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Yotsanan Meemark

Publications and source records attributed to Yotsanan Meemark.

10 recordsLinked to original sources
arXiv2025

Variance of square-full integers in short intervals and arithmetic progressions

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected order, under the assumption of a certain quasi-Riemann hypothesis. We also prove an asymptotic formula for the variance in arithmetic progressions, averaging over a quadratic residue and a nonresidue by a half, which is of smaller order of magnitude than the aforementioned bound for all primes $q\gg x^{51/114+\varepsilon}$.

math.NT↗
arXiv2024

Classical orthogonal decomposition of a modular $\mathfrak{sl}_n$

An orthogonal decomposition problem of Lie algebras over the complex numbers has been studied since the 1980s. It has many applications and relations to other areas of mathematics and sciences. In this paper, we consider this decomposition problem over a field of prime characteristic. We define a classical orthogonal decomposition of a modular Lie algebra and construct it for $\mathfrak{sl}_n$ under certain sufficient conditions. Additionally, we provide more detailed analysis of the problem when $n = 2$ and $3$.

math.RA↗
arXiv2024

An average number of square-free values of polynomials

The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Usually, it is more difficult to obtain a similar order of error term for a higher degree polynomial $f(x)$ in place of $\text{Id}(x)$. Under the Riemann hypothesis, we show that the error term, on average in a weak sense, over polynomials of arbitrary degree, is of the expected order $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$.

math.NT↗
arXiv2024

Square-full values of quadratic polynomials

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by $S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. We have known that for a relatively prime pair $(a,b)\in\mathbb N\times \mathbb N\cup\{0\}$ with a linear polynomial $f(x)=ax+b$, its counting function is $\asymp_{a,b} N^\frac{1}{2}$. Fix $\varepsilon>0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)\ll_{\varepsilon, f} N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon,f}(N^\varepsilon)$.

math.NT↗
arXiv2019

Mahler measures of a family of non-tempered polynomials and Boyd's conjectures

We prove an identity relating Mahler measures of a certain family of non-tempered polynomials to those of tempered polynomials. Evaluations of Mahler measures of some polynomials in the first family are also given in terms of special values of $L$-functions and logarithms. Finally, we prove Boyd's conjectures for conductor $30$ elliptic curves using our new identity, Brunault-Mellit-Zudilin's formula and additional functional identities for Mahler measures.

math.NT↗
arXiv2008

Nathanson's Heights and the CSS Conjecture for Cayley Graphs

Let $G$ be a finite directed graph, $β(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ is directed acyclic and $γ(G)$ the number of pairs of nonadjacent vertices in the undirected graph obtained from $G$ by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan \cite{CSS07} proved that if $G$ is triangle-free, then $β(G) \leq γ(G)$. They conjectured a sharper bound (so called the "CSS conjecture") that $β(G) \leq \dfrac{γ(G)}{2}$. Nathanson and Sullivan verified this conjecture for the directed Cayley graph $\Cay(\bbZ/N\bbZ, E_A)$ whose vertex set is the additive group $\bbZ/N\bbZ$ and whose edge set $E_A$ is determined by $E_A = {(x,x+a) : x \in \bbZ/N\bbZ, a \in A}$ when $N$ is prime in \cite{NS07} by introducing "height". In this work, we extend the definition of height and the proof of CSS conjecture for $\Cay(\bbZ/N\bbZ, E_A)$ to any positive integer $N$.

math.NT↗
arXiv2008

An Equivalence Relation on A Set of Words of Finite Length

In this work, we study several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\bf 23}, 141-147). Cardinalities of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring $\mathbb{Z}/N\mathbb{Z}$ where $N$ is a positive integer. We have arithmetic results parallel to Bacher's.

math.CO↗
arXiv2008

Hecke Operators on Drinfeld Cusp Forms

In this paper, we study the Drinfeld cusp forms for $Γ_1(T)$ and $Γ(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for $Γ_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for $Γ_1(T)$ of large weights, and not for $Γ(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $Γ(T)$ with small weights are determined and the eigenspaces characterized.

math.NT↗
arXiv2005

Ramanujan Graphs on Cosets of $PGL_2(\mathbb{F}_q)$

In this paper we study Cayley graphs on $\PGL_2(\mathbb F_q)$ mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of $\PGL_2(\mathbb F_q)$ of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here.

math.NT↗