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Exponential sums over Möbius convolutions with applications to partitions

Debmalya Basak, Nicolas Robles, Alexandru Zaharescu

TL;DR

The paper addresses the asymptotic behavior of weighted partitions with Möbius-convolution weights, focusing on the difference between counts of even and odd admissible partitions. By developing sharp exponential-sum bounds for Möbius-convolution twists and applying the Hardy-Littlewood circle method, the authors obtain nontrivial upper bounds for $p_{\mu_k}(n)$ and, in particular, prove that $\log p_{\mu*\mu}(n) = O_B(\sqrt{n}/(\log n)^B)$ for any fixed $B$. The major-arc analysis reveals an explicit connection to the zeros of the Riemann zeta-function, yielding a precise formula that mirrors classical explicit formulas in prime number theory. The work extends to general Möbius convolution levels $k$ and provides a robust framework for analyzing signed partitions with arithmetically structured weights, contributing both to partition theory and analytic number theory. The results offer asymptotic information about parity-balanced partition families and demonstrate how zeta-zero phenomenology governs major-arc contributions in this weighted setting.

Abstract

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are Möbius convolutions. We establish an upper bound for $p_w(n)$ and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.

Exponential sums over Möbius convolutions with applications to partitions

TL;DR

The paper addresses the asymptotic behavior of weighted partitions with Möbius-convolution weights, focusing on the difference between counts of even and odd admissible partitions. By developing sharp exponential-sum bounds for Möbius-convolution twists and applying the Hardy-Littlewood circle method, the authors obtain nontrivial upper bounds for and, in particular, prove that for any fixed . The major-arc analysis reveals an explicit connection to the zeros of the Riemann zeta-function, yielding a precise formula that mirrors classical explicit formulas in prime number theory. The work extends to general Möbius convolution levels and provides a robust framework for analyzing signed partitions with arithmetically structured weights, contributing both to partition theory and analytic number theory. The results offer asymptotic information about parity-balanced partition families and demonstrate how zeta-zero phenomenology governs major-arc contributions in this weighted setting.

Abstract

We consider partitions of a positive integer arising from the generating functions where the weights are Möbius convolutions. We establish an upper bound for and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.
Paper Structure (27 sections, 22 theorems, 184 equations, 3 figures, 1 table)

This paper contains 27 sections, 22 theorems, 184 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Survey of previous results
  4. Main Results
  5. Structure of the paper
  6. Preliminary Lemmas
  7. Notations
  8. Exponential sums and Dirichlet convolutions
  9. Exponential sums and the divisor function
  10. The Möbius function in arithmetic progressions
  11. A divisor sum estimate
  12. Bounding $1/\zeta(s)$ in the critical strip.
  13. Proofs of Theorems \ref{['theorem 1']}, \ref{['theorem 2']} and \ref{['theorem 3']}
  14. Proof of Theorem \ref{['theorem 1']}
  15. Proof of Theorem \ref{['theorem 2']}
  16. ...and 12 more sections

Key Result

Theorem 1.1

The number of odd admissible partitions of $n$ is asymptotic to the number of even admissible partitions of $n$, as $n$ tends to infinity.

Figures (3)

  • Figure 1.1: Plot of $\log \mathcal{A}(n)$ in blue and $\log {p}_{\mu}(n)$ in orange for $1 \leqslant n \leqslant 10000$.
  • Figure 1.2: Left: Plot of partition function $p_w(n)$ with $w(n) = \mu(n)$, Right: Plot of partition function $p_w(n)$ with $w(n) = (\mu*\mu)(n)$.
  • Figure 1.3: Left: Plot of $\operatorname{Re}(\Phi_1(X,\theta,120,800))$ in blue and $\operatorname{Re}(\Phi_2(X,\theta,20,10))$ in pink. Right: Plot of $\operatorname{Re}(\Phi_1(X,\theta,120,800))-\operatorname{Re}(\Phi_2(X,\theta,20,10))$.

Theorems & Definitions (45)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.4
  • ...and 35 more