Conjectures of Sun About Sums of Polygonal Numbers

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 ∈ N ≥3 and ∈ Z, let p m ( ) be the -th (generalized) m-gonal number For ∈ N, these count the number of dots contained in a regular polygon with m sides having 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 α ∈ N d , 1 we are interested in the solvability of the Diophantine equation 1≤ j≤d α j p m ( j ) = n (1.1) with j ∈ N 0 or j ∈ Z. We call such a sum universal if it is solvable for every n ∈ 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 1≤ j≤m p m ( 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 α ∈ 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 ⊆ N, one determines a finite subset S 0 of S such that a solution to (1.1) exists for every n ∈ S if and only if it exists for every n ∈ S 0 . Taking S = 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 ∈ Z d for all n ∈ N if and only if it is solvable for every n ≤ 8 [6], for m = 5 it is solvable with ∈ Z d for all n ∈ N if and only if it is solvable for every n ≤ 109 [13], while it is solvable with ∈ N d 0 for all n ∈ N if and only if it is solvable for every n ≤ 63 [14] and for m = 8 it is solvable for ∈ Z d for all n ∈ N if and only if it is solvable for every n ≤ 60 [15].
Here, we consider the question of universality in the case α = (1, 2, 4, 8) as one varies m. Specifically, we have the following conjecture of Sun (see [21,Conjecture 5.4]).

Remark
In this paper, we prove that Conjecture 1.1 is true for n sufficiently large. 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 we have r m, (1,2,4,8) (n) = s m,2(m−2), (1,2,4,8) Hence, since Conjecture 1.1 is equivalent to proving that r m, (1,2,4,8) (n) > 0 for every n ∈ N and m ∈ {7, 9, 10, 11, 12, 13, 14}, the conjecture is equivalent to showing that for every n ∈ N, we have 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 The paper is organized as follows. In Sect. 2, we recall properties of the theta functions r ,M,α , 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.

Modularity of the Generating Functions
In this subsection, we consider the modularity properties of the theta functions r ,M,α . To set notation, for 1 (N ) ⊆ ⊆ SL 2 (Z) (N ∈ N) and a character χ modulo N , let M k ( , χ ) be the space of modular forms of weight k with character χ . In particular, an element f in this space satisfies, for γ = a b c d ∈ , Setting N ,L := 0 (N ) ∩ 1 (L), by [7, Theorem 2.4], we have the following.

Operators on Non-holomorphic Modular Forms
For a translation-invariant function f with Fourier expansion (denoting we define the sieving operator (M, m ∈ N) As usual, we also define the V -operator (δ ∈ N) by 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. (1) For d ∈ N, the function f |V d satisfies weight k modularity on lcm(4,N d),L .
(2) For m ∈ Z and M ∈ N, the function f |S M,m satisfies weight k modularity on It is useful to determine the commutator relations between the V -operator and sieving.
Proof Recall that We may hence assume that d | m and we note that gcd(μ 1 , μ 2 ) = 1, obtaining

Decomposition Into Eisenstein Series and Cusp Forms
Comparing Fourier coefficients on both sides of (1.5), we have To obtain an upper bound for |b r ,M,α (n)|, we recall that Deligne [9] proved that for a normalized newform f (τ ) = n≥1 c f (n)q n of weight k on 0 (N ) with Nebentypus character χ (normalized so that c f (1) = 1), we have where σ k (n) := d|n d k . To obtain an explicit bound for |c f (n)| for arbitrary f ∈ S k ( 1 (N )), we combine (2.2) with a trick implemented by Blomer [5] and Duke [12].
For cusp forms f , g ∈ S k ( ), we define the Petersson inner product by a bound for |c f (n)| in terms of f may be obtained. Specifically, suppose that f is a cusp form f of weight k ∈ N on N ,L (with L | N ) and character χ 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 f . Denoting by ϕ Euler's totient function, we recall a bound from the case k = 2 below (see [1, (4.4)]).

Lemma 2.4
Suppose that f ∈ S 2 ( N ,L , χ) with L | N and χ a character modulo N . Then, we have the inequality By Lemma 2.4, in order to obtain an explicit bound for |b r ,M,α (n)|, it remains to estimate f r ,M,α , where f r ,M,α is the cusp form appearing in the decomposition in (1.5). An explicit bound for f r ,M,α was obtained in [16,Lemma 3.2]. To state the result, let α ∈ Z . For the quadratic form Q = Q α given by we define the level and the discriminant of Q α as Lemma 2.5 Let ≥ 4 be even, α ∈ N , r ∈ Z, and M ∈ N. Then,

Gauss Sums
Define the generalized quadratic Gauss sum (a, b ∈ Z, c ∈ N) 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 and we write [a] b for the inverse of a modulo b if gcd(a, b) = 1.

Lemma 2.6
For a, b ∈ Z and c, d ∈ N, the following hold.

Growth Toward the Cusps of Certain Modular Forms
In this section, we determine the growth toward the cusps of theta functions r ,M,α 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,α 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 r ,M,α toward all of the cusps, which follows by a straightforward calculation. By definition, Write n = r + Mα + Mk (α (mod k) , ∈ Z) to obtain that this equals We now recall the modular inversion formula (see [19, (2.4) We use this with τ = i 2Mα j z , r → r + Mα, M → Mk to obtain that ϑ(r + Mα, Mk; 2Mα j i z) Thus, Now assume that z ∈ R + and let z → 0 + . The contribution that is not exponentially decaying comes from ν = 0 and gives Note that Plugging back into (3.1) yields the claim.
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 Q(ζ 2 j g 0 ) for some j ∈ N 0 , where ν . 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 Q(ζ ν ) ∼ = Q[x]/ f ν , where f ν is the minimal polynomial of ζ ν over Q, which is well known to be Here, μ denotes the Möbius μ-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 with σ (n) := σ 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 ∈ Z and M 1 , M 2 ∈ N. Then, for h ∈ Z and k ∈ N with gcd(h, k) = 1, we have Proof For a translation-invariant function f , we use the presentation Applying V M 2 to this yields Plugging in f = E 2 and using the weight two modularity of E 2 , the claim follows by a standard calculation.

Eisenstein Series Component
In this section, we determine the Eisenstein series component E r ,M,α in (1.5). (2) For m = 9, we have a 9 1600)) .
Enumerating the cusps of 1 (3200) (see [ 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  As a corollary to Proposition 4.1, we obtain explicit lower bounds on the Fourier coefficients a r ,M,α (n) in these special cases.    Table 1 Bounds for a r ,M, (1,2,4,8) , f r ,M, (1,2,4,8) , and |b r ,M, (1,2,4,8) | r M Bound for a r ,M, (1,2,4,8) Bound for f r ,M, (1,2,4,8) Bound for |b r ,M, (1,2,4,8) | in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.