%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% BEGIN OF AMSTEX FILE %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input amstex.tex \magnification=1200 \documentstyle{amsppt} \refstyle A \widestnumber\key{10} \topmatter \title On some modular identities \endtitle \author Giuseppe Melfi \endauthor \address Dipartimento di Matematica, Universit\`a di Pisa, via Buonarroti 2, 56127 Pisa, Italy \endaddress \email melfi\@dm.unipi.it \endemail \abstract Using the theory of modular forms, we prove some arithmetical identities similar to certain convolution formulae for sums of divisor powers proved by Ramanujan in \cite6. In Theorem 1 we also prove a somewhat different formula involving an unusual multiplicative arithmetical function and containing an error term. \endabstract \endtopmatter \document \subheading{1. Introduction} \vskip 0.1cm Let $\sigma_m(n)$ denote the sum of the $m$-th powers of the positive divisors of $n,$ and let $\sigma_m(0)=\frac{1}{2}\zeta(-m)$ where $\zeta(s)$ is the Riemann zeta-function. In this paper, using the theory of modular forms, we prove seven identities of the following type: $$\sum_{k=0}^{[n/m]}\sigma_r(k)\sigma_s(n-mk)=P\sigma_{r+s+1}(n)+ Qn\sigma_{r+s-1}(n),\tag1 $$ which hold for every $n$ satisfying suitable congruences, for suitable integers $m\ge2$ and $r,s=1$ or 3, and for rationals $P$ and $Q$ (Theorem 2). We also prove a further identity similar to (1) but of a slightly different kind, namely $$ \sum\Sb k=0\\ k\equiv 1 \bmod 3\endSb^n\sigma_1(k)\sigma_1(n-k) = \frac19\sigma_{3}(n) \ \ \ \ \ \ \ \ \text{for every } n\equiv2\bmod3.$$ In a celebrated paper \cite6, Ramanujan, using elementary arguments, proved nine identities of the type (1) with $m=1.$ Ramanujan's nine identities can be also obtained in a natural way from the theory of modular forms for the full modular group (see \cite9). A short elementary proof of Ramanujan's identities is due to Skoruppa \cite{12}. We remark that one of the formulae we prove in Theorem 2, namely (10) below, is explicitly mentioned by Ramanujan himself in \cite6. Unfortunately, he never provided either of the two proofs he announced. The first proof of the formula (10) below was given by Masser (see \cite2) seventy years later. As far as we know, the other formulae proved in Theorem 2 appear to be new. In Theorem 1 we also provide, via modular forms, a formula for the case $r=s=1$ and any $m,$ which contains an error term. When the error term vanishes this formula yields special cases of (1), i.e. the identities (8), (11), (12), (13) and (14) below. We also give alternative proofs of the five identities (8)--(12). These proofs are based on certain formulae of Ramanujan, involving elliptic integrals of the first