报告地点：行健楼学术活动室526
报告人简介：
张天平，陕西师范大学数学与统计学院教授，博士生导师。美国数学评论评论员。主要从事数论及其应用方面的研究，在国内外期刊发表学术论文40余篇，其中部分论文发表于《Advances in Mathematics》、《中国科学：数学》、《Journal of Number Theory 》、《Acta Arithmetica》、《Mathematische Nachrichten》、《Mathematika》、《Acta Mathematica Sinica (English Series)》等。先后主持国家自然科学基金5项，陕西省自然科学杰出基金等省部级项目3项，获陕西省科学技术奖三等奖及陕西省高等学校科学技术奖二等奖各1项。
报告摘要：
We derive an asymptotic formula for the divisor function $\tau(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\Delta_{n,l}$ is defined as
$$\Delta_{n,l}=\frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}}.$$
Specifically, by setting $l=2$, we achieve $\Delta_{n,l}>3/4+5/32$, which surpasses the result obtained by Liu, Shparlinski and Zhang (Adv. Math., 2018). Meanwhile, this has also improved upon the result of Wu and Xi (Algebra Number Theory, 2021).
Notably, Hooley, Linnik and Selberg independently established that the asymptotic formula holds for $q\leq X^{2/3-\varepsilon}$. Irving (IMRN, 2015) was the first to surpass the $2/3-$barrier for certain special moduli. We break the classical $3/4-$barrier in the case of prime power moduli and extend the range of $q$. Our main ingredients borrow from Mangerel's (Forum. Math. Sigma., 2021) adaptation of Mili\'{c}evi\'{c} and Zhang's (Trans Amer Math Soc Ser B, 2023) methodology in dealing with a specific class of weighted Kloosterman sums, rather than adopting Korobov's technique employed by Liu, Shparlinski and Zhang (Adv. Math., 2018). This is a joint work with Dr. MingXuan Zhong.
