Conjectures of Sun About Sums of Polygonal Numbers

Abstract

In this paper, we consider representations of positive integers as sums of generalized m-gonal numbers, which extend the formula for the number of dots needed to make up a regular m-gon. We mainly restrict to the case where the sums contain at most four distinct generalized m-gonal numbers, with the second m-gonal number repeated twice, the third repeated four times, and the last is repeated eight times. For a number of small choices of m, Sun conjectured that every positive integer may be written in this form. By obtaining explicit quantitative bounds for Fourier coefficients related to theta functions which encode the number of such representations, we verify that Sun’s conjecture is true for sufficiently large positive integers. Since there are only finitely many choices of m appearing in Sun’s conjecture, this reduces Sun’s conjecture to a verification of finitely many cases. Moreover, the bound beyond which we prove that Sun’s conjecture holds is explicit.

Introduction and Statement of Results

For $m\in {\mathbb{N}}_{\ge 3}$ and $\ell \in \mathbb{Z}$, let $p_m(\ell )$ be the $\ell$-th (generalized) m-gonal number

$$\begin{array}{c}\hfill p_m(\ell ):=\frac{1}{2}(m-2){\ell }^2-\frac{1}{2}(m-4)\ell .\end{array}$$

For $\ell \in \mathbb{N}$, these count the number of dots contained in a regular polygon with m sides having $\ell$ dots on each side. For example, the special case $m=3$ corresponds to triangular numbers, $m=4$ gives squares, and $m=5$ corresponds to pentagonal numbers. There are several conjectures related to sums of polygonal numbers. Specifically, for $\mathit{\alpha }\in {\mathbb{N}}^d$,1 we are interested in the solvability of the Diophantine equation

$$\begin{array}{c}\hfill \sum _{1\le j\le d}{\alpha }_j{p}_m({\ell }_j)=n\end{array}$$ 1.1

with ${\ell }_j\in {\mathbb{N}}_0$ or ${\ell }_j\in \mathbb{Z}$. We call such a sum universal if it is solvable for every $n\in \mathbb{N}$. Fermat stated (his claimed proof was not found in his writings) that every positive integer is the sum of three triangular number, four squares, five pentagonal numbers, and in general at most m m-gonal numbers. In other words, he claimed that the sum ${\sum }_{1\le j\le m}{p}_m({\ell }_j)$ is universal. His claim for squares ($m=4$) was proven by Lagrange in 1770, the claim for triangular numbers ($m=3$) was shown by Gauss in 1796, and the full conjecture was proven by Cauchy in 1813. Going in another direction, Ramanujan fixed $m=4$ and conjectured a full list of choices of $\mathit{\alpha }\in {\mathbb{N}}^4$ for which the resulting sum is universal; this conjecture was later proven by Dickson [11]. Following this, the classification of universal quadratic forms was a central area of study throughout the twentieth century, culminating in the Conway–Schneeberger 15-theorem [3, 8] and the 290-theorem [4], which state that arbitrary quadratic forms whose cross terms are even (resp. are allowed to be odd) are universal if and only if they represent every integer up to 15 (resp. 290). Theorems of this type are now known as finiteness theorems. Namely, given an infinite set $S\subseteq \mathbb{N}$, one determines a finite subset $S_0$ of S such that a solution to (1.1) exists for every $n\in S$ if and only if it exists for every $n\in S_0$. Taking $S=\mathbb{N}$, one obtains a condition for universality of a given sum of polygonal numbers. For example, choosing $m=3$ or $m=6$, (1.1) is solvable with $\mathit{\ell }\in {\mathbb{Z}}^d$ for all $n\in \mathbb{N}$ if and only if it is solvable for every $n\le 8$ [6], for $m=5$ it is solvable with $\mathit{\ell }\in {\mathbb{Z}}^d$ for all $n\in \mathbb{N}$ if and only if it is solvable for every $n\le 109$ [13], while it is solvable with $\mathit{\ell }\in {\mathbb{N}}_0^d$ for all $n\in \mathbb{N}$ if and only if it is solvable for every $n\le 63$ [14] and for $m=8$ it is solvable for $\mathit{\ell }\in {\mathbb{Z}}^d$ for all $n\in \mathbb{N}$ if and only if it is solvable for every $n\le 60$ [15].

Here, we consider the question of universality in the case $\mathit{\alpha }=(1,2,4,8)$ as one varies m. Specifically, we have the following conjecture of Sun (see [21, Conjecture 5.4]).

Conjecture 1.1

For $m\in \{7,9,10,11,12,13,14\}$ and $\mathit{\ell }\in {\mathbb{Z}}^4$, the equation

$$\begin{array}{c}\hfill p_m({\ell }_1)+2p_m({\ell }_2)+4p_m({\ell }_3)+8p_m({\ell }_4)=n\end{array}$$ 1.2

is solvable for every $n\in \mathbb{N}$.

Remark

A proof of Conjecture 1.1 would give a classification of those m for which the sum (1.1) is universal in the case $\mathit{\alpha }=(1,2,4,8)$. By direct computation, one sees that $p_m(\ell )\in \{0,1\}$ or $p_m(\ell )\ge m-3$. Using this, one obtains that (1.2) is not solvable for $n=16$ for every $m\ge 20$. For $m=16$, $m=17$, $m=18$, and $m=19$, one finds directly that there is no solution for $n=29$, $n=30$, $n=16$, and $n=17$, respectively. Moreover, for $m\in \{3,6\}$, it is known by work of Liouville [18] that the sum is universal, for $m=4$, it was conjectured by Ramanujan and proven by Dickson [11] that the sum is universal, while for $m=5$ (resp. $m=8$), it was shown by Sun in [20] (resp. [21]) to be universal.

In this paper, we prove that Conjecture 1.1 is true for n sufficiently large.

Theorem 1.2

For $m\in \{7,9,10,11,12,13,14\}$, there exists an explicit constant $C_m$ (defined in Table 2) such that (1.2) is solvable with $\mathit{\ell }\in {\mathbb{Z}}^4$ for every $n\in {\mathbb{N}}_{\ge C_m}$, with the restriction that $n\not\equiv 4\left(\text{mod},,16\right)$ if $m=12$.

Table 2.

Bounds on n for $s_{m,2(m-2),(1,2,4,8)}(n)>0$ and $r_{m,(1,2,4,8)}>0$

m	Bound for $s_{m,2(m-2),(1,2,4,8)}(n)>0$	Bound $C_m$ for $r_{m,2(m-2),(1,2,4,8)}(n)>0$
7	$1.92·{10}^{82}$	$4.8·{10}^{80}$
9	$8.38·{10}^{85}$	$1.5·{10}^{84}$
10	$3.41·{10}^{87}$	$5.33·{10}^{85}$
11	$3.55·{10}^{88}$	$4.93·{10}^{86}$
12	$4.25·{10}^{89}$	$5.31·{10}^{87}$
13	$4.57·{10}^{90}$	$5.19·{10}^{88}$
14	$2.04·{10}^{91}$	$2.13·{10}^{89}$

Remark

To prove Conjecture 1.1 for $m=12$, it suffices to show that (1.2) holds for all $n\in \mathbb{N}$ with $n\not\equiv 4\left(\text{mod},,16\right)$ (see Lemma 5.1). Hence, the restriction in Theorem 1.2 is natural.

