Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.38240

Zero Harmonic Density implies Zero Density

Rafael Reno S. Cantuba
View original source ↗Read source document ↗Export metadata ↓

Abstract

A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rafael Reno S. Cantuba. 2026-09-29. Zero Harmonic Density implies Zero Density. https://arxiv.org/abs/2609.38240

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

KEEP EXPLORING

Related papers

arXiv2026

A Ramanujan Pell K3 Surface and Cube Representations

A negative Pell equation produces positive integers with $a_n^3+b_n^3+c_n^3=561N_n^3+(-1)^{n+1}$. We classify all positive solutions satisfying the natural linear constraints of the construction and prove an exact pairwise coprimality law, with exceptional factor $137$ in one class modulo $68$. The associated sextic $K=(1-13u+26u^2)^3+(6+182u^2)^3$ defines an elliptic K3 surface with six type-IV fibres. Its genus-four cubic cover has a geometrically simple Prym threefold; exact good-reduction arithmetic establishes simplicity over every finite extension of the residue field. We determine the full geometric Mordell-Weil group, the integral Neron-Severi and transcendental lattices, and the arithmetic Picard lattice over $\mathbb{Q}$, which is $U\oplus\langle-4\rangle^{\oplus4}$. Every two-cube representation by rational functions over $\mathbb{Q}$ is obtained from a multiple of one section, with numerator degrees $2m^2$ and denominator degree $2m^2-2$. Polynomial rigidity follows without a degree assumption. Along the specified Pell orbit all specialized points are nontorsion, their heights are $\frac{4}{3}|n|\log(8+\sqrt{65})+O(1)$, and infinitely many rational cubic-twist classes occur. We also identify the local zeta function at $19$ and the generic rank and rank-jump criterion for the corresponding two-parameter base-change family.

math.GM↗
arXiv2026

Generalized Pellet's theorem for analytic matrix-valued functions

Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

math.GM↗
arXiv2026

Even and Odd Integer Values of the Secondary Zeta Function

The secondary zeta function is defined by $$ Z(s)=\sum_{γ>0}\frac{1}{γ^{s}}, $$ where $\tfrac12+iγ$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $ξ$-function. Define $$ L(x)=\log\left(\frac{ξ(x+\tfrac12)}{ξ(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{π\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $ξ(x+\tfrac12)$ for reference. We also derive a limit formula for $γ_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

math.GM↗