Mathematics

Subject:

Read more about The generalised Shanks's conjecture
910 reads

Asymptotic mean square of product of higher derivatives of the zeta-function and Dirichlet polynomials

Speaker:

Mithun Das

Date:

Tue, Jul 26, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

We discuss the asymptotic behavior of the mean square of higher derivatives of the Riemann zeta function or Hardy's $Z$-function product with a Dirichlet polynomial in a short interval. As an application, we obtain a refinement of some results by Levinson--Montgomery as well as Ki--Lee on zero density estimates of higher derivatives of the Riemann zeta function near the critical line. Also, we obtain a zero distribution result for Matsumoto--Tanigawa's $\eta_k$-function. This is joint work with S. Pujahari.

Class:

Subject:

Lambert series of logarithm and a mean value theorem for $\zeta(\frac{1}{2}-it)\zeta'(\frac{1}{2}+it)$

Speaker:

Atul Dixit

Date:

Tue, Jul 26, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

An explicit transformation for the series $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny},$ Re$(y)>0$, which takes $y$ to~$\frac1y$, is obtained. This series transforms into a series containing $\psi_1(z)$, the derivative of~$R(z)$. The latter is a function studied by Christopher Deninger while obtaining an analogue of the famous Chowla--Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi_1(z)$ are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$. Our transformation readily gives the complete asymptotic expansion of $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny}$ as $y\to0$. This, in turn, gives the asymptotic expansion of $\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)e^{-\delta t}\, dt$ as $\delta\to0$. This is joint work with Soumyarup Banerjee and Shivajee Gupta.

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Subject:

Negative moments of the Riemann zeta function

Speaker:

Alexandra Florea

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. This is joint work with H. Bui.

Class:

Subject:

Read more about Negative moments of the Riemann zeta function
900 reads

The recipe for moments of $L$-functions

Speaker:

Siegfried Baluyot

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a `recipe' that leads to detailed conjectures for the asymptotic behavior of moments of various families of $L$-functions. In this talk, we will survey recent progress towards their conjectures and explore connections with different subjects.

Class:

Subject:

Read more about The recipe for moments of $L$-functions
1197 reads

One-level density of zeros of Dirichlet L-functions over function fields

Speaker:

Hua Lin

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

We compute the one-level density of zeros of order-$\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy.

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Subject:

Distributions of sums of the divisor function over function fields

Speaker:

Matilde Lalín

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

The goal of this talk is to discuss the variance of sums of the divisor function leading to certain random matrix distributions. While the knowledge of these problems is quite limited over the natural numbers, much more is known over function fields. We will start by introducing the basics of zeta functions and $L$-functions over function fields. We will then discuss the work of Keating, Rodgers, Roditty-Gershon, and Rudnick on the sums over arithmetic progressions, leading to distributions over unitary matrices by the Katz and Sarnak philosophy and a general conjecture over the natural numbers. Finally, we will present some recent work (in collaboration with Kuperberg) on sums over squares modulo a prime leading to symplectic distributions.

Class:

Subject:

Selberg's central limit theorem for quadratic Dirichlet $L$-functions over function fields

Speaker:

Allysa Lumley

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

In this talk, we will discuss the logarithm of the central value $L\left(\frac{1}{2}, \chi_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $g$ over a fixed finite field $\mathbb{F}_q$ in the limit as $g\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the full Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ and variance $\log \deg(D)$, and also $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is at least $94.27 \%$ Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.

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Subject:

Floer Homotopy 4

Speaker:

Mohammed Abouzaid

Date:

Wed, Jul 20, 2022

Location:

PIMS, University of British Columbia, Zoom, Online

Conference:

Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory

Abstract:

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 4th in a series of 4.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

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Subject:

Read more about Floer Homotopy 4
854 reads

Floer Homology Applications 1

Speaker:

Jeff Hicks

Date:

Wed, Jul 20, 2022

Location:

PIMS, University of British Columbia, Zoom, Online

Conference:

Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory

Abstract:

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 1st in a series of 3.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class:

Subject: