arXiv subjects

Zikang Dong

Publications and source records attributed to Zikang Dong.

26 records · Page 2Linked to original sources

arXiv2023

On derivatives of zeta and $L$-functions near the 1-line

We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let $\ell$ be a fixed natural number. We show that, if $|σ-1|\ll1/\log_2t$, then $|ζ^{(\ell)}(σ+ it)|$ has the same maximal order (up to the leading coefficients) as $|ζ^{(\ell)}(1+ it)|$ when $t\to\infty$. The range $1-σ\ll1/\log_2t$ is wide enough, since we also show that $(1-σ) \log_2t \to \infty\; (t \to \infty)$ implies $\limsup_{t\to\infty}|ζ^{(\ell)}(σ+ it)| / (\log_2t)^{\ell+1} = \infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell)}(σ,χ)$ with $χ\pmod q$.

math.NT↗

arXiv2023

Large zeta sums

In this article, we investigate the behaviour of values of zeta sums $\sum_{n\le x}n^{it}$ when $t$ is large. We show some asymptotic behaviour and Omega results of zeta sums, which are analogous to previous results of large character sums $\sum_{n\le x}χ(n)$.

math.NT↗

arXiv2023

Structure of large quadratic character sums

In this article, we study the distribution of large quadratic character sums. Based on the recent work of Lamzouri~\cite{La2022}, we obtain the structure results of quadratic characters with large character sums.

math.NT↗

arXiv2022

Distribution of Dirichlet $L$-functions

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of $\frac12<\RE s<1$ and $\RE s=1$ separately.

math.NT↗

arXiv2022

A note on the large values of $|ζ^{(\ell)}(1+{\rm i}t)|$

We investigate the large values of the derivatives of the Riemann zeta function $ζ(s)$ on the 1-line. We give a larger lower bound for $\max_{t\in[T,2T]}|ζ^{(\ell)}(1+{\rm i} t)|$, which improves the previous result established by Yang.

math.NT↗

arXiv2022

On Large Values of $|ζ(σ+{\rm i}t)|$

We investigate the extreme values of the Riemann zeta function $ζ(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|ζ(1+ıt)|\ge {\rm e}^γ(\log_2T+\log_3T+c),$$ with an effective constant $c$ which improves the result of Aistleitner, Mahatab and Munsch. In the half-critical strip $1/2<\re s<1$, we get an improved $c(σ)$ in the evaluation $$\max_{t\in[0,T]}\log|ζ(σ+ıt)|\ge c(σ)\frac{(\log T)^{1-σ}}{(\log_2T)^σ},$$ when $σ\searrow 1/2$, based on an improved lower bound of GCD sums. This improves the result of Bondarenko and Seip.

math.NT↗

arXiv2022

On the Distribution of large values of $|ζ(σ+{\rm i}t)|$

We investigate the distribution of large values of the Riemann zeta function $ζ(s)$ in the strip $1/2<\re s<1$. For any fixed $\re s=σ\in(1/2,1)$, we obtain an improved distribution function of large values of $|ζ(σ+ıt)|$, holding in the same range as that given by Lamzouri.

math.NT↗

arXiv2021

On the distribution of large values of $|ζ(1+{\rm i}t)|$

In this article, we study the distribution of large values of the Riemann zeta function on the 1-line. We obtain an improved density function concerning large values, holding in the same range as that given by Granville and Soundararajan.

math.NT↗