arXiv subjects
Publications and source records attributed to Shengbo Zhao.
In this paper, we investigate large values of quadratic Dirichlet character sums. We prove new Omega results for both short and long quadratic character sums under the assumption of the Generalized Riemann Hypothesis (GRH), which improve the previous results.
In this paper, we show a new lower bound for extreme values of quadratic Dirichlet $L$-functions with prime-related moduli, which generalizes the work of Darbar and Maiti in 2025, and sharpens a recent work of Gao in 2026.
In this paper, we investigate extreme values of quadratic Dirichlet $L$-functions at the central point. We provide new extreme values of $L(\frac12,\chi_d)$ as $d$ is large, which improves the recent result of Darbar and Maiti.
In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet \(L\)-functions in the range \(1/2<\sigma<1\). Compared with the work of Aistleitner, Mahatab, Munsch, and Peyrot in 2019, our result shows that derivatives of Dirichlet \(L\)-functions can attain extreme values of the same order of magnitude as the original \(L\)-functions when \(1/2<\sigma<1\) holds.
In this paper, we establish lower bounds for extreme values of the real part of the Riemann zeta function on the critical line. This work relies on the resonance method of Bondarenko and Seip, together with lower bounds for certain integrals associated with Dirichlet series with non-negative coefficients. Our results extend the work of Bondarenko and Seip (2018).
Motivated by the first author's earlier work in 2024, we use the resonance method to establish some Omega results for Dirichlet $L$-functions, extending the previous results.
In this paper, we establish joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).
Using the resonance method, we obtain refined estimates for joint extreme values of the Riemann zeta function at harmonic points, improving upon Levinson's 1972 results and providing new insight into the behavior of the Riemann zeta function. Our proof is primarily based on Dirichlet series theory and the truncated Euler product for the Riemann zeta function. As a corollary, we can recover some previously known extreme value results for the zeta function.
In this paper, we establish lower bounds for the maximum of derivatives of the Riemann zeta function on vertical homogeneous progressions. When the real part $\sigma$ lies within a suitable range, we show that the discrete case has a similar order of magnitude to the continuous case, using the resonance method.
In this article, we investigate the conditional large values of quadratic Dirichlet character sums. We prove an Omega result for quadratic character sums under the assumption of the generalized Riemann hypothesis.
In this article, we investigate conditional large values of quadratic Dirichlet character sums with multiplicative coefficients. We prove some Omega results under the assumption of the generalized Riemann hypothesis.
In this paper, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)χ(n)$. We prove a new Omega result in the region $\exp((\log q)^{\frac12+δ})\le N\le\sqrt q$, where $q$ is the prime modulus.
In this paper, we investigate large values of Dirichlet polynomials with multiplicative coefficients $\sum_{n\le N}f(n)n^{it}$, where $1\ll t\le T$ for large $T$. We prove an improved Omega result in the region $\exp((\log T)^{\frac12+\varepsilon})\le N\le\sqrt T$, where $T$ is large. We also show an Omega result when $\log N$ is around $\sqrt{\log T\log_2T}$.
In this article, we study extreme values of quadratic character sums with multiplicative coefficients $\sum_{n \le N}f(n)χ_d(n)$. For a positive number $N$ within a suitable range, we employ the resonance method to establish a conditional $Ω$-result.
In this paper, we investigate the conditional large values of the quadratic Dirichlet $L$-functions near the central point $s=1/2$. When $σ$ closes to $1/2$ within a suitable range, we show that $L(σ, χ_d)$ have the conditional large values of the similar order of magnitude as $L(1/2, χ_d)$.
In this article, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)χ(n)$. We prove an Omega result in the region $\exp((\log q)^{\frac12+\varepsilon})\le N\le\sqrt q$, where $q$ is the prime modulus.
In this paper, we establish a lower bound for the maximum of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double version convolution formula and combing special GCD sums, our result generalizes the work of Bondarenko et al. in 2023. We also set a lower bound by the resonance method when the real part is near the critical line, both of the above results refine part of Yang's work in 2022.
We establish an omega theorem for logarithmic derivative of the Riemann zeta function near the 1-line by resonance method. We show that the inequality $\left| ζ^{\prime}\left(σ_A+it\right)/ζ\left(σ_A+it\right) \right| \geqslant \left(\left(e^A-1\right)/A\right)\log_2 T + O\left(\log_2 T / \log_3 T\right)$ has a solution $t \in [T^β, T]$ for all sufficiently large $T,$ where $σ_A = 1 - A / \log_2 {T}.$Furthermore, we give a conditional lower bound for the measure of the set of $t$ for which the logarithmic derivative of the Riemann zeta function is large. Moreover, similar results can be generalized to Dirichlet $L$-functions.