List of open problems

The CPNTS open problem list is now on the arXiv.

In lieu of having a problem session during the symposium, we have created a list of open problems related to the theme of the symposium, including any of these topics:

• classical prime-counting functions

• races among primes in arithmetic progressions

• races associated with elliptic curves, number fields, and function fields

• summatory functions of arithmetic functions such as μ(n), (-1)^{Ω(n)}, τ(n), etc.

• the distribution of zeros of L-functions associated with the above races, including the linear independence of their imaginary parts

• general oscillations and frequency of sign changes of number-theoretic error terms.

The list will remain available afterwards on this symposium website.

Our hope is that this list will stimulate research and lead to future collaborations among symposium participants. In particular, problems on the list are intended to be available for anyone to decide to work on. We encourage you to contribute any open problems that you don't mind sharing, or problems that you're not actively working on (unless you are seeking collaborators). Each problem that is listed will have the name and affiliation of the submitter unless requested otherwise.

Lecture videos and slides

• Alexandre Bailleul, Unconditional comparative prime number theory over function fields

• William Banks, Shifting the ordinates of zeros of the Riemann zeta function

• Debmalya Basak, Remarks on Landau–Siegel zeros

• Chiara Bellotti, On the generalised Dirichlet divisor problem

• Bittu, Spacing statistics of the Farey sequence

• Sneha Chaubey, Bias for the κ-factor function

• Shashank Chorge, Voronoi summation formulas for the product of the generalized divisor function and the Liouville lambda function

• Lucile Devin, Distribution of Gaussian primes and zeros of L-functions

• Daniel Fiorilli, Moments of primes using comparative number theory tools

• Shivani Goel, Ramanujan sums and the Hardy–Littlewood prime tuple conjecture

• Mounir Hayani, The influence of the Galois group structure on the Chebyshev bias in number fields

• Hari Iyer, Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem

• Daniel Johnston, The average value of π(t)–li(t)

• Florent Jouve, Moments in the Chebotarev density theorem

• Amrinder Kaur, A Mertens function analogue for the Rankin–Selberg L-function

• Youness Lamzouri, The Shanks–Rényi prime number race problem

• Ethan Lee, Oscillations in Mertens’ product theorem for number fields

• Nicol Leong, Explicit estimates for the Mertens function

• Sun Kai Leung, Joint distribution of primes in multiple short intervals

• Wanlin Li, Counting “supersingularity” in arithmetic statistics

• Nathan Ng, Prime number error terms

• Gyeongwon Oh, The distribution of analytic ranks of elliptic curve over prime cyclic number fields

• Andrew Pearce-Crump, Number theory versus random matrix theory: the joint moments story

• Jagannath Sahoo, A simple proof of the Wiener–Ikehara Tauberian theorem

• Jan-Christoph Schlage-Puchta, Almost periodicity and large oscillations of prime counting functions

• Saloni Sinha, The Riemann hypothesis via the generalized von Mangoldt function

• Tim Trudgian, Fake mu’s: Make Abstracts Great Again!

• Kin Ming Tsang, A race problem arising from elliptic curves

• Tian Wang, Quantitative upper bounds related to an isogeny criterion for elliptic curves

• Peng-Jie Wong, Joint distribution of central values and orders of Sha groups of quadratic twists of an elliptic curve

• Andrew Yang, Zero-free regions of the Riemann zeta-function

• Chi Hoi (Kyle) Yip, Oscillation results for the summatory functions of fake μ’s

Annotated bibliography

In September 2023, a UBC team posted the first version of An annotated bibliography for comparative prime number theory on the arXiv. The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results, using a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.

Photos from the symposium