Class numbers of indefinite binary quadratic forms and the residual indices of integers modulo~$p$

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 179-199

Résumé

Let $h(d)$ be the class number of properly equivalent primitive binary quadratic forms $ax^2+bxy+cy^2$ with discriminant $d=b^2-4ac$. The behavior of $h(5p^2)$, where $p$ runs over primes, is studied. It is easy to show that there are few discriminants of the form $5p^2$ with large class numbers. In fact, one has the estimate $$ \#\bigl\{p\le x\mid h(5p^2)>x^{1-\delta}\bigr\}\ll x^{2\delta}, $$ where $\delta$ is an arbitrary constant number in $(0;1/2)$. Assume that $\alpha(x)$ is a positive function monotonically increasing for $x\to\infty$ and $\alpha(x)\to\infty$. If $$ \alpha(x)\le(\log x)(\log\log x)^{-3}, $$ then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved $$ \#\biggl\{p\le x\biggm|\biggl(\frac5p\biggr)=1,\ h(5p^2)>\alpha(x)\biggr\}\asymp\frac{\pi(x)}{\alpha(x)}. $$ It is also proved that for an infinite set of $p$ with $\bigl(\frac5p\bigr)=1$ one has the inequality $$ h(5p^2)\ge\frac{\log\log p}{\log_kp}, $$ where $\log_kp$ is the $k$-fold iterated logarithm ($k$ is an arbitrary integer, $k\ge3$). Results on mean values of $h(5p^2)$ are also obtained. Similar facts are true for the residual indices of an integer $a\ge2$ modulo $p$: $$ r(a,p)=\frac{p-1}{o(a,p)}, $$ where $o(a,p)$ is the order of $a$ modulo $p$.