predicting a sequence of secondary terms. Evidence is both theoretical and numerical, but the full expansion over the rationals remains open.