Schedule for: 22w2003 - L-functions in Analytic Number Theory

Note: the Lecture rooms are available after 16:00.

For those of you attending in-person, we invite you to join us in TCPL lounge. Don't forget to bring your laptop, earbuds or headphones. Tea and water is available in the foyer. Coffee is on cash honey system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Results on the asymptotic behaviour of moments of $L$-functions have deep implications regarding their size, their zero distribution, their value-distribution, and their nonvanishing at special points. Indeed, studying the moments of $L$-functions was first motivated by a conjecture of Lindelof on the size of $L$-functions in the critical strip. The $2k$-th moment of the Riemann zeta function is given by $I_{k}(T)=\int_{0}^T|\zeta(\frac12+it)|^{2k}\;dt$. It is believed that $I_{k}(T)\sim c_kT(\log T)^{k^2}$ for positive real numbers $k$. This is known to be unconditionally true for $k=1,2$ where the constants $c_1$ and $c_2$ have been determined explicitly. A major breakthrough in analytic number theory happened in 1998 when Keating and Snaith conjectured precise values for the constants $c_k$ based on considerations from random matrix theory. This was followed by the influential work of Conrey et al. in 2005 in which they developed more precise conjectural asymptotic formulae for integral moments of L-functions identifying lower order main terms. In this talk, we survey results on moments of $L$-functions starting with the classical work of Hardy-Littlewood (1918) on the asymptotic formula for the second moment of the Riemann zeta function. Moreover, we discuss developments pertaining to the discrete moments of the Riemann zeta function. We also give an overview of recent results on moments of $L$-functions associated with quadratic characters and automorphic forms. In our discussion, we highlight the important tools used in studying such moments including approximate functional equations, multiple Dirichlet series, random matrix theory, spectral theory of automorphic forms and shifted convolution sums. We are particularly interested in exploring connections between the multiple Dirichlet series approach and the approximate functional equation approach to studying moments of $L$-functions, perhaps opening up new ways to understanding some of these moments.

This talk will be an overview of explicit results in number theory, starting with Rosser and Schoenfeld who, between 1939 and 1976, proved a series of theorems about the zeros of the Riemann zeta function and the error term in the prime number theorem. The rise of computational tools has allowed us to partially verify conjectures, such as the Riemann Hypothesis, and to establish or refine statements of conjectures, and has also helped to explicitly confirm some statements known to be true only asymptotically, such as the Odd Goldbach Conjecture. The most recent years have seen an exponential increase in results of an explicit nature, with various “schools” essentially throughout Canada, US, Europe, and Australia, and with various objects of study, from primes and the Riemann zeta functions, to arithmetic progressions and Dirichlet $L$-functions, and to primes in number fields and Hecke $L$- or Dedekind zeta-functions. Explicit results are interesting on their own as they quantitatively measure the state of our understanding and the efficiency of our techniques. Their nature also allows a wide array of applications in Diophantine approximation, arithmetic, cryptography, and other fields of mathematics.

This discussion will be led and facilitated by Martha Mathurin-Moe, Executive Director, Equity, Diversity and Inclusion at the University of Lethbridge. Those attending in Banff should be prepared with their laptops and headphones to join breakout rooms on Zoom during this session. Does Equity, Diversity, and Inclusion (EDI) really work? This is a question that many persons have asked. But the challenge with EDI work is while the field is constantly evolving there are a lot of fears or misconceptions that further complicates this work. Mathematics is no exception. The Science, Technology, Engineering and Mathematics (STEM) fields continue to see huge underrepresentation by women and racialized groups (Statistics Canada, 2019). In fact, the research has shown that the playing field has not always been leveled and there continues to be systems and structures that exclude Indigenous, Black, Brown, and racialized bodies from academic spaces (Joseph-Salisbury, 2019; Lopez & Jean-Marie, 2021). So, although we speak about a post racial era, the events of 2020 have pushed to the forefront the continued harsh realities faced by marginalized bodies within the social and academic discourse. In this session, we will attempt to critically examine and unpack the importance of EDI work within the mathematics ecosystem, disrupt and open up a space for brave conversation that challenges the denial of racial and social issues within academia (Kappler, 2020; Kendi, 2019). Most importantly we will attempt to examine key practical strategies that can be done to address institutional and structural inequities within the discipline of mathematics. Racial and social inequalities are a policy and power issue that must be addressed by interrogating the systems and structures that continue to uphold these ideals (Kendi, 2019). There is journey ahead, and mathematicians have the opportunity to reshape and redefine its research, teaching, and scholarship ecosystem.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

We will first summarize the significant results and functions of interest in comparative prime number theory. Starting with the error term in the prime number theorem (and the "races" between $\psi(x)$ and $x$ and between $\pi(x)$ and $\mathop{\rm li}(x)$), we will continue with the Mertens sum and the problems of Pólya and Turán, and then move to comparing counting functions for primes in arithmetic progression (the "race" between $\pi(x;q,a)$ and $\pi(x;q,b)$), where it is also natural to compare more than two functions. All of these quantities have explicit formulas involving the zeros of $\zeta(s)$ or $L(s,\chi)$, and we describe how to derive (usually conditionally) limiting distributions for the corresponding normalized error terms; from those limiting distributions we can obtain quantitative statements such as the logarithmic density of the set for which one such function is greater than another. Finally, we will propose some major directions of ongoing and future research, including generalizations to function fields, the frequency of lead changes in these races, correlations among these normalized error terms, and the possible consequences of a quantitative linear-independence statement for the imaginary parts of the zeros of $\zeta(s)$.

2-day workshop participants are welcome to use BIRS facilities (TCPL) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 11 M. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.