By completing the square, one easily sees that representations of integers as sums of polygonal numbers are closely related to sums of squares with congruence conditions. In particular, setting

$$\begin{array}{cc}\hfill r_{m,\mathit{\alpha }}(n):& =\#\left\{\mathit{\ell },\in ,{\mathbb{Z}}^4,:,\sum _{1\le j\le 4},{\alpha }_j,p_m,({\ell }_j),=,n\right\},\hfill \\ \hfill s_{r,M,\mathit{\alpha }}(n):& =\#\left\{\mathit{x},\in ,{\mathbb{Z}}^4,:,\sum _{1\le j\le 4},{\alpha }_j,x_j^2,=,n,,,,x_j,\equiv ,r,,,\left(\text{mod},,M\right)\right\},\hfill \end{array}$$

we have

$$\begin{array}{c}\hfill r_{m,(1,2,4,8)}(n)=s_{m,2(m-2),(1,2,4,8)}\left(8,(m-2),n,+,15,{(m-4)}^2\right).\end{array}$$ 1.3

Hence, since Conjecture 1.1 is equivalent to proving that $r_{m,(1,2,4,8)}(n)>0$ for every $n\in \mathbb{N}$ and $m\in \{7,9,10,11,$ $12,13,14\}$, the conjecture is equivalent to showing that for every $n\in \mathbb{N}$, we have

$$\begin{array}{c}\hfill s_{m,2(m-2),(1,2,4,8)}\left(8,(m-2),n,+,15,{(m-4)}^2\right)>0.\end{array}$$ 1.4

We investigate the numbers $s_{r,M,\mathit{\alpha }}(n)$ by forming the generating function (setting $q:=e^{2\pi i\tau }$)

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{r,M,\mathit{\alpha }}(\tau ):=\sum _{n\ge 0}{s}_{r,M,\mathit{\alpha }}(n)q^n.\end{array}$$

It is well known that these functions are modular forms (see Lemma 2.1 for the precise statement). By the theory of modular forms, there is a natural decomposition

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{r,M,\mathit{\alpha }}=E_{r,M,\mathit{\alpha }}+f_{r,M,\mathit{\alpha }},\end{array}$$ 1.5

where $E_{r,M,\mathit{\alpha }}$ lies in the space spanned by Eisenstein series and $f_{r,M,\mathit{\alpha }}$ is a cusp form. In order to prove Theorem 1.2, we obtain in the special case $r=m$, $M=2(m-2)$, and $\mathit{\alpha }=(1,2,4,8)$ an explicit lower bound for the n-th Fourier coefficient $a_{r,M,\mathit{\alpha }}(n)$ of $E_{r,M,\mathit{\alpha }}$ in Corollary 4.2 and an explicit upper bound on the absolute value of the n-th Fourier coefficient $b_{r,M,\mathit{\alpha }}(n)$ of $f_{r,M,\mathit{\alpha }}$ in the proof of Theorem 1.2.

The paper is organized as follows. In Sect. 2, we recall properties of the theta functions ${\mathrm{\Theta }}_{r,M,\mathit{\alpha }}$, the actions of certain operators on modular forms, the decomposition of modular forms into Eisenstein series and cusp forms, and evaluate certain Gauss sums. In Sect. 3, we investigate the growth of the theta functions toward all cusps and use this to compute the Eisenstein series component of the decomposition (1.5). The Fourier coefficients of the Eisenstein series components are then explicitly computed and lower bounds are obtained in Sect. 4. We complete the paper by obtaining upper bounds on the coefficients of the cuspidal part of the decomposition (1.5) and prove Theorem 1.2 in Sect. 5.

Setup and Preliminaries

Modularity of the Generating Functions

In this subsection, we consider the modularity properties of the theta functions ${\mathrm{\Theta }}_{r,M,\mathit{\alpha }}$. To set notation, for ${\mathrm{\Gamma }}_1(N)\subseteq \mathrm{\Gamma }\subseteq {\text{SL}}_2(\mathbb{Z})$ ($N\in \mathbb{N}$) and a character $\chi$ modulo N, let $M_k(\mathrm{\Gamma },\chi )$ be the space of modular forms of weight k with character $\chi$. In particular, an element f in this space satisfies, for ,

$$\begin{array}{c}\hfill f{|}_k\gamma (\tau ):={(c\tau +d)}^{-k}f(\gamma \tau )=\chi (d)f(\tau ).\end{array}$$

Setting ${\mathrm{\Gamma }}_{N,L}:={\mathrm{\Gamma }}_0(N)\cap {\mathrm{\Gamma }}_1(L)$, by [7, Theorem 2.4], we have the following.

Lemma 2.1

For $\mathit{\alpha }\in {\mathbb{N}}^4$, we have

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{r,M,\mathit{\alpha }}\in M_2\left({\mathrm{\Gamma }}_{4\text{lcm}(\mathit{\alpha })M^2,M},,,\left(\frac{{\prod }_{j=1}^4{\alpha }_j}{·}\right)\right).\end{array}$$

Operators on Non-holomorphic Modular Forms

For a translation-invariant function f with Fourier expansion (denoting $\tau =u+iv\in \mathbb{H}$)

$$\begin{array}{c}\hfill f(\tau )=\sum _{n\ge 0}{c}_{f,v}(n)q^n,\end{array}$$

we define the sieving operator (M, $m\in \mathbb{N}$)

$$\begin{array}{c}\hfill f|S_{M,m}(\tau ):=\sum _{\begin{array}{c}n\ge 0\\ n\equiv m\left(\text{mod},,M\right)\end{array}}{c}_{f,v}(n)q^n.\end{array}$$

As usual, we also define the V-operator ($\delta \in \mathbb{N}$) by

$$\begin{array}{c}\hfill f|V_{\delta }(\tau ):=\sum _{n\ge 0}{c}_{f,\delta v}(n)q^{\delta n}.\end{array}$$

We require the modularity properties of (non-holomorphic) modular forms under the operators $S_{M,m}$ and $V_d$. Arguing via commutator relations for matrices, a standard argument (for example, see the proof of [17, Lemma 2]), one obtains the following.

Lemma 2.2

Suppose that $k\in \mathbb{Z}$, $L,N\in \mathbb{N}$ with $L\mid N$, and f satisfies weight k modularity on ${\mathrm{\Gamma }}_{N,L}$.

1. For $d\in \mathbb{N}$, the function $f|V_d$ satisfies weight k modularity on ${\mathrm{\Gamma }}_{\text{lcm}(4,Nd),L}$.

2. For $m\in \mathbb{Z}$ and $M\in \mathbb{N}$, the function $f|S_{M,m}$ satisfies weight k modularity on ${\mathrm{\Gamma }}_1(N{M}^2)$.

It is useful to determine the commutator relations between the V-operator and sieving.

Lemma 2.3

Let $m\in \mathbb{Z}$ and $M_1,M_2\in \mathbb{N}$ be given and set $d:=gcd(M_1,M_2)$ and ${\mu }_j:=\frac{M_j}{d}$. Then for any translation-invariant function f, we have

$$\begin{array}{c}\hfill f|V_{M_1}|S_{M_2,m}=\left\{\begin{array}{cc}f|S_{{\mu }_2,{\overline{\mu }}_1\frac{m}{d}}|V_{M_1}\hfill & \text{if}d\mid m,\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}\right)\end{array}$$

where ${\overline{\mu }}_1$ is the inverse of ${\mu }_1\left(\text{mod},,{\mu }_2\right)$.

Proof

Recall that

$$\begin{array}{cc}\hfill f\left|S_{M_1,m}\right|V_{M_2}(\tau )& =\sum _{n\equiv m\left(\text{mod},,M_1\right)}{c}_{f,M_{2}v}(n)q^{M_{2}n},\hfill \\ \hfill f\left|V_{M_1}\right|S_{M_2,m}(\tau )& =\sum _n{c}_{f,M_{1}v}(n)q^{M_{1}n}|S_{M_2,m}(\tau )=\sum _{n\equiv m\left(\text{mod},,M_2\right)}{\tilde{c}}_{f,v}(n)q^n\hfill \\ \hfill & =\sum _{\begin{array}{c}n\equiv m\left(\text{mod},,M_2\right)\\ M_1\mid n\end{array}}{c}_{f,M_{1}v}\left(\frac{n}{M_1}\right)q^n,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\tilde{c}}_{f,v}(n):=\left\{\begin{array}{cc}{c}_{f,M_{1}v}\left(\frac{n}{M_1}\right)\hfill & \text{if}{M}_1\mid n,\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}\right)\end{array}$$

If $d=gcd(M_1,M_2)\nmid m$, then $n\equiv m\left(\text{mod},,M_2\right)$ and $M_1\mid n$ are not consistent, and hence $f|V_{M_1}|S_{M_2,m}$ vanishes identically.

