Representation of Integers in Subsequences of the Positive Integers by Binary Quadratic Forms

Abstract

We consider positive-definite primitive binary quadratic forms of fundamental discriminant d < 0; 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 ∈ R, then almost all R-representable positive integers are C-representable. We extend this result to certain subsequences of 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 → ∞) for the sum $$\sum\limits_{\mathop {n \leqslant r}\limits_{r\left( {n;C} \right) > 0} } {\frac{1}{{r\left( {n;C} \right)}}} ,$$ 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. Bibliography: 30 titles.