kind, contained in his Notebooks \cite7. This alternative method is likely to correspond to one of the proofs that Ramanujan had in mind for the identity (10). I am greatly indebted to Don Zagier and Umberto Zannier for their illuminating comments. In particular, I am pleased to thank Don Zagier for suggesting the proof of Theorem 1. I express my gratitude to Umberto Zannier for pointing out to me the interpretation of the identities of Ramanujan's type in terms of modular forms, as well as for his constant encouragement and for several helpful suggestions. \subheading{2. Notation and definitions} Let $F(a,b;c;x)={}_2F_1(a,b;c;x)$ denote the Gauss hypergeometric series: $$ F(a,b;c;x)=\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(c)_k}\frac{x^k}{k!}, $$ where $c\neq 0,-1,-2,\ldots$ and the Pochhammer symbols $(a)_k, (b)_k, (c)_k$ are defined by $$(a)_0=1,\,\,\,\,\,\,\,(a)_k=a(a+1)(a+2)\cdots (a+k-1)\,\,\,\,\, \hbox{for }k=1,2,3,\dots.$$ As is well-known, for $a=b=\frac{1}{2}, c=1$ and $01,$ the Eisenstein series $$E_{2k}(\tau):=1+\frac2{\zeta(1-2k)}\sum_{n=1}^{\infty}\sigma_{2k-1}(n)q^n$$ are modular forms of weight $2k$ for $\Gamma.$ The function $E_2(\tau):=1-24\sum_{n=1}^\infty\sigma_1(n)q^n$ is not a modular form, but is transformed under the action of $ SL(2,\Bbb Z)$ as follows: $$E_2\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}E_2(\tau)+\frac6 {\pi i}c(c\tau+d).$$ We shall also denote $E_{2k,m}(\tau)=E_{2k}(m\tau).$ For $k>1,$ the functions $E_{2k,m}$ are modular forms of weight $2k$ for $\Gamma_0(m).$ \subheading{3. Main results} We begin this section with the following theorem: \proclaim{Theorem 1} Let $m$ be a positive integer and define $\beta(m)=m^2\prod_{p|m}(1+p^{-2}).$ For every positive integer $n$ with $(m,n)=1,$ we have $$\sum_{k=0}^{[n/m]}\sigma_1(k)\sigma_1(n-mk)= \frac5{12\beta(m)}\sigma_3(n) -\frac1{4m}n\sigma_1(n)+O\left(n^{\frac32+\epsilon}\right). $$\bigskip \endproclaim \demo{Proof} From the above-mentioned modular properties of $E_2(\tau),$ it immediately follows that $G_m(\tau):=E_2(\tau)-mE_{2,m}(\tau)$ is a modular form of weight 2 for $\Gamma_0(m).$ Hence the function $F_m(\tau):=(G_m(\tau))^2-E_4(\tau)-m^2E_{4,m}(\tau)$ is a modular form of weight 4 for $\Gamma_0(m).$ Combining the Fourier expansions of $E_2$ and $E_4$ with the first of Ramanujan's nine identities \cite6: $$ \sum_{k=0}^n\sigma_1(k)\sigma_1(n-k)=\frac{5}{12}\sigma_3(n)-\frac{1}{2}n \sigma_1(n), \tag4 $$ we find that the $n$-th Fourier coefficient of $\frac{-1}{1152m}F_m(\tau)$ is $$c(n)=\frac{n}{4m}\sigma_1(n)+\frac{n}{4}\sigma_1^*\left(\frac{n}{m}\right)+ \sum_{0\leq k\leq n/m}\sigma_1(k)\sigma_1(n-mk) , $$ where $$\sigma_1^*\left(\frac nm\right):=\left\{\aligned &\sigma_1\left(\frac nm\right)\quad\text{for } m|n\\ &0 \quad\quad\quad\quad\text{otherwise.