Past Junior Number Theory Seminar

21 June 2021
16:00
Natalie Evans
Abstract

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

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  • Junior Number Theory Seminar
14 June 2021
17:30
TBA

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  • Junior Number Theory Seminar
7 June 2021
16:00
Catherine Ray
Abstract

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

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  • Junior Number Theory Seminar
31 May 2021
16:00
Jared Duker Lichtman
Abstract

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1\approx1.14$.
A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. In joint work with C. Pomerance and T.H. Chan, we study the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In particular we prove that $\tau_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets.

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  • Junior Number Theory Seminar
24 May 2021
16:00
Abstract

Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables. While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

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  • Junior Number Theory Seminar
17 May 2021
16:00
Ayesha Hussain
Abstract

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.
 

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  • Junior Number Theory Seminar
10 May 2021
16:00
Dragos Crisan
Abstract

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.

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  • Junior Number Theory Seminar
3 May 2021
16:00
Daniele Mastrostefano
Abstract

For every positive integer N and every α ∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A of (1,...,N) is constructed by choosing independently every element of (1,...,N) with probability α. We prove that, as N → +∞, for every A in B(N, α) we have |AA| ~ |A|^2/2 with probability 1-o(1), if and only if (log(α^2(log N)^{log 4-1}))(√loglog N) → ∞. This improves on a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between |AA| and |A|^2/2 when α =o(1/√log N), and supplies a complete characterization of maximal product sets of random sets.

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  • Junior Number Theory Seminar
26 April 2021
16:00
Jay Swar
Abstract

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

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  • Junior Number Theory Seminar
29 March 2021
16:00
Jay Swar
Abstract

This talk will be the first in a spin-off series on the Lawrence-Venkatesh approach to showing that every hyperbolic curve$/K$ has finitely many $K$-points. In this talk, we will give the overall outline of the approach and prove several of  the preliminary results, such as Faltings' finiteness theorem for semisimple Galois representations.

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  • Junior Number Theory Seminar

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