We may hence assume that $d\mid m$ and we note that $gcd({\mu }_1,{\mu }_2)=1$, obtaining

$$\begin{array}{cc}\hfill f\left|V_{M_1}\right|S_{M_2,m}(\tau )& =\sum _{\begin{array}{c}\frac{n}{d}\equiv \frac{m}{d}\left(\text{mod},,{\mu }_2\right)\\ M_1\mid n\end{array}}{c}_{f,M_{1}v}\left(\frac{n}{M_1}\right)q^n\hfill \\ \hfill & \stackrel{n\mapsto M_{1}n}{=}\sum _{n\equiv {\overline{\mu }}_1\frac{m}{d}\left(\text{mod},,{\mu }_2\right)}{c}_{f,M_{1}v}(n)q^{M_{1}n}\hfill \\ \hfill & =f\left|S_{{\mu }_2,{\overline{\mu }}_1\frac{m}{d}}\right|V_{M_1}(\tau ).\hfill \end{array}$$

$\square$

Decomposition Into Eisenstein Series and Cusp Forms

Comparing Fourier coefficients on both sides of (1.5), we have

$$\begin{array}{c}\hfill s_{r,M,\mathit{\alpha }}(n)=a_{r,M,\mathit{\alpha }}(n)+b_{r,M,\mathit{\alpha }}(n).\end{array}$$ 2.1

Theorem 1.2 is equivalent to showing (1.4) for n sufficiently large (with the restriction that $n\not\equiv 4\left(\text{mod},,16\right)$ for $m=12$). Roughly speaking, the approach in this paper to proving (1.4) is to prove that for n sufficiently large with $n\equiv 15{(m-4)}^2\left(\text{mod},,8,(,m,-,2,)\right)$ (noting the congruence conditions in (1.4))

$$\begin{array}{c}\hfill a_{r,M,\mathit{\alpha }}(n)>|b_{r,M,\mathit{\alpha }}(n)|.\end{array}$$

To obtain an upper bound for $|b_{r,M,\mathit{\alpha }}(n)|$, we recall that Deligne [9] proved that for a normalized newform $f(\tau )={\sum }_{n\ge 1}{c}_f(n)q^n$ of weight k on ${\mathrm{\Gamma }}_0(N)$ with Nebentypus character $\chi$ (normalized so that $c_f(1)=1$), we have

$$\begin{array}{c}\hfill |c_f(n)|\le {\sigma }_0(n)n^{\frac{k-1}{2}},\end{array}$$ 2.2

where ${\sigma }_k(n):={\sum }_{d\mid n}{d}^k$. To obtain an explicit bound for $|c_f(n)|$ for arbitrary $f\in S_k({\mathrm{\Gamma }}_1(N))$, we combine (2.2) with a trick implemented by Blomer [5] and Duke [12]. For cusp forms $f,g\in S_k(\mathrm{\Gamma })$, we define the Petersson inner product by

$$\begin{array}{c}\hfill 〈f,,,g〉:=\frac{1}{[{\text{SL}}_2(\mathbb{Z}):\mathrm{\Gamma }]}{\int }_{\mathrm{\Gamma }\backslash \mathbb{H}}f(\tau )\overline{g(\tau )}{v}^k\frac{\mathit{dudv}}{v^2}.\end{array}$$

Letting $\Vert f\Vert :=\sqrt{〈f,,,f〉}$ denote the Petersson norm of $f\in S_k(\mathrm{\Gamma })$, a bound for $|c_f(n)|$ in terms of $\Vert f\Vert$ may be obtained. Specifically, suppose that f is a cusp form f of weight $k\in \mathbb{N}$ on ${\mathrm{\Gamma }}_{N,L}$ (with $L\mid N$) and character $\chi$ modulo N. Using Blomer’s method from [5], an explicit bound is obtained in [1, Lemma 4.1] for $|c_f(n)|$ as a function of N, L, and the Petersson norm $\Vert f\Vert$. Denoting by $\phi$ Euler’s totient function, we recall a bound from the case $k=2$ below (see [1, (4.4)]).

Lemma 2.4

Suppose that $f\in S_2({\mathrm{\Gamma }}_{N,L},\chi )$ with $L\mid N$ and $\chi$ a character modulo N. Then, we have the inequality

$$\begin{array}{c}\hfill \left|c_f,(n)\right|\le 6.95·{10}^{18}·\Vert f\Vert N^{1+2.5·{10}^{-6}}\prod _{p\mid N}{\left(1,+,\frac{1}{p}\right)}^{\frac{1}{2}}\phi (L)n^{\frac{3}{5}}.\end{array}$$

By Lemma 2.4, in order to obtain an explicit bound for $|b_{r,M,\mathit{\alpha }}(n)|$, it remains to estimate $\Vert f_{r,M,\mathit{\alpha }}\Vert$, where $f_{r,M,\mathit{\alpha }}$ is the cusp form appearing in the decomposition in (1.5). An explicit bound for $\Vert f_{r,M,\mathit{\alpha }}\Vert$ was obtained in [16, Lemma 3.2]. To state the result, let $\mathit{\alpha }\in {\mathbb{Z}}^{\ell }$. For the quadratic form $Q=Q_{\mathit{\alpha }}$ given by

$$\begin{array}{c}\hfill Q_{\mathit{\alpha }}(\mathit{x}):=\sum _{j=1}^{\ell }{\alpha }_j{x}_j^2,\end{array}$$

we define the level and the discriminant of $Q_{\mathit{\alpha }}$ as

$$\begin{array}{c}\hfill N_{\mathit{\alpha }}=4\text{lcm}(\mathit{\alpha }),D_{\mathit{\alpha }}=2^{\ell }\prod _{j=1}^{\ell }{\alpha }_j.\end{array}$$

Lemma 2.5

Let $\ell \ge 4$ be even, $\mathit{\alpha }\in {\mathbb{N}}^{\ell }$, $r\in \mathbb{Z}$, and $M\in \mathbb{N}$. Then,

$$\begin{array}{cc}\hfill \Vert f_{r,M,\mathit{\alpha }}{\Vert }^2\le & \frac{3^{2\ell -2}\left(\frac{\ell }{2},-,2\right)!}{2^{\frac{\ell }{2}-3}{\pi }^{\ell }}\frac{M^{2\ell -4}{N}_{\mathit{\alpha }}^{\ell -2}}{{\prod }_{p\mid M^2{N}_{\mathit{\alpha }}}\left(1,-,p^{-2}\right)}\hfill \\ \hfill & \times \sum _{\delta \mid M^2{N}_{\mathit{\alpha }}}\phi \left(\frac{M^2{N}_{\mathit{\alpha }}}{\delta }\right)\phi (\delta )\frac{M^2{N}_{\mathit{\alpha }}}{\delta }{\left(\frac{gcd(M^2,\delta )}{M^2}\right)}^{\ell }\hfill \\ \hfill & \times \sum _{m=0}^{\frac{\ell }{2}-2}\frac{{(2\pi )}^{-m}}{\left(\frac{\ell }{2},-,2,-,m\right)!}(\ell -m-2)!\left(\frac{9}{D_{\mathit{\alpha }}},(\ell -m-1),\frac{M^2{N}_{\mathit{\alpha }}}{\pi },+,{\ell }^2\right).\hfill \end{array}$$

Gauss Sums

Define the generalized quadratic Gauss sum ($a,b\in \mathbb{Z}$, $c\in \mathbb{N}$)

$$\begin{array}{c}\hfill G(a,b;c):=\sum _{\ell \left(\text{mod},,c\right)}{e}^{\frac{2\pi i}{c}\left(a,{\ell }^2,+,b,\ell \right)}.\end{array}$$

Background information and many properties of these sums may be found in [2]. To state the properties that we require, for d odd, we define

$$\begin{array}{c}\hfill {\epsilon }_d:=\left\{\begin{array}{cc}1\hfill & \text{if}d\equiv 1\left(\text{mod},,4\right),\hfill \\ i\hfill & \text{if}d\equiv 3\left(\text{mod},,4\right),\hfill \end{array}\right)\end{array}$$

and we write ${[a]}_b$ for the inverse of a modulo b if $gcd(a,b)=1$.

Lemma 2.6

For $a,b\in \mathbb{Z}$ and $c,d\in \mathbb{N}$, the following hold.

