Representations of integers belonging to subsequences of the positive integers by binary quadratic forms

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 15, Tome 254 (1998), pp. 165-191

Résumé

We consider positive-definite primitive binary quadratic forms of fundamental discriminant $d0$; $R$ is the genus and $C$ is the class of such forms. We obtain asymptotics for the sum of absolute values of the Fourier coefficients for the Hecke eigenforms of weight 1 and of dihedral type. In an earlier paper (Zap. Nauchn. Semin. POMI, 226 (1996)), the author showed that if $C\in R$, then almost all $R$-representable positive integers are $C$-representable. We extend this result to certain subsequences of $\mathbb N$ such as $\{a_n=p_n+l\}$, $\{a_n=n(n+1)\}$, etc. Finally, for certain genera $R$ with class number greater than one, we prove an asymptotics $(x\to\infty)$ for the sum $$ \sum_{\substack{n\le x\\ r(n;C)>0}}\frac1{r(n;C)}, $$ where $C$ is a class in $R$ and $r(n;C)$ is the number of representations of a positive integer $n$ by the class $C$.