Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32033

Efficient Computation and Congruences for Colored Partition Functions

Jean Carlos Villegas-Morales
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a positive integer $α$, let $p_α(n)$ denote the number of $α$-colored partitions of $n$. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real $α>0$, but when $α>24$ it involves several polar terms and a two-parameter family of exponential sums $A_k^{(α)}(n,m)$. For integral $α$, we prove multiplicativity and prime-power reduction formulas for these sums, expressing their local factors as classical or quadratically twisted Kloosterman sums. Together with explicit truncation and precision bounds, these formulas yield an efficient algorithm for computing $p_α(n)$ exactly. Our SageMath implementation computes the 1,113,767-digit integer $p_{100}(10^{10})$ in less than one hour. As an application, we use the algorithm to certify new Ramanujan-type congruences.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jean Carlos Villegas-Morales. 2026-09-25. Efficient Computation and Congruences for Colored Partition Functions. https://arxiv.org/abs/2609.32033

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials

Let $f$ be a polynomial of degree $d>1$ in $n$ variables over $\mathbb{Z}$. Let $f_d$ be the homogeneous part of degree $d$ of $f$ and $s$ be the dimension of the critical locus of $f_d$. In this paper, we prove Igusa's conjecture for exponential sums with the exponent $(n-s)/(2(d-1))$. This implies a weak solution for a recent conjecture raised by Cluckers and the author (2020) about an analogue of the results of Deligne (1974) and Katz (1999) for exponential sums over finite fields in the finite ring setting. Moreover, this also improves the result of Cluckers, Mustaţă and the author (2019) in case $n-s>2(d-1)$. In particular, this result improves the conditions $n-s>2^d(d-1)$ of Birch (1962) and $n-s>3(d-1)2^{d-2}$ of Browning-Prendiville (2017) on the validity of the estimation for the major arcs to $(n-s)>4(d-1)$. Therefore this result may have further applications on subjects related to the Hardy-Littlewood circle method such as the Hasse principle or distribution of rational points in algebraic varieties. On the other hand, we also improve the recent work of Cluckers, Kollár and Mustaţă (2019) on the strong monodromy conjecture in the range $(-{\rm lct}((f)+J_f^2),0]$ in case of bad reduction and bad Schwartz-Bruhat function. Namely, in the range $(-{\rm lct}((f)+J_f^2),0]$, the real part of any pole of the Igusa local zeta functions associated with $f$ and any Schwartz-Bruhat function over any $p$-adic field is a root of the Bernstein-Sato polynomial of $f$.

math.NT↗
arXiv2026

Local points on twists of $X(p)$ with applications

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to~$E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb{Q}_\ell)$ is non-empty, for all primes $\ell$. We give two different applications. First, we classify CM curves $E/\mathbb{Q}$ for which the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many~$p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes~$p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.

math.NT↗
arXiv2026

Involution on a quotient space of multiple zeta values in positive characteristic

In this paper, we introduce $*$-inverse multiple zeta values and $*$-inverse Carlitz multiple polylogarithms in positive characteristic and study their algebraic structures and relations. Using special values of Carlitz multiple polylogarithms, we show that a natural quotient of the space of multiple zeta values in positive characteristic admits a Hopf algebra structure and that the original space admits a compatible comodule structure over this quotient. These structures may be regarded as function field analogues of the Hopf algebra and comodule structures arising from motivic multiple zeta values in characteristic zero. In particular, the antipode on this quotient gives a non-trivial involution corresponding to the $*$-inverse operation. We also establish, in certain cases, the $q$-shuffle product formula and linear relations for $*$-inverse multiple zeta values, providing evidence for our conjectures on their relations. Finally, we consider another characteristic-zero analogue of our construction, based on the Hopf algebra structure given by the harmonic product and the deconcatenation coproduct, and formulate a related problem.

math.NT↗