Illinois Number Theory Seminar : Fall 2018

Bruce Reznick (University of Illinois)

Equal sums of two cubes

Ramanujan was fascinated by equal sums of two cubes, as shown in the legendary anecdote about 1729 = 10³ + 9³ = 12³ + 1³. He was also interested in the equation f_1³ + f_2³ = f_3³ + f_4³ for polynomials $f_j$ and gave several examples in which the $f_j$'s were quadratic polynomials. We will discuss the complete solution for quadratic complex polynomials, with side trips into elliptic curves and computational algebraic geometry. The simplest such identity is (x² + xy - y²)³ + (x² - xy - y²)³ = 2x⁶ -2y^{^6}. This talk is meant to be accessible to first-year graduate students.

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Florin Boca (Illinois)

-expansions with odd partial quotients

Nakada's -expansions interpolate between three classical continued fractions:  regular (obtained at ), Hurwitz singular (obtained at =little golden mean), and nearest integer (obtained at =1/2). This talk will consider -expansions in the situation where all partial quotients are asked to be odd positive integers. We will describe the natural extension of the underlying Gauss map and the ergodic properties of these transformations.  This is joint work with Claire Merriman.

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Kyle Pratt (Illinois)

Breaking the 1/2-barrier for the twisted second moment of Dirichlet L-functions

I will discuss very recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet L-functions. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.

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Scott Ahlgren (Illinois)

Maass forms and the mock theta function f(q)

Let f(q) be the well-known third order mock theta function of Ramanujan. In 1964, George Andrews proved an asymptotic formula for the Fourier coefficients of f(q), and he made two conjectures about his asymptotic series (these coefficients have an important combinatorial interpretation). The first of these conjectures was proved in 2009 by Bringmann and Ono. Here we prove the second conjecture, and we obtain a power savings bound in Andrews’ original asymptotic formula. The proofs rely on uniform bounds for sums of Kloosterman sums which follow from the spectral theory of Maass forms of half integral weight and in particular from a new estimate which we derive for the Fourier coefficients of such forms. This is joint work with Alexander Dunn.

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David Hansen (Notre Dame)

Arithmetic properties of Hurwitz numbers

Hurwitz numbers are the "-analogue" of Bernoulli numbers; they show a remarkable number of patterns and properties, and deserve to be better-known than they are.  I'll discuss some old results on these numbers due to Hurwitz and Katz, and some newer results obtained by four Columbia undergraduates during a summer REU I supervised.  No background knowledge will be assumed.

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Simon Marshall (Wisconsin)

Quantum chaos and arithmetic

If M is a compact manifold of negative curvature, Laplace eigenfunctions on M with large eigenvalue are expected to behave chaotically, reflecting the correspondence principle between classical and quantum mechanics.  I will describe this chaotic behavior, and explain what can be proved about it using methods from harmonic analysis.  I will also explain why harmonic analysis alone has a hard time giving us the complete picture, and how we can see more of it using tools from number theory.