1. If $gcd(a,c)\nmid b$, then $G(a,b;c)=0$, while if $gcd(a,c)\mid b$ then

   $$\begin{array}{c}\hfill G(a,b;c)=gcd(a,c)G\left(\frac{a}{gcd(a,c)},,,\frac{b}{gcd(a,c)},;,\frac{c}{gcd(a,c)}\right).\end{array}$$
2. If $gcd(a,c)=1$ and c is odd, then

   $$\begin{array}{c}\hfill G(a,b;c)={\epsilon }_c\sqrt{c}\left(\frac{a}{c}\right)e^{-\frac{2\pi i{[4a]}_c{b}^2}{c}}.\end{array}$$
3. If $gcd(c,d)=1$, then

   $$\begin{array}{c}\hfill G(a,b;cd)=G(ac,b;d)G(ad,b;c).\end{array}$$

4. If $gcd(a,c)=1$, $4\mid c$, and b is odd, then $G(a,b;c)=0$.

5. If a is odd, b is even, and $r\in {\mathbb{N}}_{\ge 2}$, then

   $$\begin{array}{c}\hfill G\left(a,,,b,;,2^r\right)=2^{\frac{r}{2}}(1+i)\left(\frac{-2^r}{a}\right){\epsilon }_a{e}^{\frac{2\pi i}{2^r}\left(-,{[a]}_{2^r},\frac{b^2}{4}\right)}.\end{array}$$

We require an explicit evaluation of certain Gauss sums that naturally occur in the study of theta functions (see Lemma 3.1 below). Throughout the paper, for $k,M\in \mathbb{N}$, and $r\in \mathbb{Z}$ with ${\text{ord}}_2(r)\le {\text{ord}}_2(M)$, we write $M=2^{\mu }{M}_0$, $r=2^{\varrho }{r}_0$ (with $\varrho \le \mu$), and $k=2^{\kappa }{k}_0$ with $M_0$, $r_0$, and $k_0$ odd. We furthermore set $g_0:=gcd(M_0,k_0)$ and $g_1:=gcd(g_0,\frac{k_0}{g_0})$.

Lemma 2.7

Suppose that $h\in \mathbb{Z}$, $k\in \mathbb{N}$ with $gcd(h,k)=1$, $\ell \in {\mathbb{N}}_0$, $r\in \mathbb{Z}$, $M\in \mathbb{N}$ with $gcd(M,r)\in \{1,2,4\}$, and $\varrho \le \mu$.

1. If $g_1\ne 1$ or $\varrho <min(\mu ,\kappa -\ell -\mu )-1$, then

   $$\begin{array}{c}\hfill G\left(2^{\ell },h,M^2,,,2^{\ell +1},h,r,M,;,k\right)=0.\end{array}$$
2. Suppose that $g_1=1$ and $\varrho \ge min(\mu ,\kappa -\ell -\mu )-1$. Setting $\delta :=min(\ell +2\varrho ,\kappa )$, we then have

   $$\begin{array}{cc}\hfill & e^{\frac{2\pi i{2}^{\ell }h{r}^2}{k}}G\left(2^{\ell },h,M^2,,,2^{\ell +1},h,r,M,;,k\right)\hfill \\ \hfill & =\sqrt{k{g}_0}\left\{\begin{array}{cc}{2}^{\frac{\kappa }{2}}{\epsilon }_{\frac{k_0}{g_0}}\left(\frac{2^{\ell +\kappa }h{g}_0}{\frac{k_0}{g_0}}\right)e^{\frac{2\pi ih{r}_0^2}{2^{\kappa -\delta }{g}_0}{2}^{\ell +2\varrho -\delta }{\left[\frac{k_0}{g_0}\right]}_{2^{\kappa -\delta }{g}_0}}\hfill & \begin{array}{c}\text{if}\kappa \le \ell +2\mu \hfill \\ \text{or}\kappa =\ell +2\mu +1\text{and}\varrho =\mu -1,\hfill \end{array}\hfill \\ 2^{\frac{\ell +2\mu }{2}}(1+i){\epsilon }_{h{g}_0}\left(\frac{-2^{\ell +\kappa }\frac{k_0}{g_0}}{h{g}_0}\right)e^{\frac{2\pi ih{r}_0^2}{g_0}{\left[2^{\kappa -\ell -2\mu },\frac{k_0}{g_0}\right]}_{g_0}}\hfill & \text{if}\kappa \ge \ell +2\mu +2\text{and}\varrho =\mu ,\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}\right)\hfill \end{array}$$

Proof

We evaluate G(a, b; c) for $a:=2^{\ell }h{M}^2$, $b:=2^{\ell +1}hrM$, and $c:=k$. By Lemma 2.6 (1), $G(a,b;c)=0$ unless $gcd(a,c)\mid b$. Hence, we first compute, using the fact that $gcd(h,k)=1$, $gcd(\frac{M_0}{g_0},\frac{k_0}{g_0})=1$, and $\frac{k_0}{g_0}$ is odd,

$$\begin{array}{c}\hfill gcd(a,c)=2^{min(\ell +2\mu ,\kappa )}{g}_0{g}_2,\end{array}$$ 2.3

where $g_2:=gcd(M_0,\frac{k_0}{g_0})$.

1. A direct calculation gives that $gcd(a,c)\mid b$ if and only if $g_1=1$ and $\varrho \ge min(\mu ,\kappa -\ell -\mu )-1$, which implies the claim by Lemma 2.6 (1).

2. Set $\gamma :=min(\ell +2\mu ,\kappa )$. Note that $\gamma \le \ell +\mu +\varrho +1$. From the calculation yielding (1), we see that $g_1=1$ implies $g_2=1$. Plugging $g_1=g_2=1$ into (2.3) yields $gcd(a,c)=2^{\gamma }{g}_0$ and it is not hard to see that $gcd(a,c)\mid b$. Therefore, Lemma 2.6 (1),(3) implies that

   $$\begin{array}{cc}\hfill G(a,b;c)=& 2^{\gamma }{g}_{0}G\left(2^{\ell +2\mu +\kappa -2\gamma },h,M_0,\frac{M_0}{g_0},,,2^{\ell +\mu +\varrho +1-\gamma },h,r_0,\frac{M_0}{g_0},;,\frac{k_0}{g_0}\right)\hfill \\ \hfill & \times G\left(2^{\ell +2\mu -\gamma },h,M_0,\frac{M_0}{g_0},\frac{k_0}{g_0},,,2^{\ell +\mu +\varrho +1-\gamma },h,r_0,\frac{M_0}{g_0},;,2^{\kappa -\gamma }\right).\hfill \end{array}$$

   Since $\frac{k_0}{g_0}$ is odd, we use Lemma 2.6 (2) to evaluate the first Gauss sum, yielding, after simplification,

   $$\begin{array}{cc}\hfill G(a,b;c)& =2^{\gamma }{\epsilon }_{\frac{k_0}{g_0}}\sqrt{k_0{g}_0}\left(\frac{2^{\ell +\kappa }h{g}_0}{\frac{k_0}{g_0}}\right)e^{-\frac{2\pi ih{r}_0^2}{\frac{k_0}{g_0}}{2}^{\ell +2\varrho }{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}}\hfill \\ \hfill & \times G\left(2^{\ell +2\mu -\gamma },h,M_0,\frac{M_0}{g_0},\frac{k_0}{g_0},,,2^{\ell +\mu +\varrho +1-\gamma },h,r_0,\frac{M_0}{g_0},;,2^{\kappa -\gamma }\right).\hfill \end{array}$$ 2.4

It remains to evaluate the final Gauss sum in (2.4). We use Lemma 2.6 (4) and Lemma 2.6 (5) to obtain

$$\begin{array}{cc}& G\left(2^{\ell +2\mu -\gamma },h,M_0,\frac{M_0}{g_0},\frac{k_0}{g_0},,,2^{\ell +\mu +\varrho +1-\gamma },h,r_0,\frac{M_0}{g_0},;,2^{\kappa -\gamma }\right)\hfill \\ \hfill & =\left\{\begin{array}{cc}1\hfill & \text{if}\kappa \le \ell +2\mu ,\hfill \\ 2\hfill & \text{if}\kappa =\ell +2\mu +1,\varrho =\mu -1,\hfill \\ 2^{\frac{\kappa -\ell -2\mu }{2}}(1+i)\left(\frac{-2^{\ell +\kappa }}{h{M}_0\frac{M_0}{g_0}\frac{k_0}{g_0}}\right){\epsilon }_{h{M}_0\frac{M_0}{g_0}\frac{k_0}{g_0}}{e}^{-\frac{2\pi ih{r}_0^2}{2^{\kappa -\ell -2\mu }}{\left[k_0\right]}_{2^{\kappa -\ell -2\mu }}}\hfill & \text{if}\kappa \ge \ell +2\mu +2,\varrho =\mu ,\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}\right)\hfill \end{array}$$ 2.5

