Special Session 100: Roots and trends in number theory

A new generalization of Fermat`s Last Theorem

Tianxin Cai Zhejiang University Peoples Rep of China Co-Author(s):

Abstract: There are several generalizations of Fermat`s Last Theorem, such as Bill`s conjecture, Fermat-Catalan1 conjecture. In this talk, we will present a new generalization of Fermat`s Last Theorem, it remains unproved even under the strong abc conjecture or by using the inter-universal Teichmuller theory of Mochizuki Shinichi, and meanwhile we raise some new idea on additive and multiplicative number theory.

Zeta Functions of p-Adic Analytic Varieties

Wei Cao Minnan Normal University Peoples Rep of China Co-Author(s):    Daqing Wan

Abstract: We introduce the zeta function of a $p$-adic analytic variety defined over a $p$-adic number field. This zeta function counts Techm\{u}ller points on the analytic variety. It is proven that this zeta function is a rational function. Our approach is based on the addition operation of Witt vectors, Dwork`s rationality theorem and Hilbert`s basis theorem. We also propose some problems to prompt the research on this new kind of zeta functions.

Improvements on exponential sums related to Piatetski-Shapiro primes

Zhenyu Guo Xi`an Jiaotong University Peoples Rep of China Co-Author(s):    Lingyu Guo, Li Lu

Abstract: Many problems in number theory are highly related to the bound of the following exponential sum $$ \sum_{h \sim H} \bigg| \sum_{n \sim x} \Lambda(n) \e(h n^\gamma + \alpha n) \bigg| \ll x^{1-\epsilon} $$ We give a new estimation of the type I bound to improve the admissible range of $\gamma$ to $\frac{580}{663} < \gamma

Semi-Regular Continued Fractions with Fast-Growing Partial Quotients

Aiken Kazin SDU University Kazakhstan Co-Author(s):    Shirali Kadyrov, Farukh Mashurov

Abstract: In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular continued fractions, which are produced from the sequences of alternating plus and minus ones. In this talk, attention is paid to the structure and features of semi-regular continued fractions through the lens of dimension theory. A key result is established concerning the Hausdorff dimension of number sets with partial quotients that increase more rapidly than a specified rate. Additionally, numerical analyses are conducted to highlight the distinctions between regular and semi-regular continued fractions, offering insights into potential future directions in this area.

Binary sequence family with both small cross-correlation and large family complexity

Huaning Liu Northwest University Peoples Rep of China Co-Author(s):    Huaning LIU and Xi LIU

Abstract: Ahlswede, Khachatrian, Mauduit and S\`{a}rk\{o}zy introduced the notion of family complexity, Gyarmati, Mauduit and S\`{a}rk\{o}zy introduced the cross-correlation measure for families of binary sequences. It is a challenging problem to find families of binary sequences with both small cross-correlation measure and large family complexity. In this talk we present a family of binary sequences with both small cross-correlation measure and large family complexity by using the properties of character sums and primitive normal elements in finite fields.

Random matrices and L-functions

Sheng-Chi Liu Washington State University USA Co-Author(s):

Abstract: Since the work of Montgomery and Odlyzko, there has been a significant body of literature on the similarities in the behavior of zeros of L-functions and the eigenvalues of random matrices. A major breakthrough came with the work of Katz and Sarnak, who demonstrated that while many random matrix ensembles share the same n-level correlations, there is another statistic, the n-level density, for which each ensemble has a different outcome. Moreover, most of the contribution to this statistic comes from the zeros near or at the central point, making it an ideal quantity for investigating the arithmetic of families. In this talk, we will discuss some new developments regarding the low-lying zeros of L-functions.

On the first sign change of Fourier coefficients of cusp forms

Yingnan Wang Shenzhen University Peoples Rep of China Co-Author(s):

Abstract: In the talk, we will survey some recent progress on the first sign-change of Fourier coefficients of cusp forms and present a variant of the current widely used method initiated by Choie and Kohnen in the study of the location of the first sign-change of the Fourier coefficients of a holomorphic cusp form when all the coefficients are real. This variant of Choie and Kohnen`s method applies to more cases including integral weight cusp forms on congruence subgroups of any levels as well as half-integral weight cusp forms. This is a joint work with Guohua Chen and Yuk-Kam Lau.

Rational points on elliptic curves and BSD conjecture

Shuai Zhai Shandong University Peoples Rep of China Co-Author(s):

Abstract: It is widely recognised that there is a deep connection between the analytical and algebraic aspects of elliptic curves, namely the Birch and Swinnerton-Dyer conjecture, which is one of the Millennium Prize Problems. In this talk, I will present a few classical Diophantine problems, with connections to elliptic curves and the Birch and Swinnerton-Dyer conjecture.

On a sum involving the sum-of-divisors function

Feng Zhao North China University of Water Resources and Electric Power Peoples Rep of China Co-Author(s):    Jie Wu

Abstract: Let $\sigma(n)$ be the sum of all divisors of $n$ and let $[t]$ be the integral part of $t$. In this paper, we shall prove that $$ \sum_{n\le x} \sigma\Big(\Big[\frac{x}{n}\Big]\Big) = \frac{\pi^2}{6} x\log x + O(x(\log x)^{2/3}(\log_2x)^{4/3}) $$ for $x\to\infty$, and that the error term of this asymptotic formula is $\Omega(x)$.

The classification and representations of positive definite ternary quadratic forms of level 4N

Haigang Zhou School of Mathematical Sciences, Tongji University Peoples Rep of China Co-Author(s):    Yifan Luo

Abstract: Classifications and representations are two main topics in the theory of quadratic forms. In this talk, we consider these topics of ternary quadratic forms. For a given squarefree integer N, firstly we give the classification of positive definite ternary quadratic forms of level 4N explicitly. Secondly, we give the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4N by using quaternion algebras and Jacobi forms. The formulas are involved with modified Hurwitz class number. As a corollary, we get a formula for the class number of ternary quadratic forms of level 4N. As applications, we give an explicit base of Eisenstein series space of modular forms of weight 3/2 of level 4N, and give new proofs of some interesting identities involving representation number of ternary quadratic forms.