On almost universal mixed sums of squares and triangular numbers

Abstract

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.