arXiv subjects

Seokho Jin

Publications and source records attributed to Seokho Jin.

10 recordsLinked to original sources

arXiv2026

The k-Rank Taylor Family: Canonical Harmonic Lifts, Principal Parts, and Rademacher Series

For fixed $k\ge2$, Garvan's modified $k$-rank moments are encoded by odd elliptic Taylor coefficients of an odd-level Appell function. The completed coefficient of order $2n+1$ has weight $2n+3/2$, so the family cannot be realized as a fixed-weight vector-valued modular form. Nevertheless, all non-holomorphic parts arise from scalar contractions of canonical higher Serre derivatives of a single weight-$3/2$ vector-valued harmonic Maass form. Among lifts with the prescribed unary-theta shadow, the first $d=k-1$ Appell corrections are coordinates on the weakly holomorphic ambiguity and select a unique lift; the next correction is nonzero. The initial Taylor data recover the principal part without using positive Fourier coefficients of the modified moment series, and the principal part determines the shadow. The positive coefficients of the corresponding Maass--Poincaré lift have convergent Rademacher expansions; the canonical lift differs from it by a unique cusp form. This yields exact recursions for all even modified moments. For $k=2$, the cusp-form adjustment vanishes, giving an exact formula for the numbers of $2$-marked Durfee symbols involving a convergent Kloosterman--Bessel series.

math.NT↗

arXiv2026

Odd Parts of Derivative Period Polynomials: Zero Geometry and a Logarithmic Transition

Let $f$ be a normalized level-one Hecke eigenform of even weight $k$, and let $Q_{f,m}$ be the derivative period polynomial formed from the critical values of the $m$-th derivative of its completed $L$-function. We study its odd part $Q^-_{f,m}(z)=(Q_{f,m}(z)-Q_{f,m}(-z))/2$. We prove that there is an absolute $K_0$ such that, for every even $k\ge K_0$, every normalized level-one Hecke eigenform $f$ of weight $k$, and every integer $m\ge0$, the nonzero zeros of $Q^-_{f,m}$ off the unit circle, if any, consist of four simple zeros forming a single real reciprocal quartet $\{\pm b,\pm b^{-1}\}$, where $0 1$, every nonzero zero is eventually simple and lies on the unit circle. At $θ=1$ the same real-or-unit-circle containment remains valid, and any quartet that is present consists of four simple zeros. For each fixed weight, all nonzero zeros are eventually simple and lie on the unit circle as $m\to\infty$. Consequently, the Diamantis--Rolen containment conjecture holds outside finitely many weight--derivative pairs. The proof combines an exact signed-reciprocal completion, uniform split-Mellin saddle estimates yielding a moving-sine model, and a winding count that transfers disk-zero information to the unit circle.

math.NT↗

arXiv2026

Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples

For an even or odd real entire function $H$ of order at most one, let $B_{2n+1}[H]$ denote its centered binomial sample of odd degree. After removing the zero at $z=\pm1$ forced by the parity of $H$ and writing $x=z+z^{-1}$, one obtains a real quotient $C_n(x)$. We prove uniform strip conditions on the zeros of $H$ under which $C_n$ and $C_{n+1}$ generate a real-rooted pencil for every $n$. For even $H$ the optimal uniform half-width is $\sqrt{15/28}$, whereas for odd $H$ the half-width $1$ is sufficient. This conclusion is genuinely stronger than separate unit-circle-rootedness of the two sampled polynomials: the latter may hold while the adjacent quotients fail to interlace. The structural result is a theorem for Jacobi spectral multipliers. For every $0<ν<2$, the quotient problem becomes preservation of the endpoint pencil $y^n(y+t)$. We obtain explicit fixed-$n$ and uniform strip thresholds and determine the exact threshold for $n=1$. The proof combines Bernstein variation diminution with total nonnegativity of finite Jacobi matrices attached to the zero orbits of $H$; a possible unpaired outer real pair, which is not covered by the full defect-class argument, is treated directly on the endpoint pencil. For nonpolynomial even sources satisfying the fixed-$n$ strip condition, a remote-zero-orbit deformation removes common zeros whenever the adjacent images have simple zeros in $(0,4)$. This yields strict interlacing for quotient families associated with Dedekind zeta derivatives and with nested critical-value blocks of self-dual newforms.

math.CV↗

arXiv2026

Sharp Circular Sampling and Derivative Period Polynomials

We determine the exact maximal reflected zero region that forces centered binomial samples of a balanced entire function to have all zeros on the unit circle. In degree $d\ge2$, this region is \[ Ω_d=\left\{a+ib:\ a^2-\frac{b^2}{d-1}\le\frac d4\right\}. \] The finite theorem is sharp already for a single reflected zero pair, and a phase-preserving canonical-product approximation extends it to balanced entire functions of order at most one. De Bruijn strip contraction and projective Hermite--Kakeya--Obreschkoff theory then give the exact common-zero obstruction, simplicity, strict interlacing of consecutive derivative samples, and a monotone real-pencil root flow. As an application, we prove the derivative-period-polynomial unit-circle theorem for completed $L$-functions of primitive holomorphic newforms, in every derivative order and for arbitrary level and nebentypus. After the standard normalization, every zero of \[ \sum_{j=0}^{k-2}\binom{k-2}{j}Λ^{(m)}(f,j+1)z^j \] lies on the unit circle for every weight $k\ge4$, level, nebentypus, and derivative order $m\ge0$. In particular, this proves the full-polynomial unit-circle conjecture of Diamantis and Rolen in its original level-one setting and extends it to arbitrary level and nebentypus. The same source-side theorem also gives simplicity, strict interlacing, and, for each fixed derivative order, conductor-uniform quantitative localization in the weight aspect.

