Searcharxiv⌕ Search

arXiv subjects

Taylor Daniels

Publications and source records attributed to Taylor Daniels.

7 recordsLinked to original sources
arXiv2026

Vanishing Coefficients in Products of Quintuple Products

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $\alpha = \alpha(m,n,p)$ such that $a_{pt+ \alpha}=0$. Our results are proven using involutive transformations on integer lattices.

math.NT↗
arXiv2026

The $p$-Dissection of a Product of Quintuple Products

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given.

math.NT↗
arXiv2026

Periodic vanishings of the Legendre-17 signed partition numbers

For $f : \mathbb{N} \to \{0,\pm 1\}$ the $f$-signed partition numbers $\mathfrak{p}(n,f)$ are defined to be the weighted partition sums \[ \mathfrak{p}(n,f) = \sum_{\substack{x_{1}+\cdots+x_{k} = n \\ x_{1} \geq \cdots \geq x_{k} > 0 \\ k \geq 1}} f(x_{1})f(x_{2})\cdots f(x_{k}). \] For prime $p > 2$, let $(\frac{\cdot}{p})$ denote the Legendre symbol modulo $p$. The first half of this paper derives Rademacher-style series formulae for the quantities $\mathfrak{p}(n,\pm(\frac{\cdot}{p}))$ for $p < 24$ satisfying $p \equiv 1 \pmod{4}$ (that is, for $p=5,13,17$), and the extensions to general $p \equiv 1 \pmod{4}$ are made apparent in our derivations. In the second half of this paper, the series formulae for $\mathfrak{p}(n,\pm(\frac{\cdot}{17}))$, as well as various properties of Dedekind sums and their "character-twisted" analogues, are used to establish that these two quantities are identically zero on certain (mod $34$)-arithmetic progressions.

math.NT↗
arXiv2024

Legendre-signed partition numbers

Let $f:\mathbb{N}\to\{0,\pm 1\}$, for $n \in \mathbb{N}$ let $\Pi[n]$ be the set of partitions of $n$, and for all partitions $\pi = (a_1,a_2,\ldots,a_k) \in \Pi[n]$ let \[ f(\pi) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the $f$-signed partition numbers \[ \mathfrak{p}(n,f) = \sum_{\pi\in\Pi[n]} f(\pi). \] In this paper, for odd primes $p$ we derive asymptotic formulae for $\mathfrak{p}(n,\chi_p)$ as $n\to\infty$, where $\chi_p(n)$ is the Legendre symbol $(\frac{n}{p})$ associated $p$. A similar asymptotic formula for $\mathfrak{p}(n,\chi_2)$ is also established, where $\chi_2(n)$ is the Kronecker symbol $(\frac{n}{2})$. Special attention is paid to the sequence $(\mathfrak{p}(n,\chi_5))_\mathbb{N}$, and a formula for $\mathfrak{p}(n,\chi_5)$ supporting the recent discovery that $\mathfrak{p}(10j+2,\chi_5)=0$ for all $j\geq 0$ is discussed. Our main results imply, as a corollary, that the periodic vanishing displayed by $(\mathfrak{p}(n,\chi_5))_\mathbb{N}$ does not occur in any sequence $(\mathfrak{p}(n,\chi_p))_\mathbb{N}$ for $p \neq 5$ such that $p\not\equiv 1\,\,(\mathrm{mod}\,8)$. In addition, work of Montgomery and Vaughan on exponential sums with multiplicative coefficients is applied to establish an upper bound on certain doubly infinite series involving multiplicative functions $f$ with $|f| \leq 1$.

math.NT↗
arXiv2023

Biasymptotics of the Möbius- and Liouville-signed partition numbers

For $n \in \mathbb{N}$ let $Π[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb{N}\to\{0,\pm 1\}$, for $π= (x_1,x_2,\ldots,x_k) \in Π[n]$ let $f(π) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{π\inΠ[n]} f(π). \] Building on the author's previous work on the quantities $p(n,μ)$ and $p(n,λ)$, where $μ$ and $λ$ are the Möbius and Liouville functions of prime number theory, respectively, on assumptions about the zeros of the Riemann zeta function we establish an alternation of the terms $p(n,μ)$ between two asymptotic behaviors as $n\to\infty$. Similar results for the quantities $p(n,λ)$ are established. However, it is also demonstrated that if the Riemann Hypothesis (RH) holds, then it is possible that the quantities $p(n,λ)$ maintain a single asymptotic behavior as $n\to\infty$. In particular, this stable asymptotic behavior occurs if, in addition to RH, it holds that all zeros of $ζ(s)$ in the critical strip $\{0 < \Re(s) < 1\}$ are simple and the residues of $1/ζ(s)$ at these zeros are not too large. To formally describe these stable and alternating behaviors, the notions of asymptotic and biasymptotic sequences are introduced using a modification of the real logarithm.

math.NT↗
arXiv2023

Bounds on the M\"obius-signed partition numbers

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb{N}\to\{0,\pm 1\}$, for $\pi = (x_1,x_2,\ldots,x_k) \in \Pi[n]$ let $f(\pi) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{\pi\in\Pi[n]} f(\pi). \] Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for quantities $p(n,\mu)$ and $p(n,\lambda)$, where $\mu$ and $\lambda$ denote the M\"obius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities $p(n,f)$ generalize the classical notion of restricted partitions.

math.NT↗