Plugging (2.5) into (2.4) and then simplifying yields that G(a, b; c) equals

$$\begin{array}{cc}& {\epsilon }_{\frac{k_0}{g_0}}\sqrt{k{g}_0}\left(\frac{h{g}_0}{\frac{k_0}{g_0}}\right)\left\{\begin{array}{cc}{2}^{\frac{\kappa }{2}}\left(\frac{2^{\ell +\kappa }}{\frac{k_0}{g_0}}\right)e^{-\frac{2\pi ih{r}_0^2}{\frac{k_0}{g_0}}{2}^{\ell +2\varrho }{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}}\hfill & \begin{array}{c}\text{if}\kappa \le \ell +2\mu \hfill \\ \text{or}\kappa =\ell +2\mu +1,\varrho =\mu -1,\hfill \end{array}\hfill \\ 2^{\frac{\ell +2\mu }{2}}(1+i){\epsilon }_{h{M}_0\frac{M_0}{g_0}\frac{k_0}{g_0}}\left(\frac{-2^{\ell +\kappa }}{h{M}_0\frac{M_0}{g_0}\frac{k_0}{g_0}}\right)\left(\frac{2^{\ell +\kappa }}{\frac{k_0}{g_0}}\right)\hfill & \text{if}\kappa \ge \ell +2\mu +2,\varrho =\mu ,\hfill \\ \times e^{-\frac{2\pi ih{r}_0^2}{\frac{k_0}{g_0}}{2}^{\ell +2\mu }{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}}{e}^{-\frac{2\pi ih{r}_0^2}{2^{\kappa -\ell -2\mu }}{\left[k_0\right]}_{2^{\kappa -\ell -2\mu }}}\hfill & \hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}\right)\hfill \end{array}$$

To obtain the claim, we multiply by $e^{\frac{2\pi i{2}^{\ell }h{r}^2}{k}}$ and simplify by using the Chinese Remainder Theorem to combine the exponentials. For example, if $\kappa \le \ell +2\mu$ or ($\kappa =\ell +2\mu +1$ and $\varrho =\mu -1$), then the exponential becomes

$$\begin{array}{c}\hfill e^{\frac{2\pi ih{r}_0^2}{2^{\kappa -\delta }{k}_0}{2}^{\ell +2\varrho -\delta }\left(1,-,2^{\kappa },g_0,{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}\right)}.\end{array}$$

Since $gcd(g_0,\frac{k_0}{g_0})=g_1=1$ and $k_0$ is odd, to determine $1-2^{\kappa }{g}_0{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}\left(\text{mod},,2^{\kappa -\delta },k_0\right)$ the Chinese Remainder Theorem implies that it suffices to compute

$$\begin{array}{cc}\hfill 1-2^{\kappa }{g}_0{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}& \equiv 1\left(\text{mod},,g_0\right),1-2^{\kappa }{g}_0{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}\equiv 0\left(\text{mod},,\frac{k_0}{g_0}\right),\hfill \\ \hfill 1-2^{\kappa }{g}_0{\left[2^{\kappa },g_0\right]}_{\frac{k_0}{g_0}}& \equiv 1\left(\text{mod},,2^{\kappa -\delta }\right).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill 1-2^{\kappa }{g}_0{\left[g_0\right]}_{\frac{k_0}{g_0}}\equiv \frac{k_0}{g_0}{\left[\frac{k_0}{g_0}\right]}_{2^{\kappa -\delta }{g}_0}\left(\text{mod},,2^{\kappa -\delta },k_0\right).\end{array}$$

So the exponential simplifies in this case as $e^{\frac{2\pi ih{r}_0^2}{2^{\kappa -\delta }{g}_0}{2}^{\ell +2\varrho -\delta }{[\frac{k_0}{g_0}]}_{2^{\kappa -\delta }{g}_0}}$.

The remaining case $\kappa \ge \ell +2\mu +2$ and $\varrho =\mu$ follows by a similar but longer and more tedious calculation. $\square$

Growth Toward the Cusps of Certain Modular Forms

In this section, we determine the growth toward the cusps of theta functions ${\mathrm{\Theta }}_{r,M,\mathit{\alpha }}$ and certain (non-holomorphic) Eisenstein series. The purpose of this calculation is to compare the growth in order to determine the unique Eisenstein series $E_{r,M,\mathit{\alpha }}$ in (1.5) whose growth toward the cusps matches that of the theta function.

Growth of the Theta Functions at the Cusps

In order to obtain the Eisenstein series, we determine the growth of ${\mathrm{\Theta }}_{r,M,\mathit{\alpha }}$ toward all of the cusps, which follows by a straightforward calculation.

Lemma 3.1

Let $m\in {\mathbb{N}}_{\ge 3}$ and $\mathit{\alpha }\in {\mathbb{N}}^4$ be given. For $h\in \mathbb{Z}$ and $k\in \mathbb{N}$ with $gcd(h,k)=1$, we have

$$\begin{array}{cc}\hfill & -\underset{z\to 0^{+}}{lim}{z}^2{\mathrm{\Theta }}_{r,M,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\hfill \\ \hfill & =-\frac{1}{4k^2{M}^4{\prod }_{j=1}^4\sqrt{{\alpha }_j}}\prod _{j=1}^4{e}^{\frac{2\pi i{r}^{2}h{\alpha }_j}{k}}G\left(h,{\alpha }_j,M^2,,,2,h,r,{\alpha }_j,M,;,k\right).\hfill \end{array}$$

Proof

We have

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{r,M,\mathit{\alpha }}(\tau )=\sum _{\begin{array}{c}\mathit{x}\in {\mathbb{Z}}^d\\ x_j\equiv r\left(\text{mod},,M\right)\end{array}}{q}^{{\sum }_{j=1}^4{\alpha }_j{x}_j^2}=\prod _{j=1}^4\vartheta (r,M;2M{\alpha }_j\tau ),\end{array}$$ 3.1

where

$$\begin{array}{c}\hfill \vartheta (r,M;\tau ):=\sum _{n\equiv r\left(\text{mod},,M\right)}{q}^{\frac{n^2}{2M}}.\end{array}$$

By definition,

$$\begin{array}{c}\hfill \vartheta \left(r,,,M,;,2,M,{\alpha }_j,\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\right)=\sum _{\begin{array}{c}n\equiv r\left(\text{mod},,M\right)\end{array}}{e}^{2\pi i{\alpha }_j{n}^2\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)}.\end{array}$$

Write $n=r+M\alpha +Mk\ell$ ($\alpha \left(\text{mod},,k\right),\ell \in \mathbb{Z}$) to obtain that this equals

$$\begin{array}{cc}\hfill & \sum _{\alpha \left(\text{mod},,k\right)}{e}^{2\pi i{\alpha }_j{(r+M\alpha )}^2\frac{h}{k}}\sum _{\ell \in \mathbb{Z}}{e}^{-2\pi {(r+M\alpha +Mk\ell )}^2\frac{{\alpha }_j}{k}z}\hfill \\ \hfill & =\sum _{\alpha \left(\text{mod},,k\right)}{e}^{2\pi i{\alpha }_j{(r+M\alpha )}^2\frac{h}{k}}\vartheta (r+M\alpha ,Mk;2M{\alpha }_{j}iz).\hfill \end{array}$$

We now recall the modular inversion formula (see [19, (2.4)])

$$\begin{array}{c}\hfill \vartheta \left(r,,,M,;,-,\frac{1}{\tau }\right)=M^{-\frac{1}{2}}{(-i\tau )}^{\frac{1}{2}}\sum _{\nu \left(\text{mod},,M\right)}{e}^{\frac{2\pi i\nu k}{M}}\vartheta (\nu ,M;\tau ).\end{array}$$