} \endaligned\right. \,\,\,\, $$ Notice that $c(n)$ is defined also for $(m,n)\neq 1.$ Let $r$ be any positive integer. Using modular properties of $E_2(\tau)$ and $E_4(\tau),$ we can find the expansions of $E_{4,r}$ and $F_m$ at the cusps. Let $(a,c)=1.$ In the usual topology of $\frak H\cup\{$cusps$\}$ (see \cite{5, p. 103--105}) we have, for $\varepsilon\in\frak H$, $\varepsilon\rightarrow0$, $$E_{4,r}\left(\frac ac+\varepsilon\right)\sim \frac{(c,r)^4}{r^4c^4\varepsilon^4},\,\,\,\,\,\,\,\,\,\,\,\,\, F_m\left(\frac ac+\varepsilon\right)\sim-\frac{2(c,m)^2}{mc^4\varepsilon^4}. $$ In fact if $b,d$ are integers such that $ad-bc=1,$ then $$ \xi:=-\frac dc-\frac1{\varepsilon c^2}\in\frak H,$$ $$\frac ac+\varepsilon =\frac{a\xi+b}{c\xi+d}, $$ $$ c\xi+d=-\frac1{\varepsilon c}.$$ Hence $$E_4\left(\frac ac+\varepsilon\right)=E_4\left(\frac{a\xi+b}{c\xi+d}\right)= (c\xi+d)^4E_4(\xi).$$ Since, for $\varepsilon\rightarrow 0$ in the usual topology of $\frak H\cup\{$cusps$\},$ \, $\text{Im }\xi\rightarrow+\infty,$ it follows that $E_4(\xi)\rightarrow1,$ i.e. $E_4(\frac ac+\varepsilon)\sim\frac1{\varepsilon^4c^4}.$ If $r$ is any positive integer, denoting $a'=ra/(r,c), $ $ c'=c/(r,c),$ $\varepsilon'=r\varepsilon,$ we have $(a',c')=1,$ whence $$E_{4,r}\left( \frac ac+\varepsilon\right)=E_4\left(\frac{a'}{c'}+ \varepsilon'\right)\sim\frac1{{{\varepsilon'}^4} {c'}^4}= \frac{(c,r)^4}{r^4c^4\varepsilon^4}.$$ In a similar manner one can find the asymptotic formula for $F_m(a/c+\varepsilon)$. With the same notation as above, for a modular form $f$ of weight $2k$ for $\Gamma_0(m)$ we have $$f\left(\frac{a}{c}\right)=\lim_{\tau\rightarrow i\infty} (c\tau+d)^{-2k}f\left(\frac{a\tau+b}{c\tau+d}\right)= \lim_{\varepsilon\rightarrow 0} (c\varepsilon)^{2k} f\left(\frac{a}{c}+\varepsilon\right) . $$ For $(a,c)=1,$ we have $E_{4,r}(a/c)=E_{4,r}(1/c)$ and $F_m(a/c)=F_m(1/c).$ This will allow us to simplify the study of the behaviour at the cusps in a sense that will be clear below. We now show that, for $r|m$ and $\alpha_m(r):=r^2\prod_{p|(r,m/r)}(1-p^{-2}),$ the modular form $$F_{0,m}(\tau):=F_m(\tau)+\frac{2m}{\beta(m)}\sum_{r|m}\alpha_m(r) E_{4,r}(\tau)=\sum_{n=0}^{\infty}c_{0,m}(n)q^n\tag{5}$$ is a cusp form for $\Gamma_0(m)$ \, (obviously of weight 4). To prove this we seek, for any $r|m$, a coefficient $u_r$ such that $F_m(\tau)+\sum_{r|m}u_rE_{4,r}(\tau)$ vanishes at every cusp. This can be done by looking at one prime number at a time. Assuming $m=p^\mu$ one has $$ F_{p^\mu}(\infty)=-2p^\mu, \,\,\,\,\,\,\,F_{p^\mu}\left(-\frac1{kp}\right)= -2p^{2+2v_p(k)-\mu},\,\,\,\,\,\,\,\,\, F_{p^\mu}(0)=-2p^{-\mu}, $$ $$ E_{4,p^i}(\infty)=1, \,\,\,\,\,\,\,\,\, E_{4,p^i}\left(-\frac1{kp}\right)=p^{4\min\{0,v_p(k)-i+1\}},\,\,\,\,\,\,\,\,\, \,E_{4,p^i}(0)=p^{-4i}, $$ where $v_p(k)$ is the exponent of the prime $p$ in the factorization of $k.