English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

This article discusses the classical problem of how to calculate $r_n(m)$,the number of ways to represent an integer $m$ by a sum of $n$ squares from acomputational efficiency viewpoint. Although this problem has been studied ingreat detail, there are very few formulas given for the purpose of computing$r_n(m)$ quickly. More precisely, for fixed $n$, we want a formula for $r_n(m)$that computes in log-polynomial time (with respect to $m$) when the primefactorization of $m$ is given. Restricting to even $n$, we can view$\theta_n(q)$, the theta function associated to sums of $n$ squares, as amodular form of weight $n/2$ on $\Gamma_1(4)$. In particular, we show that foronly a small finite list of $n$ can $\theta_n$ be written as a linearcombination consisting entirely of Eisenstein series and cusp forms withcomplex multiplication. These are the only $n$ that give rise to "elementary"formulas for $r_n(m)$, i.e. formulas such that for a prime $p$, $r_n(p)$ can becalculated in $\cO(\log(p))$-time. Viewing $\theta_n(q)$ as one of the simplerexamples of modular forms that are not strictly Eisenstein, this resultmotivates the necessity of a log-polynomial time algorithm that directlycalculates the Fourier coefficients of modular forms in the generic situationwhen there is no such formula, as described in Couveignes and Edixhoven'sforthcoming book (for level 1 cases) and Peter Bruin's Ph.D. thesis (for higherlevel, including 4).

Finding elementary formulas for theta functions associated to even sums of squares