We use this with $\tau =\frac{i}{2M{\alpha }_{j}z}$, $r\mapsto r+M\alpha$, $M\mapsto Mk$ to obtain that

$$\begin{array}{cc}\hfill & \vartheta (r+M\alpha ,Mk;2M{\alpha }_{j}iz)\hfill \\ \hfill & ={(Mk)}^{-\frac{1}{2}}{\left(\frac{1}{2M{\alpha }_{j}z}\right)}^{\frac{1}{2}}\sum _{\nu \left(\text{mod},,M,k\right)}{e}^{\frac{2\pi i}{\mathit{Mk}}(r+M\alpha )\nu }\vartheta \left(\nu ,,,M,k,;,\frac{i}{2M{\alpha }_{j}z}\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \vartheta \left(r,,,M,;,2,M,{\alpha }_j,\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\right)=& \frac{1}{M}\sqrt{\frac{1}{2k{\alpha }_{j}z}}\sum _{\alpha \left(\text{mod},,k\right)}{e}^{2\pi i{\alpha }_j{(r+M\alpha )}^2\frac{h}{k}}\hfill \\ \hfill & \times \sum _{\nu \left(\text{mod},,M,k\right)}{e}^{\frac{2\pi i}{\mathit{Mk}}(r+M\alpha )\nu }\vartheta \left(\nu ,,,M,k,;,\frac{i}{2M{\alpha }_{j}z}\right).\hfill \end{array}$$

Now assume that $z\in {\mathbb{R}}^{+}$ and let $z\to 0^{+}$. The contribution that is not exponentially decaying comes from $\nu =0$ and gives

$$\begin{array}{c}\hfill \underset{z\to 0^{+}}{lim}\sqrt{z}\vartheta \left(r,,,M,;,2,M,{\alpha }_j,\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\right)=\frac{1}{M}\sqrt{\frac{1}{2k{\alpha }_j}}\sum _{\alpha \left(\text{mod},,k\right)}{e}^{2\pi i{\alpha }_j{(r+M\alpha )}^2\frac{h}{k}}.\end{array}$$

Note that

$$\begin{array}{cc}\hfill \sum _{\alpha \left(\text{mod},,k\right)}{e}^{2\pi i{\alpha }_j{(r+M\alpha )}^2\frac{h}{k}}& =e^{\frac{2\pi i{\alpha }_j{r}^{2}h}{k}}\sum _{\alpha \left(\text{mod},,k\right)}{e}^{\frac{2\pi i}{k}\left({\alpha }_j,M^2,{\alpha }^2,+,2,r,{\alpha }_j,M,\alpha \right)h}\hfill \\ \hfill & =e^{\frac{2\pi i{\alpha }_j{r}^{2}h}{k}}G\left(h,{\alpha }_j,M^2,,,2,h,r,{\alpha }_j,M,;,k\right).\hfill \end{array}$$

Plugging back into (3.1) yields the claim. $\square$

We next use Lemma 2.7 to evaluate the right-hand side of Lemma 3.1. Since the theta function ${\mathrm{\Theta }}_{r,M,\mathit{\alpha }}$ only depends on r modulo M, we may assume without loss of generality that

$$\begin{array}{c}\hfill \varrho ={\text{ord}}_2(r)\le {\text{ord}}_2(M)=\mu \end{array}$$

by replacing r with $r+M$ in Lemma 3.1 if $\varrho >\mu$. A direct calculation gives the following.

Corollary 3.2

Suppose that $h\in \mathbb{Z}$ and $k\in \mathbb{N}$ with $gcd(h,k)=1$, $\mathit{\alpha }=(1,2,4,8)$, $r\in \mathbb{Z}$, and $M\in \mathbb{N}$ with $gcd(M,r)\in \{1,2,4\}$ and ${\text{ord}}_2(r)\le {\text{ord}}_2(M)$. If $g_1\ne 1$ or $\varrho <min(\mu ,k-\ell -\mu )-1$, then

$$\begin{array}{c}\hfill -\underset{z\to 0^{+}}{lim}{z}^2{\mathrm{\Theta }}_{r,M,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)=0.\end{array}$$

If $g_1=1$ and $\varrho \ge min(\mu ,k-\ell -\mu )-1$, then, setting ${\delta }_0:=min(\kappa ,2\varrho )$,

$$\begin{array}{cc}& -\underset{z\to 0^{+}}{lim}{z}^2{\mathrm{\Theta }}_{r,M,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\hfill \\ \hfill & =\left\{\begin{array}{cc}-\frac{2^{2\kappa -4\mu -5}}{M_0^4}{g}_0^2{e}^{\frac{2\pi ih{r}_0^2}{2^{\kappa -{\delta }_0}{g}_0}{2}^{2\varrho -{\delta }_0}15{\left[\frac{k_0}{g_0}\right]}_{2^{\kappa -{\delta }_0}{g}_0}}\hfill & \text{if}\kappa \le 2\mu ,\hfill \\ \hfill & \text{or}\kappa =2\mu +1\text{and}\varrho =\mu -1,\hfill \\ \frac{g_0^2}{M_0^4}{e}^{\frac{2\pi ih{r}_0^2}{g_0}15{\left[2^{\kappa -2\mu },\frac{k_0}{g_0}\right]}_{g_0}}\hfill & \text{if}\kappa \ge 2\mu +5\text{and}\varrho =\mu ,\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}\right)\hfill \end{array}$$

Remark

Although the right-hand side of Corollary 3.2 splits into a number of cases, we obtain an explicit element of the cyclotomic field $\mathbb{Q}({\zeta }_{2^j{g}_0})$ for some $j\in {\mathbb{N}}_0$, where ${\zeta }_{\nu }:=e^{\frac{2\pi i}{\nu }}$. To use Corollary 3.2 for practical purposes, one can evaluate the right-hand side of Corollary 3.2 with a computer as an element of $\mathbb{Q}({\zeta }_{\nu })\cong \mathbb{Q}[x]/〈f_{\nu }〉$, where $f_{\nu }$ is the minimal polynomial of ${\zeta }_{\nu }$ over $\mathbb{Q}$, which is well known to be

$$\begin{array}{c}\hfill f_{\nu }(x)=\prod _{\begin{array}{c}1\le k\le \nu \\ gcd(k,\nu )=1\end{array}}\left(x,-,{\zeta }_{\nu }^k\right)=\prod _{d\mid \nu }{\left(x^d,-,1\right)}^{\mu \left(\frac{\nu }{d}\right)}.\end{array}$$

Here, $\mu$ denotes the Möbius $\mu$-function.

Growth of Eisenstein Series Toward the Cusps

The goal of this section is to obtain the growth of certain weight two Eisenstein series toward the cusps. These are formed by applying certain sieving and V-operators to the (non-holomorphic but modular) weight two Eisenstein series

$$\begin{array}{c}\hfill {\widehat{E}}_2(\tau ):=E_2(\tau )-\frac{3}{\pi v},\text{where}{E}_2(\tau ):=1-24\sum _{n\ge 1}\sigma (n)q^n\end{array}$$

with $\sigma (n):={\sigma }_1(n)$. In light of Lemma 2.3, we may furthermore always assume without loss of generality that sieving is applied before the V-operator. The growth toward the cusps of such functions is given in the following lemma.

Lemma 3.3

Let $m\in \mathbb{Z}$ and $M_1,M_2\in \mathbb{N}$. Then, for $h\in \mathbb{Z}$ and $k\in \mathbb{N}$ with $gcd(h,k)=1$, we have

$$\begin{array}{cc}\hfill & -\underset{z\to 0^{+}}{lim}{z}^2{\widehat{E}}_2|S_{M_1,m}|V_{M_2}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\hfill \\ \hfill & =\frac{1}{M_1^3{M}_2^2}\sum _{j\left(\text{mod},,M_1\right)}gcd{\left(h,M_1,M_2,+,j,k,,,M_1,k\right)}^2{\zeta }_{M_1}^{-jm}.\hfill \end{array}$$

Proof

For a translation-invariant function f, we use the presentation

$$\begin{array}{c}\hfill f|S_{M_1,m}(\tau )=\frac{1}{M_1}\sum _{j=0}^{M_1-1}{\zeta }_{M_1}^{-jm}f\left(\tau ,+,\frac{j}{M_1}\right).\end{array}$$

Applying $V_{M_2}$ to this yields

$$\begin{array}{c}\hfill f|S_{M_1,m}|V_{M_2}(\tau )=\frac{1}{M_1}\sum _{j=0}^{M_1-1}{\zeta }_{M_1}^{-jm}f\left(M_2,\tau ,+,\frac{j}{M_1}\right).\end{array}$$

Plugging in $f={\widehat{E}}_2$ and using the weight two modularity of ${\widehat{E}}_2$, the claim follows by a standard calculation. $\square$

Eisenstein Series Component

In this section, we determine the Eisenstein series component $E_{r,M,\mathit{\alpha }}$ in (1.5).

