Illinois Number Theory Seminar 2020-2021

Jesse Thorner (UIUC)

Large class groups

Abstract: For a number field F of degree  over the rationals, let  be the absolute discriminant.  In 1956, Ankeny, Brauer, and Chowla proved that for a given degree d, there exist infinitely many number fields of degree d such that for any fixed , the class group of F has size at least .  This was conditionally refined by Duke in 2003:  assuming Artin's holomorphy conjecture and the generalized Riemann hypothesis, there exist infinitely many number fields F of degree d such that the class group of F has size

In particular, given $d \ge 2$, there are (conditionally) infinitely many number fields of degree d whose class group has maximal asymptotic order.  In 2014, Cho showed that Artin's holomorphy conjecture and the generalized Riemann hypothesis can be replaced with the single assumption that Artin representations are automorphic (which implies Artin's holomorphy conjecture), unconditionally establishing Duke's conclusion for .

I will discuss joint work with Robert Lemke Oliver and Asif Zaman in which we unconditionally establish Duke's conclusion for all  (among many other things).

Jake Chinis (McGill)

On the Liouville function in short intervals

Abstract: Let λ denote the Liouville function. Assuming Riemann's Hypothesis, we'll prove that

as , provided . The proof uses a simple variation of the methods developed by Matomäki and Radziwiłł in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.

Kevin Ford (UIUC math)

Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros

Abstract: Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x.  Our bound depends on a property of the interval sieve which is not well understood.  We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size.  Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally.  Much of this work is joint with Bill Banks and Terry Tao.

Amita Malik (AIM)

Partitions into primes in arithmetic progression

Abstract:  In this talk, we discuss the number of ways to write a given integer as a sum of primes in an arithmetic progression. While the study of asymptotics for the number of ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. If time permits, we compare our results with some known estimates in special cases and discuss connections to certain classical results in analytic number theory.

Daniel Fiorilli (CNRS and Université Paris-Saclay)

Higher moments of primes in progressions

Abstract: Since the work of Barban, Davenport and Halberstam, the variance of primes in arithmetic progressions has been widely studied and continues to be an active topic of research. However, much less is known about higher moments. Hooley established a bound on the third moment, which was later sharpened by Vaughan for a variant involving a major arcs approximation. Little is known for moments of order four or higher, other than a conjecture of Hooley. In this talk I will discuss recent joint work with Régis de la Bretèche on weighted moments of moments of primes in progressions. In particular we will show how to deduce sharp unconditional omega results on all weighted even moments in certain ranges.

Scott Ahlgren (Illinois)

Scarcity of congruences for the partition function

Abstract: The arithmetic properties of the ordinary partition function  have been the topic of intensive study. Ramanujan proved that there  are linear congruences of the form for the primes , and it is known that there are no others of this form.  On the other hand, there are many examples of congruences of the form where is prime and . Here we prove that such congruences are very scarce in the case when or . The proofs rely on a variety of tools from the theory of modular forms and from analytic number theory. This is joint work with Olivia Beckwith and Martin Raum.

Maria Siskaki (Illinois)

Distribution of reduced quadratic irrationals arising from even and of backward CF expansions

Abstract: The reduced quadratic irrationals (RQIs) coming from the regular continued Fraction (CF) expansion, when ordered by their length, are known to be uniformly distributed with respect to the Gauss probability measure. In this talk I will present the corresponding result for the RQIs arising from the even and backwards CF expansions, where the invariant measure is infinite. I will also be mentioning their connection with the Pell equation. This is joint work with F. Boca. 

Kyle Pratt (Oxford)

Landau-Siegel zeros and central values of L-functions

Abstract: Researchers have tried for many years to eliminate the possibility of Landau-Siegel zeros---certain exceptional counterexamples to the Generalized Riemann Hypothesis. Often one thinks of these zeros as being a severe nuisance, but there are many situations in which their existence allows one to prove spectacular, though illusory, results. I will review some of this history and some of these results. In the latter portion of the talk I will discuss recent work, joint with H. M. Bui and Alexandru Zaharescu, in which we show that the existence of Landau-Siegel zeros has implications for the behavior of L-functions at the central point.

Joni Teravainen (Oxford)

On the Liouville function at polynomial arguments

Abstract: Let be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence  changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.

Edna Jones (Rutgers)

An Asymptotic Local-Global Principle for Integral Kleinian Sphere Packings

Abstract: We will discuss an asymptotic local-global principle for certain integral Kleinian sphere packings. Examples of Kleinian sphere packings include Apollonian circle packings and Soddy sphere packings. Sometimes each sphere in a Kleinian sphere packing has a bend (1/radius) that is an integer. When all the bends are integral, which integers appear as bends? For certain Kleinian sphere packings, we expect that every sufficiently large integer locally represented as a bend of the packing is a bend of the packing. We will discuss ongoing work towards proving this for certain Kleinian sphere packings. This work uses the circle method, quadratic forms, spectral theory, and expander graphs.

Sarah Peluse (Princeton)

Modular zeros in the character table of the symmetric group

Abstract: In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.

