arXiv subjects

Li Lai

Publications and source records attributed to Li Lai.

15 recordsLinked to original sources

arXiv2026

$\mathbb{F}_q$-linear relations among Thakur's multiple zeta values in positive characteristic

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang--Chen--Mishiba and independently by Im--Kim--Le--Ngo Dac--Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im--Kim--Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.

math.NT↗

arXiv2026

On the $P(t)$-adic Littlewood conjecture in odd characteristics

The $P(t)$-adic Littlewood conjecture is a function field analogue of the famous $p$-adic Littlewood conjecture in Diophantine approximation. In this paper, we prove that the $P(t)$-adic Littlewood conjecture fails for any irreducible polynomial $P(t)$ over any ground field of odd characteristic.

math.NT↗

arXiv2025

On the irrationality of certain $p$-adic zeta values

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value $\zeta_p(i)$ is irrational.

math.NT↗

arXiv2025

A partial result towards the Chowla--Milnor conjecture

The Chowla--Milnor conjecture predicts the linear independence of certain Hurwitz zeta values. In this paper, we prove that for any fixed integer $k \geqslant 2$, the dimension of the $\mathbb{Q}$-linear span of $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) is at least $(c -o(1)) \cdot \log q$ as the positive integer $q \to +\infty$ for some absolute constant $c>0$. It is well known that $\zeta(k,a/q)+(-1)^{k}\zeta(k,1-a/q) \in \overline{\mathbb{Q}}\pi^k$, but much less is known previously for $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$. Our proof is similar to those of Ball--Rivoal (2001) and Zudilin (2002) concerning the linear independence of Riemann zeta values. However, we use a new type of rational functions to construct linear forms.

math.NT↗

arXiv2025

A note on the irrationality of $\zeta_2(5)$

In a spirit of Ap\'ery's proof of the irrationality of $\zeta(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $\zeta_2(5)$ which satisfy $0 < |\zeta_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-\delta}$ for an explicit constant $\delta>0$. This leads to a new proof of the irrationality of $\zeta_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $\mu(\zeta_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$.

math.NT↗

arXiv2025

A note on the number of irrational odd zeta values, II

We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\ldots$, $\zeta(s-1)$ for any sufficiently large even integer $s$. This result improves upon the previous finding by a constant factor. The proof combines the elimination technique of Fischler-Sprang-Zudilin (2019) with the $\Phi_n$ factor method of Zudilin (2001).

math.NT↗

arXiv2024

Small improvements on the Ball-Rivoal theorem and its $p$-adic variant

We prove that the dimension of the $\mathbb{Q}$-linear span of $1,\zeta(3),\zeta(5),\ldots,\zeta(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in $1$ and odd zeta values. Moreover, we establish the $p$-adic variant: for any prime number $p$, the dimension of the $\mathbb{Q}$-linear span of $1,\zeta_p(3),\zeta_p(5),\ldots,\zeta_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020).

math.NT↗

arXiv2023

Many $p$-adic odd zeta values are irrational

For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.

math.NT↗

arXiv2023

On the irrationality of certain $2$-adic zeta values

Let $\zeta_2(\cdot)$ be the Kubota-Leopoldt $2$-adic zeta function. We prove that, for every nonnegative integer $s$, there exists an odd integer $j$ in the interval $[s+3,3s+5]$ such that $\zeta_2(j)$ is irrational. In particular, at least one of $\zeta_2(7),\zeta_2(9),\zeta_2(11),\zeta_2(13)$ is irrational. Our approach is inspired by the recent work of Sprang. We construct explicit rational functions. The Volkenborn integrals of these rational functions' (higher-order) derivatives produce good linear combinations of $1$ and $2$-adic Hurwitz zeta values. The most difficult step is proving that certain Volkenborn integrals are nonzero, which is resolved by carefully manipulating the binomial coefficients.

math.NT↗

arXiv2022

On two conjectures of Sun concerning Ap\'ery-like series

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Ap\'ery-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few highly non-trivial functional equations of the polylogarithms of weight $3$ and $4$.

math.NT↗

arXiv2022

Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman element and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple $t$-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $t$-values of the same weight involving $2$'s and $3$'s.

math.NT↗

arXiv2021

At least two of $ζ(5),ζ(7),\ldots,ζ(35)$ are irrational

Let $ζ(s)$ be the Riemann zeta function. We prove the statement in the title, which improves a recent result of Rivoal and Zudilin by lowering $69$ to $35$. We also prove that at least one of $β(2),β(4),\ldots,β(10)$ is irrational, where $β(s) = L(s,χ_4)$ and $χ_4$ is the Dirichlet character with conductor $4$.

math.NT↗

arXiv2021

On the largest prime divisor of $n!+1$

For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup_{n \rightarrow +\infty} P(n!+f(n))/n \geqslant 1+9\log2$. This improves a result of Luca and Shparlinski. These improvements come from an additional combinatoric idea to the works mentioned above.

math.NT↗

arXiv2020

A note on the number of irrational odd zeta values

It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values: $ζ(3),ζ(5),ζ(7),\cdots,ζ(s)$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse of Euler totient funtion.

math.NT↗

arXiv2019

On the rigidity of stationary charged black holes: small perturbations of the non-extremal Kerr-Newman family

We prove a perturbative result concerning the uniqueness of Kerr-Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons, if it agrees with a non-extremal Kerr-Newman space-time asymptotically flat at infinity and it is sufficiently close to the Kerr-Newman family, then the space-time must be one of the Kerr-Newman solutions. The closeness to the Kerr-Newman family is measured by the smallness of a pair of Mars-Simon type tensors, which were introduced by Wong in \cite{Wong_09} to detect the Kerr-Newmann family.

gr-qc↗