Proposition 4.1

For $n\in \mathbb{N}$, we have the following.

1. For $m=7$, we have $a_{7,10,(1,2,4,8)}(n)=0$ unless $n\equiv 15\left(\text{mod},,40\right)$, in which case we have

   $$\begin{array}{c}\hfill a_{7,10,(1,2,4,8)}(n)=\frac{1}{240}\left(\sigma ,(n),-,\sigma ,\left(\frac{n}{5}\right)\right).\end{array}$$
2. For $m=9$, we have $a_{9,14,(1,2,4,8)}(n)=0$ unless $n\equiv 39\left(\text{mod},,56\right)$, in which case we have

   $$\begin{array}{c}\hfill a_{9,14,(1,2,4,8)}(n)=\frac{1}{672}\sigma (n).\end{array}$$
3. For $m=10$, we have $a_{10,16,(1,2,4,8)}(n)=0$ unless $n\equiv 28\left(\text{mod},,64\right)$, in which case we have

   $$\begin{array}{c}\hfill a_{10,16,(1,2,4,8)}(n)=\frac{1}{256}\sigma \left(\frac{n}{4}\right).\end{array}$$
4. For $m=11$, we have $a_{11,18,(1,2,4,8)}(n)=0$ unless $n\equiv 15\left(\text{mod},,72\right)$, in which case we have

   $$\begin{array}{c}\hfill a_{11,18,(1,2,4,8)}(n)=\frac{1}{1728}\sigma (n).\end{array}$$
5. For $m=12$, we have $a_{12,20,(1,2,4,8)}(n)=0$ unless $80\mid n$, in which case we have

   $$\begin{array}{cc}\hfill a_{12,20,(1,2,4,8)}(n)=& \frac{1}{120}\left(\sigma ,\left(\frac{n}{16}\right),-,\sigma ,\left(\frac{n}{32}\right),-,\sigma ,\left(\frac{n}{80}\right),+,\sigma ,\left(\frac{n}{160}\right)\right)\hfill \\ \hfill & \left(+,8,\sigma ,\left(\frac{n}{256}\right),-,32,\sigma ,\left(\frac{n}{512}\right)\right)\hfill \\ \hfill & \left(-,8,\sigma ,\left(\frac{n}{1280}\right),+,32,\sigma ,\left(\frac{n}{2560}\right)\right).\hfill \end{array}$$
6. For $m=13$, we have $a_{13,22,(1,2,4,8)}(n)=0$ unless $n\equiv 71\left(\text{mod},,88\right)$, in which case we have

   $$\begin{array}{c}\hfill a_{13,22,(1,2,4,8)}(n)=\frac{1}{2640}\sigma (n).\end{array}$$
7. For $m=14$, we have

   $$\begin{array}{c}\hfill a_{14,24,(1,2,4,8)}(n)=\left\{\begin{array}{cc}\frac{1}{768}\left(\sigma ,\left(\frac{n}{4}\right),-,\sigma ,\left(\frac{n}{12}\right)\right)\hfill & \text{if}n\equiv 60\left(\text{mod},,96\right),\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}\right)\end{array}$$

Proof

(1) By comparing Fourier coefficients, we see that the identity is equivalent to

$$\begin{array}{c}\hfill E_{7,10,(1,2,4,8)}=-\frac{1}{5760}{E}_2|\left(1,-,V_5\right)|S_{40,15}.\end{array}$$ 4.1

Lemma 2.1 and (1.5) give that

$$\begin{array}{c}\hfill E_{7,10,\left(1,,,2,,,4,,,8\right)}\in M_2({\mathrm{\Gamma }}_{3200,10}),\end{array}$$

while Lemma 2.2 implies that

$$\begin{array}{c}\hfill E_2|\left(1,-,V_5\right)|S_{40,15}\in M_2\left({\mathrm{\Gamma }}_1,\left(1600\right)\right).\end{array}$$

Enumerating the cusps of ${\mathrm{\Gamma }}_1(3200)$ (see [10, Proposition 3.8.3]), we then use a computer together with Lemma 3.3 and Corollary 3.2 to verify that the growth toward every cusp of both sides of (4.1) agrees, yielding the claim.

To see this in more details note that by [10, Proposition 3.8.3], two cusps $\frac{a}{c}$ and $\frac{\alpha }{\gamma }$ are equivalent modulo the action of ${\mathrm{\Gamma }}_1(N)$ if and only if there exists $j\in \mathbb{Z}$ such that $(\alpha ,\gamma )\equiv \pm (a+jc,c)\left(\text{mod},,N\right)$ (for some choice of ±). As in [10, p. 102], by taking $d:=gcd(c,N)$, we may write a set of representatives of the inequivalent cusps in the form $\frac{a}{d\gamma }$ with $d\mid N$, a running modulo d with $gcd(a,d)=1$, and $1\le \gamma \le \lceil \frac{N}{2d}\rceil$ with $gcd(\gamma ,\frac{N}{d})=1$. Since both sides of (4.1) are elements of $M_2({\mathrm{\Gamma }}_1(3200))$, we thus need to compute the constant term at every cusp $\frac{h}{k}$ with $h,k\in \mathbb{Z}$, $gcd(h,k)=1$, and $k=d\gamma$ with $d\mid 3200$ and $1\le \gamma \le \frac{1600}{d}$ with $gcd(\gamma ,\frac{3200}{d})=1$. For each such representative $\frac{h}{k}$ of the cusps of ${\mathrm{\Gamma }}_1(3200)$, we use Lemma 3.3 together with a computer to evaluate

$$\begin{array}{c}\hfill -\frac{1}{5760}\underset{z\to 0^{+}}{lim}{z}^2\left(E_2,|,S_{8,3},|,V_5,\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right),-,E_2,|,S_{40,15},\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\right)\end{array}$$

as an element of $\mathbb{Q}({\zeta }_{40})$. Comparing this with Corollary 3.2 in the case $r=7$ and $M=10$, we then verify with a computer that

$$\begin{array}{cc}\hfill & -\frac{1}{5760}\underset{z\to 0^{+}}{lim}{z}^2\left(E_2,|,S_{8,3},|,V_5,\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right),-,E_2,|,S_{40,15},\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)\right)\hfill \\ \hfill & =-\underset{z\to 0^{+}}{lim}{z}^2{\mathrm{\Theta }}_{7,10,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right).\hfill \end{array}$$ 4.2

Since $f_{7,10,(1,2,4,8)}$ is a cusp form, we have

$$\begin{array}{c}\hfill -\underset{z\to 0^{+}}{lim}{z}^2{f}_{7,10,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)=0\end{array}$$

and hence

$$\begin{array}{c}\hfill -\underset{z\to 0^{+}}{lim}{z}^2{\mathrm{\Theta }}_{7,10,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right)=-\underset{z\to 0^{+}}{lim}{z}^2{E}_{7,10,\mathit{\alpha }}\left(\frac{h}{k},+,\frac{\mathit{iz}}{k}\right).\end{array}$$

Therefore, (4.2) implies that

$$\begin{array}{c}\hfill E_{7,10,(1,2,4,8)}+\frac{1}{5760}{E}_2|(1-V_5)S_{40,15}\end{array}$$

vanishes toward all cusps, and is hence a cusp form. Since it is also in the subspace of Eisenstein series, it is orthogonal to all cusp forms and therefore vanishes, implying (4.1), and hence the claim.

For the remaining cases, the argument is similar, but we provide the identities analogous to (4.1) for the convenience of the reader.