$ As a set of representatives of the cusps for $\Gamma_0(m)$ we can take the cusps $\infty,$ $0,$ and $-1/kp$ for $k=1,2,\dots,p^{\mu-1}-1$ (see \cite{5, p. 107--108}). Therefore $F_{p^\mu}(\tau)+\sum_{i=0}^\mu x_iE_{4,p^i}(\tau)$, with suitable coefficients $x_i$ to be determined, is a cusp form if it vanishes at the above cusps. Notice that there are only $\mu-1$ distinct conditions at the cusps $-1/kp,$ since for $(h',p^\mu)=(k',p^\mu)$ the condition at $-1/h'p$ is the same as the condition at $-1/k'p.$ Thus one gets the following linear system of $\mu+1$ equations in the $\mu+1$ unknowns $x_0,x_1,\dots,x_\mu:$ $$\left\{\alignedat7 &x_0&&+p^{-4}x_1&&+p^{-8}x_2&&+\dots&&+p^{-4\mu+4}x_{\mu-1}&&+p^{-4\mu}x_\mu&& =2p^{-\mu}\\ &x_0&&+x_1&&+p^{-4}x_2&&+\dots&&+p^{-4\mu+8}x_{\mu-1}&&+p^{-4\mu+4}x_\mu&& =2p^{-\mu+2}\\ &\vdots && && && \cdots\cdots && && && \,\,\,\,\,\,\,\,\,\vdots \\ &x_0 &&+x_1&&+x_2&&+\dots &&+x_{\mu-1} &&+p^{-4}x_{\mu}&&=2p^{\mu-2}\\ &x_0 &&+x_1 &&+x_2&&+\dots && +x_{\mu-1} && +x_\mu && =2p^\mu, \endalignedat\right. $$ whose solution is $$x_0=\frac2{p^\mu+p^{\mu-2}}, \,\,\,\,\,\, x_\mu=\frac{2p^{\mu+2}}{p^2+1},$$ and for $i=1,\dots,\mu-1$ $$x_i=2\frac{p^{2i}-p^{2i-2}}{p^\mu+p^{\mu-2}}.$$ Denoting $u_{p^i}=x_i,$ we may express the above solution as $$u_r=\frac{2p^\mu}{\beta(p^\mu)}\alpha_{p^\mu}(r) \,\,\,\,\,\, \,\,\,\,\,\,\,\,\text{for each } r|p^\mu.\tag6$$ This proves that $F_{0,m}(\tau)$ defined by (5) is a cusp form when $m=p^\mu$. In the general case, for an arbitrary positive integer $m=p_1^{\mu_1}\cdots p_h^{\mu_h}$ one finds a similar system of $d(m)$ \, (the number of positive divisors of $m$) linear equations in the $d(m)$ unknowns $u_r$ for $r|m$, namely $$\sum_{r|m}\frac{(r,n)^4}{r^4}u_r=2\frac{n^2}{m}\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,{\text{for each }}n|m,$$ whose solution is $$u_r=\frac{2m}{\beta(m)}\alpha_m(r),$$ as is easily seen using (6) for each $p_j^{\mu_j}$ $(j=1,\dots,h).$ This proves that $F_{0,m}(\tau)$ is a cusp form for any $m$. Since the weight of the cusp form $F_{0,m}$ is 4, the $n$-th Fourier coefficient of $\frac{-1}{1152m}F_{0,m}$ is bounded by $O(n^{\frac32+\epsilon})$ (see for example \cite8), i.e. $$-\frac{1}{1152m}c_{0,m}(n)=c(n)-\frac{5}{12\beta(m)}\sum_{r|(m,n)} \alpha_m(r)\sigma_3\left(\frac{n}{r} \right)=O\left(n^{\frac32+\epsilon}\right)$$ for all $n.$ For $(m,n)=1$ this is our theorem.