Number Theory Test
A list of members of the group and research intersts is available here
The Number Theory Seminar takes place on Mondays from 3 pm to 4 pm (or on Tuesday if Monday is a bank holiday).
Seminars are held in B3.02 of the Zeeman building (MS.03 if Tuesday) and live-streamed on Teams; sometimes the speaker is online but the talk will still be streamed in B3.02. Sometimes a different room is used, see the entry below.
Organisers: Simon Myerson, Harry Schmidt and Ju-Feng Wu
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08 December 2025 at 15:00 in B3.02
Speaker: Lasse Grimmelt (University of Cambridge)
Title: Sums, Sieves, and Power Saving
Abstract: A modern perspective on studying primes in additive contexts is the following: Replace the difficult prime indicator by a simpler model, while not changing the count you are interested about. Clearly the model must have some similarity with the primes, as for example the sum of three primes is generically odd, the same must hold for the model. This swap inevitably introduces an error term and one exciting area is to push this error term into the so-called power saving range. This is remarkable, given we do not even have such a saving for the prime counting function itself.
In the first part, I will explain from a modern point of view how Montgomery and Vaughan proved a power-saving exceptional-set result for the binary Goldbach problem, and introduce the model Green used for his power-saving version of Sárközy in shifted primes. Afterwards, I’ll describe joint work with J. Teräväinen where we use numbers free of small prime factors as a model to study sums of almost twin-primes. -
01 December 2025 at 15:00 in B302
Speaker: Alex Bartel (Glasgow)
Title: Isospectral manifolds via orders in quaternion algebras
Abstract: I will report on joint work with Aurel Page on a number/representation theoretic approach to the question "Can you hear the shape of a drum". We use quaternion algebras over number fields to construct pairs of manifolds that "sound the same", but differ from each other in subtle ways. I will not assume that you already care about this question.
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24 November 2025 at 15:00 in B3.02
Speaker: Adam Morgan (University of Cambridge)
Title: TBA
Abstract: TBA
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17 November 2025 at 15:00 in B3.02
Speaker: David Hokken (Universiteit Utrecht)
Title: TBA
Abstract: TBA
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10 November 2025 at 15:00 in B3.02
Speaker: Cathy Swaenepoel (Paris Cite)
Title: Prime numbers with an almost prime reverse
Abstract: Let b ≥ 2 be an integer. For any integer n ≥ 0, we call `reverse' of n in base b the integer obtained by reversing the digits of n. The existence of infinitely many prime numbers whose reverse is also prime is an open problem. In this talk, we will present a joint work with Cécile Dartyge and Joël Rivat, in which we show that there are infinitely many primes with an almost prime reverse. More precisely, we show that there exist an explicit integer \Omega_b > 0 and c_b > 0 such that, for at least c_b b^ℓ / ℓ^2 primes p ∈ [b^{ℓ-1},b^ℓ[, the reverse of p has at most \Omega_b prime factors. Our proof is based on sieve methods and on obtaining a result in the spirit of the Bombieri-Vinogradov theorem concerning the distribution in arithmetic progressions of the reverse of prime numbers.
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03 November 2025 at 15:00 in B302
Speaker: Ross Paterson (University of Bristol)
Title: Quadratic Twists as Random Variables
Abstract: For each square-free integer D, and each elliptic curve E, the 2-Selmer groups of E and its quadratic twist E_D naturally live in the same space. We are motivated to study their independence as E varies. We shall present a heuristic in this direction, and some results in support of it.
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27 October 2025 at 15:00 in B3.02
Speaker: Hung Bui (University of Manchester)
Title: Weighted central limit theorem for central values of L-functions.
Abstract: A classical result of Selberg says that \log|\zeta(1/2 + it)| has a Gaussian limit distribution. We expect the same thing holds for \log|L(1/2, \chi)| for \chi being over the primitive Dirichlet characters modulo q, as q tends to infinity. Proving such a result remains completely out of reach, as it would imply 100% of these central L-values are non-zero, which is a well-known open conjecture. In this talk, I will describe how one can establish a weighted central limit theorem for the central values of Dirichlet L-functions. Under the Generalized Riemann Hypothesis, one can also obtain a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. This is joint work with Natalie Evans, Stephen Lester and Kyle Pratt.
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20 October 2025 at 15:00 in B3.02
Speaker: Holly Krieger (University of Cambridge)
Title: Uniformity in arithmetic dynamics
Abstract: The periodic points of a discrete algebraic dynamical system control its local and global dynamical behaviour. When we impose an arithmetic structure on such a system, we do not generally expect periodic points to be rational. The central open conjecture in arithmetic dynamics asks whether this arithmetic structure imposes uniform constraints on the possible periods of points for families of algebraic dynamical systems. In this talk, we will discuss this conjecture, how it generalizes the torsion conjecture—in particular, the celebrated theorems of Mazur and Merel on rational torsion of elliptic curves—and survey some recent progress on and strategies for attacking this problem.
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13 October 2025 at 15:00 in B3.02
Speaker: Thomas Bloom (University of Manchester)
Title: TBA
Abstract: TBA
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06 October 2025 at 15:00 in B3.02
Speaker: Chris Hughes (University of York)
Title: Discrete moments of the Riemann zeta function
Abstract: I will discuss some new results on moments of zeta'(rho), the derivative of the Riemann zeta function evaluated at the zeta zeros. Despite being a complex function evaluated at complex points, it turns out to be real and positive on average. We will discuss this from both theoretical and heuristic viewpoints.
Click on a title to view the abstract!