• (2)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{9,14,(1,2,4,8)}=-\frac{1}{16128}{E}_2|S_{56,39}.\end{array}$$
• (3)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{10,16,(1,2,4,8)}=-\frac{1}{6144}{E}_2|S_{16,7}|V_4.\end{array}$$
• (4)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{11,18,(1,2,4,8)}=-\frac{1}{41472}{E}_2|S_{72,15}.\end{array}$$
• (5)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{12,20,(1,2,4,8)}=-\frac{1}{2880}{E}_2|\left(S_{5,0},-,V_5\right)|\left(1,-,V_2,+,8,V_{16},-,32,V_{32}\right)|V_{16}.\end{array}$$
• (6)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{13,22,(1,2,4,8)}=-\frac{1}{63360}{E}_2|S_{88,71}.\end{array}$$
• (7)
  The claim is equivalent to

  $$\begin{array}{c}\hfill E_{14,24,(1,2,4,8)}=-\frac{1}{18432}{E}_2|\left(1,-,V_3\right)|S_{24,15}|V_4.\end{array}$$

  $\square$

As a corollary to Proposition 4.1, we obtain explicit lower bounds on the Fourier coefficients $a_{r,M,\mathit{\alpha }}(n)$ in these special cases.

Corollary 4.2

Let $n\in \mathbb{N}$.

1. If $n\equiv 15\left(\text{mod},,40\right)$, then we have

   $$\begin{array}{c}\hfill a_{7,10,(1,2,4,8)}(n)\ge \frac{n}{240}.\end{array}$$
2. If $n\equiv 39\left(\text{mod},,56\right)$, then we have

   $$\begin{array}{c}\hfill a_{9,14,(1,2,4,8)}(n)\ge \frac{n}{672}.\end{array}$$
3. If $n\equiv 28\left(\text{mod},,64\right)$, then we have

   $$\begin{array}{c}\hfill a_{10,16,(1,2,4,8)}(n)\ge \frac{n}{1024}.\end{array}$$
4. If $n\equiv 15\left(\text{mod},,72\right)$, then we have

   $$\begin{array}{c}\hfill a_{11,18,(1,2,4,8)}(n)\ge \frac{n}{1728}.\end{array}$$
5. Assume that $80\mid n$ and write $n=2^a{5}^{b}c$ with $gcd(10,c)=1$. We have

   $$\begin{array}{c}\hfill a_{12,20,(1,2,4,8)}(n)\ge \frac{5^{b}c}{120}\left\{\begin{array}{cc}{2}^{a-4}\hfill & \text{if}4\le a\le 7,\hfill \\ 24\hfill & \text{if}a\ge 8.\hfill \end{array}\right)\end{array}$$
6. If $n\equiv 71\left(\text{mod},,88\right)$, then we have

   $$\begin{array}{c}\hfill a_{13,22,(1,2,4,8)}(n)\ge \frac{n}{2640}.\end{array}$$
7. If $n\equiv 60\left(\text{mod},,96\right)$, then we have

   $$\begin{array}{c}\hfill a_{14,24,(1,2,4,8)}(n)\ge \frac{n}{3072}.\end{array}$$

Proof

For $m\ne 12$, the claims with the exception of (5) follow directly from Proposition 4.1. For (5), a direct simplification yields that the right-hand side of Proposition 4.1 (5) simplifies as

$$\begin{array}{c}\hfill \frac{5^b\sigma (c)}{120}\left\{\begin{array}{cc}{2}^{a-4}\hfill & \text{if}4\le a\le 7,\hfill \\ 24\hfill & \text{if}a\ge 8,\hfill \end{array}\right)\end{array}$$

which gives the claim.$\square$

Proof of Theorem 1.2

In this section, we prove Theorem 1.2. The constants $C_m$ from the theorem statement may be found in Table 2.

Proof of Theorem 1.2

We require the case $\ell =4$ of Lemma 2.5. Since the inner sum only has a single term namely $m=0$ in this case, Lemma 2.5 simplifies as

$$\begin{array}{cc}\hfill & \Vert f_{r,M,\mathit{\alpha }}{\Vert }^2\le \frac{2·3^6}{{\pi }^4}\frac{M^4{N}_{\mathit{\alpha }}^2}{{\prod }_{p\mid M^2{N}_{\mathit{\alpha }}}\left(1,-,p^{-2}\right)}\hfill \\ \hfill & \times \sum _{\delta \mid M^2{N}_{\mathit{\alpha }}}\phi \left(\frac{M^2{N}_{\mathit{\alpha }}}{\delta }\right)\phi (\delta )\frac{M^2{N}_{\mathit{\alpha }}}{\delta }{\left(\frac{gcd(M^2,\delta )}{M^2}\right)}^{4}2\left(\frac{27}{D_{\mathit{\alpha }}},\frac{M^2{N}_{\mathit{\alpha }}}{\pi },+,16\right).\hfill \end{array}$$ 5.1

For ${\theta }_{r,M,(1,2,4,8)}$, we obtain a lower bound for $a_{r,M,(1,2,4,8)}(n)$ (for n in an appropriate congruence class) from Corollary 4.2 (see the third column of Table 1 for a list of the bounds for individual choices of r and M).

Table 1.

Bounds for $a_{r,M,(1,2,4,8)}$, $\Vert f_{r,M,(1,2,4,8)}\Vert$, and $|b_{r,M,(1,2,4,8)}|$

r	M	Bound for $a_{r,M,(1,2,4,8)}$	Bound for $\Vert f_{r,M,(1,2,4,8)}\Vert$	Bound for $|b_{r,M,(1,2,4,8)}|$
7	10	$\frac{n}{240}$	$8.11·{10}^{14}$	$3.41·{10}^{30}{n}^{\frac{3}{5}}$
9	14	$\frac{n}{672}$	$1.03·{10}^{16}$	$3.48·{10}^{31}{n}^{\frac{3}{5}}$
10	16	$\frac{n}{1024}$	$3.2·{10}^{16}$	$9.98·{10}^{31}{n}^{\frac{3}{5}}$
11	18	$\frac{n}{1728}$	$6.1·{10}^{16}$	$1.52·{10}^{32}{n}^{\frac{3}{5}}$
12	20	$\frac{n}{1920}$	$1.49·{10}^{17}$	$3.69·{10}^{32}{n}^{\frac{3}{5}}$
13	22	$\frac{n}{2640}$	$2.55·{10}^{17}$	$6.96·{10}^{32}{n}^{\frac{3}{5}}$
14	24	$\frac{n}{3072}$	$5.63·{10}^{17}$	$1.09·{10}^{33}{n}^{\frac{3}{5}}$

Computing the constants in (5.1) explicitly for fixed M yields an upper bound for $\Vert f_{r,M,(1,2,4,8)}{\Vert }^2$ (see the fourth column of Table 1 for the explicit bounds), which plugged into Lemma 2.4 yields an upper bound for $|b_{r,M,(1,2,4,8)}(n)|$ (see the final column of Table 1 for the explicit bounds). Plugging the bounds for $a_{r,M,(1,2,4,8)}(n)$ and $|b_{r,M,(1,2,4,8)}(n)|$ into (2.1), we see that $s_{r,M,(1,2,4,8)}(n)>0$ for n sufficiently large in an appropriate congruence class (see Table 2 for the explicit constants).

We then conclude that $r_{m,(1,2,4,8)}>0$ for n sufficiently large by using (1.3), yielding the claim. $\square$

In order to explain why it is sufficient to assume that $n\not\equiv 4\left(\text{mod},,16\right)$ for $m=12$ in Theorem 1.2, we require the following lemma combined with (1.3).

Lemma 5.1

Let $n\in \mathbb{N}$ be given. If the equation

$$\begin{array}{c}\hfill x_1^2+2x_2^2+4x_3^2+8x_4^2=n\end{array}$$

is solvable with $x_j\equiv 12\left(\text{mod},,20\right)$, then the equation

$$\begin{array}{c}\hfill x_1^2+2x_2^2+4x_3^2+8x_4^2=256n\end{array}$$

is also solvable with $x_j\equiv 12\left(\text{mod},,20\right)$.

Funding

Open Access funding enabled and organized by Projekt DEAL.

Declarations

Conflict of interest

On behalf of all authors, the corresponding author, Kathrin Bringmann, states that there is no conflict of interest.