\qed\enddemo In some cases the error term vanishes, and this yields special cases of identities (1). In the following theorem we obtain eight identities. \proclaim{Theorem 2} If $n\equiv2\bmod 3,$ then $$ \sum\Sb k=0\\ k\equiv 1 \bmod 3\endSb^n\sigma_1(k)\sigma_1(n-k) = \frac19\sigma_{3}(n). \tag7$$ If $n$ is a positive odd integer, then $$ \sum_{k=0}^{[n/2]}\sigma_1(k)\sigma_1(n-2k) = \frac{1}{12}\sigma_{3}(n) -\frac{1}{8}n\sigma_1(n), \tag8$$ $$ \sum_{k=0}^{[n/2]}\sigma_1(k)\sigma_3(n-2k) = \frac{1}{48}\sigma_5(n) -\frac{1}{16}n\sigma_3(n), \tag9$$ $$\sum_{k=0}^{[n/2]}\sigma_3(k)\sigma_1(n-2k) = \frac{1}{240}\sigma_{5}(n), \tag10$$ $$\sum_{k=0}^{[n/4]}\sigma_1(k)\sigma_1(n-4k)= \frac{1}{48}\sigma_3(n) -\frac{1}{16}n\sigma_1(n). \tag11$$ If $n\not\equiv 0\bmod 3,$ then $$ \sum_{k=0}^{[n/3]}\sigma_1(k)\sigma_1(n-3k)=\frac{1}{24}\sigma_3(n) -\frac{1}{12}n\sigma_1(n).\tag12 $$ If $n\equiv8\bmod 16$ and $n\not\equiv0\bmod 5 ,$ then $$ \sum_{k=0}^{[n/5]}\sigma_1(k)\sigma_1(n-5k) = \frac5{312}\sigma_3(n) -\frac1{20}n\sigma_1(n). \tag13$$ If $n$ satisfies one of the following conditions: (i) $n\equiv2\bmod3$; (ii) $n\equiv1\bmod3$ and there exists a prime $p\equiv2\bmod 3$ such that $p|n$ but $p^2\nmid n$; \noindent then $$\sum_{k=0}^{[n/9]}\sigma_1(k)\sigma_1(n-9k) = \frac1{216}\sigma_3(n)-\frac1{36}n\sigma_1(n).\tag14 $$ \endproclaim \demo{Proof} Let $f(\tau)$ be a non-vanishing modular form of weight $2k$ for $\Gamma(m).$ We consider the Riemann surface $\Cal S$ associated with $\Gamma(m).$ We associate with $f$ the differential $\omega:=f(\tau)(d\tau)^k$ of weight $k$ on $\frak H$. Then $\omega$ corresponds to a differential $\omega^*$ of weight $k$ on $\Cal S$ (see also \cite{11}). We want to compare the order $n_p(f)$ of $f$ at the point $p$ with the order $\nu_{p^*}(\omega^*)$ of $\omega^*$ at the point $p^*$ of $\Cal S$ corresponding to $p$. This appears in \cite{11, Prop. 3.7, p. 28}. We distinguish three cases, according to the nature of $p$:\smallskip (i) $p$ is a {\sl regular point}, namely it is not a fixed point of some non-identical transformation in $\Gamma(m)$. In this case $n_p(f)=\nu_{p^*}(\omega^*)$; \smallskip (ii) $p$ is an \, {\sl elliptic fixed point} \, of period \, $e_p\in\{2,3\}$ \, (the points in this set \, are $\Gamma$-equivalent either to $i=\sqrt{-1}$ or to $\rho =e^{2\pi i/3}$). Now $n_p(f)=e_p\nu_{p^*}(\omega^*)+k(e_p-1)$;\smallskip (iii) $p$ is either $\infty$ or a finite parabolic point. We have $n_p(f)=\nu_{p^*}(\omega^*)+k$.\medskip We let $F\subset \frak H\cup\{$cusps$\}$ be a system of representatives for the action of $\Gamma(m)$, so the map $p\mapsto p^*$ is 1-1. Also, let $F_1,$ $F_2$ and $F_3$ be the subsets of $F$ made up of the points satisfying the above properties (i), (ii) and (iii) respectively. We now apply the Riemann-Roch theorem to $\omega^*$; this implies that $$\sum_{p^*\in{\Cal S}}\nu_{p^*}(\omega^*)=2k(g-1),$$ where $g$ is the genus of $\Cal S$. We get $$\sum_{p\in F_1}n_p(f)+\sum_{p\in F_2}{1\over e_p}n_p(f)+\sum_{p\in F_3}n_p(f)=k\left( 2g-2+\sum_{p\in F_2}(1-1/e_p)+\# F_3 \right),$$ where $\# F_3$ denotes the cardinality of $F_3.$ We point out that, in each concrete case, the above formula may also be obtained by complex integration on the boundary of a fundamental domain for $\Gamma(m)$, similarly to \cite{9, Ch. VII, Th. 3, p. 139}. Let $p\in F_1\cup F_3.$ Since $n_{p'}(f)\ge 0 $ for every $p'\in F,$ we have $$\spreadlines{0.1cm}\align n_p(f) & \le \left(\sum_{p'\in F_1}n_{p'}(f)+\sum_{p'\in F_2}\frac{1}{e_{p'}} n_{p'}(f)+\sum_{p'\in F_3}n_{p'}(f)\right) \\ & \leq k\left(2g-2+\sum_{p'\in F_2}(1-1/e_{p'})+ \# F_3\right). \endalign$$ For $ m\ge2$ we also have, as is easy to see, $\# F_2=0$ (\cite{10, Prop. 1.39, p. 22}) and $\# F_3\le m^2,$ whence $$ n_p(f) \leq k(2g-2+m^2). $$ As an application of this formula, we now prove (7). For any $A=\left( \matrix a & b \\ c & d \endmatrix \right)\in\Gamma$ denote $\dsize A(\tau)=\frac{a\tau+b}{c\tau+d}\,\,$ and $J_A(\tau)=c\tau+d.$ Let $m\ge2$ and $A\in\Gamma(m).$ Define $T_0,T_1,\dots,T_{m-1}$ by $$ T_j=\left( \matrix a+jc & (b+jd-ja-j^2c)/m \\ mc & d-jc \endmatrix \right) .$$ Since $a\equiv d \bmod m$ and $b\equiv c\equiv 0 \bmod m,$ $(b+jd-ja-j^2c)/m $ is an integer, and it is easy to show that $T_j\in\Gamma.$ Moreover $$ \frac{A(\tau)+j}{m}=T_j\left(\frac{\tau+j}{m}\right). $$ For $2k=2,4$ and $j=0,1,\dots,m-1,$ let $$E_{2k,j}^*(\tau)=E_{2k}\left(\frac{\tau+j}m\right).$$ If $2k=4$ we have $$\spreadlines{0.1cm}\align E_{4,j}^*\bigl(A(\tau)\bigr) & = E_4\left(\frac{A(\tau)+j}{m}\right) \\ & = E_4\left(T_j\left(\frac{\tau+j}{m}\right)\right)\\ & = \left(J_{T_j}\left(\frac{\tau+j}{m}\right) \right)^4\cdot E_4\left(\frac{\tau+j}{m}\right) \\ & = J_A(\tau)^4E_{4,j}^*(\tau) , \endalign$$ and it is straightforward to check that $E_{4,j}^*$ is holomorphic at every cusp. Hence $E_{4,j}^*$ is a modular form of weight $4$ for $\Gamma(m).$ If $2k=2,$ we have $$\spreadlines{0.1cm}\align E_{2,j}^*\bigl(A(\tau)\bigr)&=E_2\left(T_j\left(\frac{\tau+j}m\right)\right)\\ & = \frac6{\pi i}mcJ_{T_j}\left(\frac{\tau+j}{m}\right) +\left(J_{T_j}\left(\frac{\tau+j}{m}\right) \right)^2E_{2,j}^*(\tau) \\ & = \frac6{\pi i}mcJ_A(\tau)+J_A(\tau)^2 E_{2,j}^*(\tau). \endalign$$ If $\bold a=(a_0,\dots,a_{m-1})\in\Bbb C^m$ is such that $\sum a_j=0,$ one easily gets that the function $E_{\bold a}$ defined by $$E_{\bold a}(\tau)=\sum_{j=0}^{m-1}a_jE_{2,j}^*(\tau)$$ is a modular form of weight 2 for $\Gamma(m).$ For $1\le t\le m-1$ let $$ E_{2k,t}^{(m)}(\tau)=\sum\Sb n>0 \\ n\equiv t \bmod m \endSb \sigma_{2k-1}(n)q_m^n\qquad (2k=2 \text{ or }4).