Jesse Thorner (UIUC)

Bombieri-Vinogradov for higher rank groups

Abstract: The classical Bombieri-Vinogradov theorem implies that primes up to X in progressions to moduli q are as well-distributed when q ≤ x^{1/2-\epsilon}, which is as good as the generalized Riemann hypothesis implies.  I will describe an extension of this problem which asks for similar results on the distribution of Hecke eigenvalues of GL(n) cusp forms in progressions modulo q.  For GL(n) cusp forms with n ≤ 4, we can produce an exact analogue of the classical Bombieri-Vinogradov theorem, and we achieve nontrivial progress when n ≥ 5.  These results constitute the limits of what is known, and we prove these results without recourse to any unproven hypothesis.  This is joint work with Yujiao Jiang, Guangshi Lü, and Zihao Wang.

Michael Griffin (BYU)

Class pairings and elliptic curves

Abstract: Ideal class pairings map the rational points of an elliptic curve to the ideal class groups of certain imaginary quadratic fields, by means of explicit maps to SL2(Z)-equivalence classes of integral binary quadratic forms. Such pairings have been studied by Buell, Call, Soleng and others. In recent work with Ono and Tsai, we used such pairings to study the class group and give explicit lower bounds on the class numbers. In many cases our result give an improvement to the lower bound of Goldfeld, Gross and Zagier. Conversely, using classical upper bounds on class numbers h(-D) for some discriminant -D represented by the equation of the elliptic curve, these pairings imply effective lower bounds for the canonical heights of non-torsion points.  I will also discuss recent work by REU students wherein the authors prove instances where the torsion subgroup of an elliptic curve injects into the class group of an imaginary quadratic field. Using this result, they demonstrate several infinite families of class groups with subgroups whose orders are divisible by various factors.

Kunjakanan Nath (U. Montreal)

Bombieri-Vinogradov type theorems for primes with a missing digit

Abstract: One of the fundamental questions in number theory is to find primes in any subset of the natural numbers. In general, it's a difficult question and leads to open problems like the twin prime conjecture, Landau's problem and many more. Recently, Maynard considered the set of natural numbers with a missing digit and showed that it contains infinitely many primes whenever the base b ≥ 10. In fact, he has established the right order of the upper and the lower bounds when the base b = 10 and an asymptotic formula whenever b is large (say 2 × 10⁶). In this talk, we will consider the distribution of primes with a missing digit in arithmetic progressions for base b large enough. In particular, we will show an analog of the Bombieri-Vinogradov type theorems for primes with a missing digit for large base b. The proof relies on the circle method, which in turn is based on the Fourier structure of the digital set and the Fourier transform of primes over arithmetic progressions on an average. Finally, we will give its application to count the primes of the form p = 1 + m² + n² with a missing digit for a large base.

Robert Schneider (UGA)

Work in progress: a multiplicative theory of integer partitions

Abstract: 

I give an incomplete survey of a meta-theory of partitions I began developing in graduate school with my pair of papers on partition zeta functions (2016) and the $q$-bracket (2017), that is still under construction by myself and my collaborators (Akande, Dawsey, Hendon, Jamerson, Just, Ono, Pulagam, Rolen, M. Schneider, Sellers, Sills, Wagner), my research students, and other authors.

Much like the natural numbers $\mathbb N$, the set $\mathcal P$ of integer partitions ripples with interesting patterns and relations. Now, Euler's product formula for the zeta function as well as his generating function formula for the partition function $p(n)$ share a common theme, despite their analytic dissimilarity: expand a product of geometric series, collect terms and exploit arithmetic structures in the terms of the resulting series. Works of Alladi-Erd\H{o}s and Andrews hint at algebraic and analytic super-structures unifying aspects of partition theory and arithmetic. One wonders then: might some theorems of classical multiplicative number theory arise as images in $\mathbb N$ of greater algebraic or set-theoretic structures in $\mathcal P$? 

We show that many well-known objects from elementary and analytic number theory can be viewed as special cases of phenomena in partition theory such as: a multiplicative arithmetic of partitions that specializes to many theorems of elementary number theory; a class of ``partition zeta functions'' containing the Riemann zeta function and other Dirichlet series (as well as exotic non-classical cases); partition-theoretic methods to compute arithmetic densities and Abelian-type theorems as limiting cases of $q$-series; and other phenomena at the intersection of the additive and multiplicative branches of number theory.

Demi Allen (Bristol)

Weighted Diophantine approximation on manifolds

Abstract: 

Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. Consequently, studying the size of sets of numbers that can be approximated to within a given precision is very important. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. From a measure-theoretic point of view, this classical set of $\psi$-well-approximable numbers in Euclidean space is well understood. However, it is far less clear what happens when this set, or its various generalisations or higher dimensional analogues, is intersected with other natural sets such as curves, manifolds, or fractals. Studying such questions has become a hugely popular topic of interest in Diophantine Approximation in recent years. In this talk I will discuss recent joint work with Baowei Wang (HUST, China) concerning the particular topic of weighted Diophantine Approximation on manifolds.