Generalized polygonal number representations

Technical Summary: Generalized Polygonal Number Representations

Problem Statement
The paper addresses the problem of counting the number of representations of a positive integer $N$ as a sum of $n$ generalized $k$-gonal numbers, denoted $r_k^n(N)$. A generalized $k$-gonal number is defined by the polynomial $p_k(x) = \frac{(k-2)x^2 - (k-4)x}{2}$ for $x \in \mathbb{Z}$. This contrasts with "ordinary" $k$-gonal numbers, where the argument is restricted to non-negative integers ($x \in \mathbb{N}_0$), denoted $r_{k,+,n}(N)$.

While the existence of such representations is a well-studied area (e.g., the Fermat–Cauchy polygonal number theorem), the asymptotic behavior of the representation functions themselves has received less attention. The primary goal is to establish a closed-form asymptotic relation between $r_k^n(N)$ and $r_n^\square(N)$, the number of representations of $N$ as a sum of $n$ squares. The author notes that while $k=3$ (triangular) and $k=4$ (squares) possess symmetries allowing for direct relationships between ordinary and generalized counts, these symmetries vanish for $k \ge 5$, making the derivation of such relations difficult.

Methodology
The core methodology involves modifying the Heath-Brown circle method, originally developed for quadratic forms, to count solutions to the sum of $n$ squares equation subject to fixed congruence conditions.

1. Reduction to Congruence Conditions: By completing the square on the equation $\sum_{i=1}^n p_k(x_i) = N$, the problem is transformed into counting solutions to $\sum_{i=1}^n x_i^2 = X_N$ (where $X_N = 8(k-2)N + n(k-4)^2$) subject to the congruence $x_i \equiv 2b \pmod{4a}$ (specifically $x_i \equiv k \pmod{2(k-2)}$).
2. Heath-Brown Circle Method Adaptation: The author adapts the weighted counting function approach from Heath-Brown's work on quadratic forms. By imposing the congruence condition $x \equiv \mathbf{k} \pmod M$ on the solution set, the author derives an asymptotic formula for the number of such solutions, $r_n^\equiv(N)$.
3. Singular Series and Densities: The asymptotic formula separates into a singular integral (which remains invariant under the congruence modification) and a singular series composed of $p$-adic densities. The author computes these densities explicitly:
   • Square Densities ($\delta_p^\square$): Calculated using standard Gauss sum evaluations.
   • Polygonal Densities ($\delta_p^k$): Calculated using generalized quadratic Gauss sums. The author shows that for odd primes $p | (k-2)$, the density is uniform, while the case $p=2$ requires careful case analysis based on $k \pmod 4$.
4. Moment Estimation: Using the derived asymptotic relation, the paper estimates the second moment of $r_k^4(N)$ by reducing the problem to estimating moments of the sum-of-squares function $r_4^\square(X_N)$, utilizing Dirichlet series and Perron's formula.

Key Contributions and Results

• Theorem 1.1 (Main Asymptotic Relation): For $n \ge 4$ and $k \ge 3$, the paper proves a closed-form asymptotic relation:
  $$r_k^n(N) = \frac{1 + (-1)^{\lfloor n/4 \rfloor} 4^{n-2} T(X_N, n, k)}{2^{n-3}(k-2)^{n-1} N_n(X_N, k-2)} r_n^\square(X_N) + O(N^{(n-1)/4 + \varepsilon})$$
  Here, $T$ and $N_n$ are explicit functions involving $p$-adic densities. This result generalizes the known relationship between sums of squares and sums of polygonal numbers to arbitrary $k \ge 3$ and $n \ge 4$.

• Corollary 1.2 (Second Moment): The paper derives the asymptotic behavior of the second moment of the representation function for four variables:
  $$\sum_{N \le x} r_k^4(N)^2 = C_k \zeta(3) \frac{1}{(k-2)^4} \prod_{p_i | (k-2)} \frac{1 - p_i^{-3}}{1 - p_i^{-4}} x^3 + O(x^{11/4 + \varepsilon})$$
  where $C_k$ depends on the divisibility of $k$ by 4.

• Corollary 1.3 (Ordinary Polygonal Numbers): By combining Theorem 1.1 with a recent result by Bringmann et al. relating ordinary and generalized counts ($r_{k,+,n}(N) \approx 2^{-n}r_k^n(N)$), the paper provides an asymptotic formula for $r_{k,+,4}(N)$. This recovers known results for triangular ($k=3$) and pentagonal ($k=5$) numbers as special cases.

• Corollary 1.4 (Boundedness of Subsequences): The paper analyzes the behavior of $r_{k,+,4}(N)$ on subsequences.

  • If $4 \nmid k$, there is no strictly increasing infinite subsequence on which $r_{k,+,4}(N)$ is bounded.
  • If $4 \mid k$, any strictly increasing infinite subsequence on which $r_{k,+,4}(N)$ is bounded must converge 2-adically to $(k-4)^2/(4-2k)$. This supplements a previous result by Meng and Sun regarding specific subsequences where the count is zero.

• Exact Identities: The paper notes that for specific cases, such as $n=4$ and $k=6$, the error term in the asymptotic relation vanishes, yielding an exact identity: $r_6^4(N) = \frac{1}{24} r_4^\square(32N + 16)$.

Significance and Claims
The paper claims to provide the first closed-form asymptotic relation between generalized $k$-gonal number representations and sums of squares for general $k \ge 3$ and $n \ge 4$. The author emphasizes that while the existence of representations is well-understood, the quantitative study of representation functions has been less explored.

The significance lies in:

1. Unification: Connecting the study of polygonal numbers to the classical theory of sums of squares via the Heath-Brown circle method.
2. Explicit Formulas: Providing explicit constants and error terms for the asymptotic behavior, which allows for the calculation of moments (as demonstrated for the second moment).
3. Refinement of Boundedness: Offering a 2-adic characterization of subsequences where the representation count remains bounded, extending previous work on the non-existence of such representations for specific forms.

The author modestly notes that the error term $O(N^{(n-1)/4 + \varepsilon})$ might be improvable, particularly for specific $k$ and $n$ (as seen in the $k=6$ case), and suggests that generalizing the relation to ordinary polygonal numbers for $n > 4$ remains a challenging open problem. The work is presented as an undergraduate thesis contribution to the active field of additive number theory.