$$ Denoting $c_2=-1/24$, $c_4=1/240$, we seek a vector $\bold a=(a_0,\dots,a_{m-1})\in\Bbb C^m$ such that $$E_{2k,t}^{(m)}(\tau)=c_{2k}\sum_{j=0}^{m-1}a_jE_{2k,j}^*(\tau).$$ This immediately leads to the following linear system: $$\left\{\alignedat6 &a_0 &&+a_1 &&+a_2 &&+\dots &&+a_{m-1}&&=0\\ &a_0 &&+\omega a_1 &&+\omega^2 a_2 &&+\dots &&+\omega^{m-1}a_{m-1}&&=0 \\ &\dots&&\dots\dots\dots . \, . &&\dots\dots\dots\dots . &&\dots\dots &&\dots\dots\dots\dots\dots . && \dots\\ &a_0 &&+\omega^t a_1 &&+\omega^{2t} a_2 &&+\dots &&+\omega^{(m-1)t}a_{m-1}&&=1 \\ &\dots&&\dots\dots\dots . \, . &&\dots\dots\dots\dots . &&\dots\dots &&\dots\dots\dots\dots\dots . && \dots\\ &a_0 &&+\omega^{m-1} a_1 &&+\omega^{2(m-1)} a_2 &&+\dots && +\omega^{(m-1)^2}a_{m-1}&&=0, \\ \endalignedat\right.$$ where $\omega=e^{2\pi i/m}$. The matrix $\Omega$ of the coefficients is a Vandermonde matrix, whence $$\det\Omega=\prod_{0\le j14.$ Therefore $f(\tau)\equiv 0,$ i.e. (7) holds. We now consider $F_{0,m}(\tau)$ for $m=2,3,4$. As we saw in the proof of Theorem~1, $F_{0,m}(\tau)$ is a cusp form of weight 4 for $\Gamma_0(m).$ Further, it is easy to check that $\dim S_4(\Gamma_0(m))=0$ for $m=2,3,4$ (see \cite{3, Th\'eor\`eme 1}). Hence $F_{0,m}(\tau)\equiv0,$ thus proving (8), (12) and (11) respectively. Again by \cite{3, Th\'eor\`eme 1} we have $\dim S_6(\Gamma_0(2))=0,$ whence the identities (9) and (10) follow in a similar manner. Consider now $F_{0,5}(\tau)=\sum_{n=0}^{\infty}c_{0,5}(n)q^n.$ In this case, $\dim S_4(\Gamma_0(5))=1$ and $(\eta(\tau)\eta(5\tau))^4\in S_4(\Gamma_0(5)),$ where $\eta(\tau):=e^{\pi i\tau/12}\prod_{n=1}^\infty(1-q^n)$ is the Dedekind eta-function \cite{5, Prop. 19, p. 130}, so $F_{0,5}(\tau)$ is a Hecke eigenform. In particular for every $m, n$ with $(m,n)=1$ we have $c_{0,5}(m)c_{0,5}(n)=c_{0,5}(mn)$. Since $c_{0,5}(8)=0,$ for any odd $n$ we get $c_{0,5}(8n)=0,$ and this proves (13). Since $\dim S_4(\Gamma_0(9))=1$ we have that $F_{0,9}(\tau)=\sum_{n=0}^{\infty}c_{0,9}(n)q^n$ is also a Hecke eigenform. Moreover, $\eta(3\tau)^8=q\prod_{n=1}^\infty(1-q^{3n})^8\in S_4(\Gamma_0(9))$, whence $c_{0,9}(n)=0$ for $n\equiv2\bmod3$ and, by multiplicativity, this immediately implies (14). \qed \enddemo \medskip Five of the preceding eight identities, i.e. (8)--(12), can be also proved by using certain formulae from Ramanujan's Notebooks. To see this, let $w=e^{-y},$ where $y$ is defined by (2), and let $z$ be defined by (3). Let $$ L(w)=1-24\sum_{k=1}^{\infty}\frac{kw^k}{1-w^k}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, M(w)=1+240\sum_{k=1}^{\infty}\frac{k^3w^k}{1-w^k},$$ $$N(w)=1-504\sum_{k=1}^{\infty}\frac{k^5w^k}{1-w^k}. $$ Since $$ \sum_{k=1}^{\infty}\frac{k^m w^k}{1-w^k}=\sum_{k=1}^{\infty} \sum_{h=1}^{\infty} k^m w^{hk}=\sum_{n=1}^{\infty}w^n\sum_{hk=n}k^m=\sum_{n=1}^{\infty} \sigma_m(n)w^n, $$ we have $$ L(w)=-24\sum_{n=0}^{\infty}\sigma_1(n)w^n, \,\,\, M(w)=240\sum_{n=0}^{\infty}\sigma_3(n)w^n, \,\,\, N(w)=-504\sum_{n=0}^{\infty}\sigma_5(n)w^n. $$ By \cite{1, Ch. 17, Entry 13, p. 126--127} one can easily deduce $$ \left(2L(w^2)-L(w)\right)^2=\frac{4}{5}M(w^2)+\frac{1}{5}M(w), \tag{15} $$ $$ M(w)\left(2L(w^2)-L(w)\right)=\frac{32}{21} N(w^2)-\frac{11}{21}N(w) \tag{16} $$ and $$ M(w^2)\left(2L(w^2)-L(w)\right) =\frac{22}{21}N(w^2)-\frac{1}{21}N(w). \tag{17} $$ Recall that (\cite6) $$ \sum_{k=0}^n\sigma_1(k)\sigma_3(n-k)=\frac{7}{80}\sigma_5(n)-\frac{1}{8}n \sigma_3(n).\tag{18} $$ Equating coefficients of $w^n$ for $n$ odd in (15), (16) and (17) and then using (4) and (18), we obtain (8), (9) and (10) respectively. In a similar manner the identities (11) and (12) can be obtained using \cite{1, Example (ii) and (iii), p. 139} and \cite{1, Entry 3 (i), p. 460} respectively. \Refs \ref \key 1 \by B. C. Berndt \book Ramanujan's notebooks part III \publaddr Springer-Verlag \yr 1991 \endref \ref \key 2 \by B. C. Berndt, R. J. Evans \paper Chapter 15 of Ramanujan's Second Notebook: Part 2, Modular forms \jour Acta Arith. \vol 47 \yr 1986 \pages 123--142 \endref \ref \key3 \by H. Cohen, J. Oesterl\'e \paper Dimensions des espaces de formes modulaires \jour Lecture Notes in Mathematics \vol 627 \yr 1976 \publaddr Springer-Verlag \pages 69--78 \endref \ref \key4 \by R. C. Gunning \book Lectures on modular forms \publaddr Princeton University Press \yr 1962 \endref \ref \key 5 \by N. Koblitz \book Introduction to elliptic curves and modular forms \publaddr Springer-Verlag, GTM 97 \yr 1982\endref \ref \key6 \by S. Ramanujan \paper On certain arithmetical functions \jour Trans. Cambridge Philos. Soc. \vol 22 \yr 1916 \pages 159--184 \endref \ref \key7 \by S. Ramanujan \book Notebooks (2 volumes) \publaddr Tata Institute of Fundamental Research, Bombay \yr 1957 \endref \ref \key8 \by P. Sarnak \book Some applications of modular forms \publaddr Cambridge University Press \yr 1990 \endref \ref \key9 \by J. P. Serre \book Cours d'arithm\'etique \publaddr Presses Universitaires de France, Paris \yr 1970 \endref \ref \key10 \by G. Shimura \book Introduction to the arithmetic theory of automorphic functions \publaddr Princeton University Press \yr 1971 \endref \ref \key11 \by J. H. Silverman \book Advanced topics in the Arithmetic of Elliptic Curves \publaddr Springer-Verlag GTM 151 \yr 1994\endref \ref \key12 \by N-P. Skoruppa \paper A quick combinatorial proof of Eisenstein series identities \jour J. Number Theory \vol 43 \yr 1993 \pages 68--73 \endref \endRefs \enddocument %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% END OF AMSTEX FILE %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%