math.NT↗

arXiv2025

Persode: Personalized Visual Journaling with Episodic Memory-Aware AI Agent

Reflective journaling often lacks personalization and fails to engage Generation Alpha and Z, who prefer visually immersive and fast-paced interactions over traditional text-heavy methods. Visual storytelling enhances emotional recall and offers an engaging way to process personal expe- riences. Designed with these digital-native generations in mind, this paper introduces Persode, a journaling system that integrates personalized onboarding, memory-aware conversational agents, and automated visual storytelling. Persode captures user demographics and stylistic preferences through a tailored onboarding process, ensuring outputs resonate with individual identities. Using a Retrieval-Augmented Generation (RAG) framework, it prioritizes emotionally significant memories to provide meaningful, context-rich interactions. Additionally, Persode dynamically transforms user experiences into visually engaging narratives by generating prompts for advanced text-to-image models, adapting characters, backgrounds, and styles to user preferences. By addressing the need for personalization, visual engagement, and responsiveness, Persode bridges the gap between traditional journaling and the evolving preferences of Gen Alpha and Z.

cs.HC↗

arXiv2025

A truncated inner product formula in the geometry of numbers

We study the statistical distribution of primitive sublattices in the space of lattices $\mathrm{SL}(n,\mathbb Z)\backslash\mathrm{SL}(n,\mathbb R)$. A central difficulty in this area is that the second moment of the counting function for rank $k$ sublattices, where $2 \le k \le n-2$, diverges. To overcome this, we analyze the inner product of truncated pseudo-Eisenstein series of the form $E_{f}(g) = \sum_{L} f(\det Lg)$, where the sum is over primitive rank $k$ sublattices of $\mathbb Z^n$. We establish an asymptotic formula for this inner product for both the standard Arthur truncation and a "harsh" truncation that vanishes outside a compact set. Our analysis relies on several technical results of independent interest, including a proof of the uniform moderate growth (UMG) property for these pseudo-Eisenstein series and a new method for resolving singularities in the Maass-Selberg relations. As a primary application, we obtain a significant improvement on the discrepancy bound for the number of primitive sublattices. For almost every lattice, we improve the error term in counting rank $k$ sublattices with determinant up to $p$ to $O(p^{n-1/7+ε})$, surpassing classical bounds for $\min(k, n-k) \ge 8$.

math.NT↗

arXiv2024

Certain infinite products in terms of MacMahon type series

Recently, Ono and the third author discovered that the reciprocals of the theta series $(q;q)_\infty^3$ and $(q^2;q^2)_\infty(q;q^2)_\infty^2$ have infinitely many closed formulas in terms of MacMahon's quasimodular forms $A_k(q)$ and $C_k(q)$. In this article, we use the well-known infinite product identities due to Jacobi, Watson, and Hirschhorn to derive further such closed formulas for reciprocals of other interesting infinite products. Moreover, with these formulas, we approximate these reciprocals to arbitrary order simply using MacMahon's functions and {\it MacMahon type} functions. For example, let $Θ_{6}(q):=\frac{1}{2}\sum_{n\in\mathbb{Z}} χ_6(n) n q^{\frac{n^2-1}{24}}$ be the theta function corresponding to the odd quadratic character modulo $6$. Then for any positive integer $n$, we have $$\frac{1}{Θ_{6}(q)}= q^{-\frac{3n^2+n}{2}}\sum_{\substack{k=r_1\\ k\equiv n\hspace{-0.2cm}\pmod{2}}}^{r_2}(-1)^{\frac{n-k}{2}}A_{k}(q)C_{\frac{3n-k}{2}}(q)+O(q^{n+1}),$$ where $r_1:=\lfloor\frac{3n-1-\sqrt{12n+13}}{3}\rfloor+1$ and $r_2:=\lceil\frac{3n-1+\sqrt{12n+13}}{3}\rceil-1$.

math.NT↗

arXiv2016

The Riemann Hypothesis For Period Polynomials Of Modular Forms

The period polynomial $r_f(z)$ for an even weight $k\geq 4$ newform $f\in S_k(Γ_0(N))$ is the generating function for the critical values of $L(f,s)$. It has a functional equation relating $r_f(z)$ to $r_f\left(-\frac{1}{Nz}\right)$. We prove the Riemann Hypothesis for these polynomials: that the zeros of $r_f(z)$ lie on the circle $|z|=\frac{1}{\sqrt{N}}$ . We prove that these zeros are equidistributed when either $k$ or $N$ is large.

math.NT↗

arXiv2016

A Note On Non-ordinary Primes

Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\sum_{n=1}^\infty a_f(n)q^n\in S_k\cap O_L[[q]]$ (note: $q := e^{2πiz}$) is a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbb{Z})$. We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\mathfrak{p}\subset O_L$ above $p$ for which $a_f(p)\equiv 0 \ (mod\ {\mathfrak{p}})$. For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\in S$. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author.

math.NT↗

arXiv2013

Periods of Jacobi forms and Hecke operator

A Hecke action on the space of periods of cusp forms, which is compatible with that on the space of cusp forms, was first computed using continued fraction and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions. As an application an elementary proof of the Eichler-Selberg trace formula was derived. Similar modification has been applied to period space of Maass cusp forms with spectral parameter $s$. In this paper we study the space of period functions of Jacobi forms by means of Jacobi integral and give an explicit description of Hecke operator acting on this space. a Jacobi Eisenstein series $E_{2,1}(τ,z)$ of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals are already appeared as a disguised form in the work of Zwegers to study Mordell integral coming from Lerch sums and mock Jacobi forms are typical example of Jacobi integral.

math.NT↗