Events

Note: The time of this talk is different from the usual Number Theory Seminar time.

Trevor Wooley, Purdue University

Freiman proved that when $(k_i)$ is an increasing sequence of positive integers, then for each $j$, there exists $s = s(j)$ having the property that all large integers $n$ are represented as a sum of positive integral $k_i$-th powers (with $i\in \{ j,j + 1,...,s\}$) if and only if $1/k_1 + 1/k_2 + · · ·$ diverges. We describe recent work joint with Joerg Bruedern making Freiman's theorem effective. Some concrete examples will be described, as well as the underlying progress on Waring's problem.

Zoom only: https://uwaterloo.zoom.us/j/98425417240?pwd=YUpVd2x5ejJKd3JZb2xvamlXeDZFUT09

Cameron Stewart, Department of Pure Mathematics, University of Waterloo

We shall discuss estimates for the greatest prime factor of terms of binary recurrence sequences.

MC 5479

J.C. Saunders, Middle Tennessee State University

In this talk, we look at numbers of the form $H_{r,k} := F_{k-1}F_{r-k+2} +F_kF_{r-k},$ where $F_n$ denotes the $n$th Fibonacci number. These numbers are the entries of a triangular array called the {\it determinant Hosoya triangle}\/ which we denote by $\mathcal{H}.$ We discuss the divisibility properties of the above numbers and their primality. We give a small sieve to illustrate the density of prime numbers in $\mathcal{H}.$ Since the Fibonacci and Lucas numbers appear as entries in $\mathcal{H},$ this research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. We prove that $\mathcal{H}$ has arbitrarily large neighbourhoods of composite entries. Finally we present an abundance of data indicating a very high density of primes in $\mathcal{H}.$ This is joint work with Hsin-Yun Ching, Rigoberto Fl\'orez, Florian Luca, and Antara Mukherjee.

MC 5417

Alan Talmage, Department of Pure Mathematics, University of Waterloo

We present a result showing existence of prime solutions to systems of polynomial equations with sufficiently strong mean value bounds. This result serves to bring our knowledge of prime solutions of Diophantine systems closer to the state of our knowledge regarding integer solutions to such systems as new mean value bounds are proved. Results for Vinogradov systems and conditional results for the Waring-Goldbach problem for cubes will be presented as corollaries of this work.

MC 5417

Zhenchao Ge, Department of Pure Mathematics, University of Waterloo

The integral of Hardy's Z-function from $0$ to $T$ measures the occurrence of its sign changes. Hardy proved that this integral is $o(T)$ from which he deduced that the Riemann zeta-function has infinitely many zeros on the critical line. A. Ivić conjectured this integral is $O(T^{1/4})$ and $\Omega_{\pm}(T^{1/4})$ as $T\to\infty$. These estimates were proved, independently, by M.A. Korolev and M. Jutila.

In this talk, we will show that the analogous conjecture is false for the Z-functions of certain "special" Dirichlet L-functions. In particular, we show that the integral of the Z-function of a Dirichlet L-functions from $0$ to $T$ is asymptotic to $c_\chi T^{3/4}$ and we classify precisely when the constant $c_\chi$ is nonzero. Somewhat surprisingly, numerical evidence seems to suggest that the unexpectedly large mean value is caused by a currently unexplained parity bias in the gaps between the zeros of these "special" Dirichlet L-functions.

This is joint work with Jonathan Bober and Micah Milinovich.

MC 5501

**Note special time for this seminar**

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

In this talk, we first introduce a numerical criterion which bounds the degree of any algebraic integer in short intervals. Then we use the numerical tool to classify all the unicritical polynomials defined over the maximal totally real extension of the field of rational numbers. We also classify all the quadratic unicritical polynomials defined over the field of rational numbers in which they have only finitely many totally real preperiodic points. Using the numerical tool, we are able to explicitly compute totally real preperiodic points of some quadratic unicritical polynomials. Our approach uses tools from complex and p-adic potential theory, particularly, the Fekete-Szego theorem. This is based on joint work with Clay Petsche.

MC 5501

M. Ram Murty, Queen's University

We will survey several results in the theory of arithmetical functions of several variables, a subject that is not often treated in number theory textbooks though it should be!

MC 5501

Shivani Goel, IIIT Delhi and Queen's University

The Hardy and Littlewood k-tuple prime conjecture is one of the most enduring unsolved problems in mathematics. In 1999, Gadiyar and Padma presented a heuristic derivation of the 2-tuples conjecture by employing the orthogonality principle of Ramanujan sums. Building upon their work, we explore triple convolution Ramanujan sums and use this approach to provide a heuristic derivation of the Hardy-Littlewood conjecture concerning prime 3-tuples. Furthermore, we estimate the triple convolution of the Jordan totient function using Ramanujan sums.

MC 5501

Michael Lipnowski, Ohio State University

Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a p-adic analogue of Borcherds’ singular theta lift. The values of rigid meromorphic cocycles at special points of an associated p-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.

MC 5501

Sun Woo Park, University of Wisconsin-Madison

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer, Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erd\"os-Kac theorem, and the geometric ergodicity of Markov chains.

MC 5501