Proofs of the Conjectures on $SOME(n)$ and $DSOME(n)$ Functions Related to Integer Partitions

Technical Summary of "Proofs of the Conjectures on SOME(n) and DSOME(n) Functions Related to Integer Partitions"

Problem Statement
This paper addresses two specific conjectures regarding the arithmetic properties of two partition-related functions, $SOME(n)$ and $DSOME(n)$, introduced by Andrews and Dastidar (2026) and Baruah and Gogoi (2026), respectively.

• $SOME(n)$ is defined as the sum of all odd parts minus the sum of all even parts across all partitions of a positive integer $n$.
• $DSOME(n)$ is the analogous difference (sum of odd parts minus sum of even parts) restricted to partitions of $n$ into distinct parts.

The primary objectives are to prove:

1. The Andrews-Dastidar conjecture: For integers $\alpha \ge 1$ and $\lambda \ge 0$ such that $24\lambda \equiv 1 \pmod{5^\alpha}$, $SOME(\lambda) \equiv 0 \pmod{5^\alpha}$.
2. The Baruah-Gogoi conjecture: $DSOME(50n + 21) \equiv 0 \pmod 8$ for all integers $n \ge 0$.

Additionally, the authors aim to establish new infinite families of congruences for $DSOME(n)$ modulo 2, 4, and 8.

Methodology
The authors employ techniques from the theory of $q$-series, modular forms, and partition identities. The methodology relies heavily on generating functions and the manipulation of $q$-Pochhammer symbols, denoted as $(a; q)_\infty$.

• Generating Functions: The paper utilizes the known generating functions for $SOME(n)$ and $DSOME(n)$, expressed in terms of infinite products and sums involving $q$-series.
• Modular Arithmetic and Congruences: The proofs involve extracting coefficients from generating functions and analyzing them modulo specific integers (2, 4, 5, 8, and powers of 5).
• Ramanujan's Theta Functions: The authors utilize Ramanujan's general theta function $f(\alpha, \beta)$ and its special cases $\phi(q)$, $\psi(q)$, and $f(-q)$ to transform complex series into more manageable forms.
• Operator Theory: An operator $U_2$ is defined to extract terms with even indices from power series, facilitating the derivation of congruences for specific arithmetic progressions (e.g., $DSOME(2n)$).
• Eisenstein Series and Divisor Sums: The proof for the $SOME(n)$ conjecture connects the function to the classical quasimodular Eisenstein series $E_2(q)$ and the sum of divisors function $\sigma(n)$. It leverages known congruences for the partition function $p(n)$ and a related coefficient sequence $c(n)$ established by Wang and Yang.
• Combinatorial Analysis: For the $DSOME(n)$ proofs, the authors analyze the parity of coefficients in series expansions involving sums of squares and generalized pentagonal numbers, often reducing the problem to counting solutions to Diophantine equations modulo specific primes.

Key Contributions and Results

1. Proof of the Andrews-Dastidar Conjecture:
   The authors establish that $SOME(\lambda) \equiv 0 \pmod{5^\alpha}$ whenever $24\lambda \equiv 1 \pmod{5^\alpha}$.

   • Mechanism: They derive a relation linking $SOME(n)$ to a coefficient $c(2n)$ and the partition function $p(n)$: $12SOME(n) = c(2n) - (12n+1)p(n)$.
   • Logic: By utilizing known congruences for $c(n)$ (derived from Wang and Yang's work on quasimodular forms) and the classical Ramanujan congruences for $p(n)$, they demonstrate that both terms on the right-hand side vanish modulo $5^\alpha$ under the given condition. Since $\gcd(12, 5^\alpha)=1$, the result follows.

2. Proof of the Baruah-Gogoi Conjecture:
   The paper proves that $DSOME(50n + 21) \equiv 0 \pmod 8$.

   • Mechanism: This is established as a specific case ($t=2$) of a more general theorem (Theorem 4.11) stating that $DSOME(50n + 10t + 1) \equiv 0 \pmod 8$ for $1 \le t \le 4$.
   • Logic: The proof involves expanding the generating function of $DSOME(n)$ modulo 8 into components involving $f_1$, $X(q)$, and their derivatives. The authors analyze the coefficients of $q^N$ where $N = 50n + 10t + 1$ by relating them to the 5-adic valuation of $M = 24N + 1$. They show that the relevant coefficients vanish due to the non-existence of certain integer solutions to quadratic forms modulo 5.

3. New Infinite Families of Congruences:
   The paper establishes several new congruences for $DSOME(n)$:

   • Modulo 2: A complete characterization of when $DSOME(n) \equiv 1 \pmod 2$ is provided, showing it occurs if and only if $n$ is of the form $24j^2 + 10j + 1$ or $24j^2 + 22j + 5$. Corollaries provide conditions for $DSOME(Mn+r) \equiv 0 \pmod 2$ based on Legendre symbols.
   • Modulo 4: Similar characterizations are derived for $DSOME(2n) \pmod 4$, identifying specific quadratic forms ($48k^2 + 14k + 1$ and $48k^2 + 46k + 11$) where the function is non-zero.
   • Modulo 8: A convolution formula is derived expressing $DSOME(n)$ modulo 8 in terms of the partition function into distinct parts $pd(n)$ and divisor sums $\sigma_3(n)$ and $\sigma(n)$.

Significance
The paper claims significance primarily in the resolution of open conjectures regarding the arithmetic behavior of $SOME(n)$ and $DSOME(n)$. By proving these conjectures, the authors confirm the existence of deep structural patterns in these partition functions that mirror Ramanujan-type congruences found in the standard partition function $p(n)$. The work extends the understanding of how parity and modular constraints apply to weighted sums of partition parts, specifically for distinct partitions. The establishment of new infinite families of congruences modulo 2, 4, and 8 provides a broader framework for future investigations into the arithmetic properties of partition-related functions.