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Sums of Five, Seven and Nine Squares

Authors:

Institution:

(1) Institute of Information and Mathematical Sciences, Massey University—Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand

Abstract:

Let r _k(n) denote the number of representations of an integer n as a sum of k squares. We prove that

$\begin{gathered} r_5 \left( n \right) = r_5 \left( {n\prime } \right)\left {\frac{{2^{3\left\lfloor {\lambda /2} \right\rfloor + 3} - 1}}{{2^3 - 1}} - \varepsilon _5 \left( {n\prime } \right)\frac{{2^{3\left\lfloor {\lambda /2} \right\rfloor } - 1}}{{2^3 - 1}}} \right] \hfill \\ {\text{ }} \times \mathop \prod \limits_p \left {\frac{{p^{3\left\lfloor {\lambda p/2} \right\rfloor + 3} - 1}}{{p^3 - 1}} - p\left( {\frac{{n\prime }}{p}} \right)\frac{{p^{3\left\lfloor {\lambda p/2} \right\rfloor } - 1}}{{p^3 - 1}}} \right], \hfill \\ \end{gathered}$

where

$\varepsilon _5 \left( {n\prime } \right) = \left\{ \begin{gathered} {\text{0 if }}n\prime \equiv 1 \left( {\bmod {\text{ }}8} \right) \hfill \\ 4{\text{ if }}n\prime \equiv 2{\text{ or 3 }}\left( {\bmod {\text{ 4}}} \right) \hfill \\ 16/7{\text{ if }}n\prime \equiv 5{\text{ }}\left( {\bmod {\text{ 8}}} \right). \hfill \\ \end{gathered} \right.$

Here n = 2 prod

_p p _p is the prime factorisation of n, n prime

is the square-free part of n, the products are taken over the odd primes p, and ( $\frac{n}{p}$ ) is the Legendre symbol.Some similar formulas for r ₇(n) and r ₉(n) are also proved.

Keywords:

Hecke operator modular forms of half integer weight sums of squares

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