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Some relations between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$

Volume 175 / 2016

Zhi-Hong Sun Acta Arithmetica 175 (2016), 269-289 MSC: 11D85, 11E25, 30B10, 33E20. DOI: 10.4064/aa8418-5-2016 Published online: 5 August 2016

Abstract

For given positive integers $a,b,c,d$ and $n$ let $N(a,b,c,d;n)$ be the number of representations of $n$ as $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ as $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). By using Ramanujan’s theta functions we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$.

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