Oberseminar Zahlentheorie

Bruce Berndt
Balanced Derivatives, Identities, and Bounds for Trigonometric Sums and Bessel Series
Abstract: Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, my co-authors and I derived representations for certain sums of products of trigonometric functions as double series of Bessel functions. These series are generalized by introducing the novel notion of balanced derivatives, leading to further theorems. The regions of convergence in the unbalanced case are different from those in the balanced case. If \(x\) denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as \(x\to\infty\) are established. This is joint work Martino Fassina, Sun Kim, and Alexandru Zaharescu.
Slides

Oberseminar Zahlentheorie

Scott Ahlgren
Congruences for the partition function
Abstract: The arithmetic properties of the ordinary partition function have been the topic of intensive study for many years. Much of the interest (and the difficulty) in this problem arises from the fact that values of the partition function are given by the coefficients of a weakly holomorphic modular form of half integral weight. I’ll describe some new work with Olivia Beckwith and Martin Raum and some new work with Patrick Allen and Shiang Tang which goes a long way towards explaining exactly when congruences for the partition function can occur. The main tools are techniques from the theory of modular forms, Galois representations, and analytic number theory.

Oberseminar Zahlentheorie

Danylo Radchenko
Fourier interpolation from zeros of the Riemann zeta function
Abstract: I will talk about a recent result that shows that any sufficiently nice even analytic function can be recovered from its values at the nontrivial zeros of \(\zeta(1/2+is)\) and the values of its Fourier transform at logarithms of integers. The proof uses an explicit linear interpolation formula, whose construction involves modular integrals for the theta group. The talk is based on a joint work with Andriy Bondarenko and Kristian Seip.
Slides

Oberseminar Zahlentheorie

George Andrews
How Ramanujan May Have Discovered of the Mock Theta Functions
Abstract: The mock theta functions made their first public appearance in Ramanujan's last letter to Hardy. Ramanujan explains that he is trying to find functions apart from theta functions that behave like theta functions near the unit circle. Where did he ever get the idea that such functions might exist? Why in the world would he consider the special q-series that he lists in his last letter? The object of this talk is to provide a plausible explanation for the discovery of mock theta functions.

Oberseminar Zahlentheorie

Jan Manschot
Modularity in Topological Field Theory
Abstract: Partition functions of topological quantum field theories are of interest in both physics and mathematics. A remarkable phenomenon is that these partition functions can often be expressed in terms of modular forms thanks to physical dualities of the theories. This talk will focus on the modularity of a theory known as N=2* super Yang-Mills with gauge group SU(2). I will explain how explicit evaluation of the partition function of the topologically twist of this theory on a smooth, compact 4-manifold gives rise to bi-modular forms, mock modular forms and generalizations. Based on joint work with G. W. Moore.

Oberseminar Zahlentheorie

Frank Garvan
The Unimodal Sequence Conjectures
Abstract: In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4 congruences for the coefficients of these generating functions. In 2016 Kim, Lim and Lovejoy obtained similar results for odd-balanced unimodal sequences and made similar mod 4 conjectures. In 2017 the speaker made some similar conjectures for the Andrews spt-function.
In this talk we sketch the proof of one of these conjectures. The proof involves connecting the Hurwitz class number function with one of Ramanujan's mock theta functions.
If time permits we describe the necessary ingredients for approaching the other conjectures.
This is joint work with Rong Chen.

Oberseminar Zahlentheorie

Oberseminar Zahlentheorie

Oberseminar Zahlentheorie

Stephen Kudla
The case of the N-gon
Abstract: In joint work with Jens Funke, we construct indefinite theta series for the data proposed in S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline, Multiple D3-instantons and mock modular forms II. This data can be viewed as defining an N-gon \(\gamma\) in the symmetric space D of oriented negative 2-planes in an inner product space of signature (m-2,2). As in our earlier work, the resulting theta series is defined by integrating the KM theta 2-form over a surface S in D with boundary \(\gamma\). The problem of actually constructing such a surface S is avoided by the introduction of a homotopy argument. This new method provides an interpretation of the subtle sign invariant as a linking number and should be applicable in more general situations.

Oberseminar Zahlentheorie

Jeremy Lovejoy
Parity bias in partitions
Abstract: By parity bias in partitions, we mean the tendency of partitions to have more odd parts than even parts. In this talk we will discuss exact and asymptotic results for \(p_e(n)\) and \(p_o(n)\), which denote the number of partitions of n with more even parts than odd parts and the number of partitions of n with more odd parts than even parts, respectively. We also discuss some open problems, one of which concerns a q-series with an "almost regular" sign pattern, reminiscent of some notorious q-series found in Ramanujan's lost notebook. This is joint work with Byungchan Kim and Eunmi Kim.
Slides

Oberseminar Zahlentheorie

Ken Ono
Variants of Lehmer's Conjecture on Ramanujan's tau-function
Abstract: In the spirit of Lehmer's unresolved speculation on the non-vanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of \(\tau(n)\), or is a Fourier coefficient of any given modular form. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained the first results for such questions. This lecture will describe the latest results on such questions.