Date: April 10, 2025
Time: 12.00
Location: Digital Research Lab, Baku State University, Main Building, 3rd floor
An Introduction to Arithmetic Functions II
Ertan Elma, Mathematics Research Lab
Abstract:
In this second talk of the series, we will continue with some further properties of the Möbius function such as the Möbius Inversion Formula. Then we will see some other arithmetic functions such as the divisor function and the Euler totient function. If time permits, we will start considering the summatory functions of these functions and cover a technique called partial summation.
Date: April 9, 2025
Time: 11.00
Location: ADA University
Number of Prime Factors with a Given Multiplicity
Ertan Elma, Mathematics Research Lab
Abstract:
For natural numbers $k,n\geqslant 1$, let $\omega_k(n)$ be the number of prime factors of $n$ with multiplicity $k$. The functions $\omega_k(n)$ with $k\geqslant 1$ are refined versions of the well-known function $\omega(n)$ counting the number of distinct prime factors of $n$ without any conditions on the multiplicities.
In this talk, we will cover several elementary, analytic and probabilistic results about the functions $\omega_k(n)$ with $k\geqslant 1$ and their function field analogues in polynomial rings with coefficients from a finite field. In particular, we will see that the function $\omega_1(n)$ and its function field analogue satisfy the Erdős-Kac Theorem. The results we will see in this talk are based on joint works with Yu-Ru Liu, with Sourabhashis Das, Wentang Kuo and Yu-Ru Liu, and with Greg Martin.
Date: March 17, 2025
Time: 9.30
Location:
Towards the Bloch-Kato conjecture for GSp6.
Waqar Ali Shah, University of California, Santa Barbara
Abstract:
One of the central problems in number theory is the Birch and Swinnerton-Dyer conjecture, which asserts that the order of vanishing of the -function of a rational elliptic curve at the central value coincides with the rank of its Mordell-Weil group. A far-reaching generalization of this is the Bloch-Kato conjecture, which posits a similar relationship between the order of vanishing of the -functions associated with Galois representations and the dimension of their Selmer groups. In recent years, significant progress has been made in establishing new cases of this conjecture for automorphic Galois representations arising in the cohomology of Shimura varieties, most notably in the work of Skinner, Loeffler, and Zerbes for -Shimura varieties. A crucial ingredient in these developments is the construction of an Euler system, a powerful tool for studying the dimension of Selmer groups.
In this talk, we recall the Bloch-Kato conjecture in the setting of -Shimura varieties and present the construction of an Euler system using a novel method that overcomes a major obstacle. As a consequence, we obtain the first non-trivial result towards the Bloch-Kato conjecture in this setting.
Date: March 11, 2025
Time: 10.30
Location: Digital Research Lab, Baku State University, Main Building, 3rd floor
Multiplicative functions $k$-additive on hexagonal
numbers
Elchin Hasanalizade, ADA University
Abstract
Characterization of the identity function using functional equations has been actively studied by many authors. In 1992, Claudia Spiro introduced the concept of additive uniqueness. A set $E\subseteq\mathbb{N}$ is called an {\it additive uniqueness set} of a set of arithmetic functions $\mathcal{F}$ if there is exactly one element $f\in\mathcal{F}$ which satisfies
\[f(m+n)=f(m)+f(n) \ \text{for all} \ m,n\in E.\]
For example, $\mathbb{N}$ and $\{1\}\cup2\mathbb{N}$ are trivially additive uniqueness sets of the set of multiplicative functions.
She proved that the set of primes is an additive uniqueness set of the collection of multiplicative functions. Many mathematicians have generalized her result. If the additive condition is replaced by multivariate Cauchy’s equation this gives rise to the notion of a k-additive uniqueness set. It is known that squares, cubes, triangular numbers, generalized pentagonal numbers, practical numbers, odious numbers are k-additive uniqueness sets for the set of multiplicative functions. In the present talk, we add another example to the growing list of k-additive uniqueness sets. Namely, we will show that for $k\ge3$ the set of all nonzero hexagonal numbers is a new k-additive uniqueness set for the collection of multiplicative functions. This extends B. Kim et al.’s work for $k = 2$. This is a joint work with Poo-Sung Park (Kyungnam University, Republic of Korea) and Emil Inochkin (ADA University, Azerbaijan)
Date: December 26, 2024
Time: 10.30
Location: Digital Research Lab, Baku State University, Main Building, 3rd floor
A Discrete Mean Value of the Riemann Zeta Function and its Derivatives
Ertan Elma, MRL
Abstract
In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the \(2k\)-th moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.