arXiv subjects

Hwanyup Jung

Publications and source records attributed to Hwanyup Jung.

7 recordsLinked to original sources

arXiv2026

The first moment of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble

We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ over $\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula with truncation levels $g$ and $g-1$; combining the two truncations before the square-dual contour shifts leaves the three cubic points $z^3 = q^{-4}$ as the only secondary poles, and they contribute the secondary term $q^{2g/3+2}(a_g g + b_g)$ with an error $O_\varepsilon(q^{g(1+\varepsilon)/2})$. The coefficients are real and $m \mapsto (a_m, b_m)$ has minimal period exactly three, so the coefficient of $g$ at the order $q^{2g/3}$ depends on $g$ modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.

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arXiv2026

Mertens products in arithmetic progressions over function fields

We prove a function field analogue of Mertens' formula for Euler products over prime polynomials in arithmetic progressions in $\mathbb{F}_q[t]$, the counterpart of a formula of Languasco and Zaccagnini over the integers. An elementary argument shows that the product over the prime polynomials of degree at most $n$ equals $e^{-H_n}$, where $H_n$ is the $n$-th harmonic number, up to an exponentially small relative error. Combined with the unconditional Riemann hypothesis for Dirichlet $L$-functions, this gives the main result: the product over a reduced residue class is an explicit Euler-product constant times a power of the normalizing function $\mathcal{L}_q(n)=e^{H_n-γ}\log q$, the exponent being the reciprocal of the number of reduced classes, again with an exponentially small error and uniformly for all moduli of degree at most $n$. It is $\mathcal{L}_q(n)$, and not the naive analogue $n\log q$ of $\log x$ under $x=q^{n}$, that is the right normalization; substituting the latter turns the formula into an expansion in powers of $1/n$ whose leading coefficients we compute, the leading correction being combinatorial rather than arithmetic.

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arXiv2026

The Period-Three Secondary Term in the Mean Value of $L(1/2,χ_D)$ in the Hyperelliptic Ensemble

Let $q$ be an odd prime power. We revisit Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the odd-degree hyperelliptic ensemble $\mathcal H_{2g+1}$, and show that its secondary term is not the complete one. The square-dual generating function has three double poles of the same modulus on the secondary circle: the positive real one reproduces Florea's polynomial, while the two nonreal conjugate poles contribute at the same order. The complete secondary term is $q^{2g/3}(α_{g\bmod3}g+β_{g\bmod3})$, with real coefficients whose dependence on the genus has minimal period exactly three, and on at least two of the three residue classes the nonreal poles contribute a term of order $gq^{2g/3}$. We also prove the formula for every odd prime power $q$, whereas Florea assumes $q\equiv1\bmod4$ throughout; this is achieved by a phase-normalized quadratic Gauss sum that restores the multiplicativity and the local evaluations her argument uses. Exact finite-ensemble moments for $q=3$ exhibit the three residue-class trends.

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arXiv2019

Mean values of derivatives of $L$-functions in function fields: IV

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $μ\geq1$ an integer, we compute the mean value of the $μ$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.

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arXiv2018

The Integral Moments and Ratios of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_{q}[T]$

In this paper we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of $L$-functions. We also adapt to the function setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of $L$-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet $L$-functions $L(s,χ_{P})$ where the character $χ$ is defined by the Legendre symbol for polynomials in $\mathbb{F}_{q}[T]$ with $\mathbb{F}_{q}$ a finite field of odd cardinality and the averages are taken over all monic and irreducible polynomials $P$ of a given odd degree. As an application we also compute the formula for the one-level density for the zeros of these $L$-functions.

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arXiv2017

Average values of L-functions in even characteristic

Let $k = \mathbb{F}_{q}(T)$ be the rational function field over a finite field $\mathbb{F}_{q}$, where $q$ is a power of $2$. In this paper we solve the problem of averaging the quadratic $L$-functions $L(s, χ_{u})$ over fundamental discriminants. Any separable quadratic extension $K$ of $k$ is of the form $K = k(x_{u})$, where $x_{u}$ is a zero of $X^2+X+u=0$ for some $u\in k$. We characterize the family $\mathcal I$ (resp. $\mathcal F$, $\mathcal F'$) of rational functions $u\in k$ such that any separable quadratic extension $K$ of $k$ in which the infinite prime $\infty = (1/T)$ of $k$ ramifies (resp. splits, is inert) can be written as $K = k(x_{u})$ with a unique $u\in\mathcal I$ (resp. $u\in\mathcal F$, $u\in\mathcal F'$). For almost all $s\in\mathbb C$ with ${\rm Re}(s)\ge \frac{1}2$, we obtain the asymptotic formulas for the summation of $L(s,χ_{u})$ over all $k(x_{u})$ with $u\in \mathcal I$, all $k(x_{u})$ with $u\in \mathcal F$ or all $k(x_{u})$ with $u\in \mathcal F'$ of given genus. As applications, we obtain the asymptotic mean value formulas of $L$-functions at $s=\frac{1}2$ and $s=1$ and the asymptotic mean value formulas of the class number $h_{u}$ or the class number times regulator $h_{u} R_{u}$.

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arXiv2016

Average Values of $L$-series for Real Characters in Function Fields

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet $L$--functions, at the centre of the critical strip, associated to the real quadratic function field $k(\sqrt{P})$ and inert imaginary quadratic function field $k(\sqrt{γP})$ with $P$ being a monic irreducible polynomial over a fixed finite field $\mathbb{F}_{q}$ of odd cardinality $q$ and $γ$ a generator of $\mathbb{F}_{q}^{\times}$. We also study mean values for the class number and for the cardinality of the second $K$-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over $\mathbb{F}_{q}$. One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet $L$-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average $L$-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of $L$-functions over function fields.

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