Dutch Analytic Number Theory Symposium

Friday, November 13 in Minnaert 2.01 at Utrecht University

Schedule

13:00 - 14:00 Stephanie Chan

14:00 - 14:30 Justin Uhlemann

14:30 - 15:00 Coffee Break

15:00 - 16:00 Youness Lamzouri

16:00 - 16:30 Coffee Break

16:30 - 17:30 Ade Irma Suriajaya

18:30 - 20:30   Dinner*

*If you would like to join for the dinner, please e-mail L.Thompson@uu.nl

Abstracts

Stephanie Chan (University College London)

Title: Large Selmer groups in families of elliptic curves

Abstract: Building on earlier work of Klagsbrun–Lemke Oliver, we study the distribution of Tamagawa ratios in certain families of elliptic curves admitting a rational isogeny of prime degree. We prove a central limit theorem and establish tail estimates that give lower bounds for the number of curves with large p-Selmer groups in special families, for p=2,3,5,7,13. This is joint work with Matteo Verzobio.

Justin Uhlemann (Utrecht University)

Title: Counting powerful solutions of generalised diagonal forms 

Abstract: We present some results on asymptotic counts for the number of powerful solutions of bounded height on high-dimensional hypersurfaces defined by a generalised class of diagonal forms.

Youness Lamzouri (Université de Lorraine)

Title: A Hilbert space approach to the zeros of the Riemann zeta function and applications

Abstract: The Riemann hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. In the absence of a proof of the Riemann hypothesis, a fundamental problem is to determine what proportion of the zeros can be shown to lie on the critical line.

In this talk, I will introduce a new Hilbert space approach that establishes a direct connection between Montgomery's pair correlation method and the study of zeros of the Riemann zeta function on the critical line. This approach yields a new, conceptually simpler unconditional proof that more than two thirds of the zeros are simple and lie on the critical line.

I will also discuss several further applications of this method, including the discovery of a surprising rigidity phenomenon linking the simplicity and location of the zeros, a proof that more than 13/20 of the zeros are isolated, and new results on consecutive gaps between their ordinates. Finally, I will explain how these ideas can be used to prove that at least 3/10 of the zeros of certain linear combinations of two primitive Dirichlet L-functions lie on the critical line.

Ade Irma Suriajaya (Kyushu University)

Title: Pair correlation of zeros of the Riemann zeta-function without the Riemann Hypothesis

Abstract: H. L. Montgomery in 1973 suggested the pair correlation method of nontrivial zeros of the Riemann zeta-function, where he proved an asymptotic formula for the appropriate F-function within a limited range under the assumption of the Riemann Hypothesis (RH). The behavior of Montgomery's F-function in the remaining range is completely unknown, but is conjectured to satisfy a simple asymptotic formula which would imply the famous Montgomery Pair Correlation Conjecture (PCC) for these zeros.

In recent work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, Junghun Lee, Jordan Schettler, and Caroline L. Turnage-Butterbaugh, we showed how to remove RH in Montgomery's pair correlation method and recover known results on the proportion of simple zeros under weaker hypotheses than previously known results. Additionally, by not assuming RH, our method clarifies that Montgomery's pair correlation method can be used to show that the zeros captured are not only simple but also on the critical line, which was the first result of this type.

More precisely, we obtained that 

(1) 67.25% of the zeros are simple and on the critical line under hypotheses weaker than RH.

 This recovers Montgomery and Taylor's 1975 conditional result on RH, with the addition that the zeros detected are indeed on the critical line.

(2) asymptotically 100% of the zeros are simple and on the critical line assuming only either PCC or an appropriate form of the Alternative Hypothesis (AH) which contradicts PCC but has yet been disproved.

The former recovers Gallagher and Mueller's 1978 conditional result on both RH and PCC, using only PCC by showing that PCC alone forces the zeros to be on the critical line.

The recent work by the Anthropic AI Claude and, independently, Y. Lamzouri has made our result (1) completely unconditional. Their idea is to treat Montgomery's inequality to bound the number of simple zeros within the framework of a specific vector space or, more generally, a Hilbert space. In particular, Lamzouri's approach has opened new possibilities to apply Montgomery's pair correlation method to various problems. In this talk, I will introduce some of the main ideas for studying pairs of zeta-zeros without RH.