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\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
$$\det\Omega=\prod_{0\le j*14.$ Therefore $f(\tau)\equiv 0,$ i.e. (7) holds.
We now consider $F_{0,m}(\tau)$ for $m=2,3,4$. As we saw in the proof of
Theorem~1, $F_{0,m}(\tau)$ is a cusp form of weight 4 for $\Gamma_0(m).$
Further, it is easy to check that $\dim S_4(\Gamma_0(m))=0$ for $m=2,3,4$ (see
\cite{3, Th\'eor\`eme 1}). Hence $F_{0,m}(\tau)\equiv0,$ thus proving (8),
(12) and (11) respectively. Again by \cite{3, Th\'eor\`eme 1} we have
$\dim S_6(\Gamma_0(2))=0,$ whence the identities (9) and (10) follow in a
similar manner.
Consider now $F_{0,5}(\tau)=\sum_{n=0}^{\infty}c_{0,5}(n)q^n.$ In this case,
$\dim S_4(\Gamma_0(5))=1$ and $(\eta(\tau)\eta(5\tau))^4\in S_4(\Gamma_0(5)),$
where
$\eta(\tau):=e^{\pi i\tau/12}\prod_{n=1}^\infty(1-q^n)$ is the Dedekind
eta-function \cite{5, Prop. 19, p. 130},
so $F_{0,5}(\tau)$ is a
Hecke eigenform. In particular for every $m, n$ with $(m,n)=1$ we have
$c_{0,5}(m)c_{0,5}(n)=c_{0,5}(mn)$. Since $c_{0,5}(8)=0,$
for any odd $n$ we get $c_{0,5}(8n)=0,$
and this proves (13).
Since $\dim S_4(\Gamma_0(9))=1$
we have that $F_{0,9}(\tau)=\sum_{n=0}^{\infty}c_{0,9}(n)q^n$
is also a Hecke eigenform.
Moreover, $\eta(3\tau)^8=q\prod_{n=1}^\infty(1-q^{3n})^8\in S_4(\Gamma_0(9))$,
whence $c_{0,9}(n)=0$ for $n\equiv2\bmod3$ and,
by multiplicativity, this immediately implies (14).
\qed
\enddemo
\medskip
Five of the preceding eight identities, i.e. (8)--(12),
can be also proved by using certain
formulae from Ramanujan's Notebooks.
To see this, let $w=e^{-y},$ where $y$ is defined by (2), and
let $z$ be defined by (3). Let
$$
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.
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\ref \key 1 \by B. C. Berndt \book Ramanujan's notebooks part III
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\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
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\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
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\endRefs
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