# Ninth Bucharest Number Theory Days June 1-3, 2021In honor of Alexandru Zaharescu's 60th Birthday

 Home Schedule & Abstracts Registration

## Schedule

Chicago timeJune 1stJune 2ndJune 3rdBucharest time
8:00-8:40Berndt Shparlinski Kim 16:00-16:40
8:50-9:30Murty Xiong Dixit16:50-17:30
9:40-10:20Rudnick Yee Matomäki17:40-18:20
10:30-11:10Buium Diaconu Marklof18:30-19:10
11:10-11:40BreakBreakBreak19:10-19:40
11:40-12:20Sarnak Ford Kontorovich19:40-20:20
12:30-13:10 Granville Soundararajan Athreya20:30-21:10
14:10-14:50 Remarks/speeches Popescu Cobeli22:10-22:50

## Abstracts

Motivated by some problems from theoretical physics, we describe a toy model problem of counting isometric immersions of certain graphs in the flat torus and related lattice point counting problems. This is joint work with David Aulicino and Harry Richman. There will be lots of pictures, and hopefully, some discussion of the motivations.

Bruce Berndt, Balanced Derivatives, Identities, and Bounds for Trigonometric Sums and Bessel Series
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in a paper published in J. Reine Angew. Math. in 2013, Zaharescu, Sun Kim, and the speaker derived representations for certain sums of products of trigonometric functions as double series of Bessel functions. With Martino Fassina, in a recently submitted paper, we returned to the ideas from our Crelle paper. The aforementioned 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 finding more efficient proofs of the identities mentioned above, we establish theorems and conjectures for upper and lower bounds for the trig sums as $$x\to\infty$$.

Alexandru Buium, Arithmetic PDEs
Some of the classical ODEs (Riccati, Weierstrass, Painleve', etc.) have arithmetic analogues in which derivatives of functions are replaced by Fermat quotients of numbers. These lead to Diophantine applications (related to the Manin-Mumford conjecture, Heegner points, etc.) Recently, in joint work with L. Miller, arithmetic analogues of some classical PDEs were introduced. An application to a new type of reciprocity law will be presented.

Cristian Cobeli, TBA

Atul Dixit, TBA

Kevin Ford, TBA

Andrew Granville, TBA

Sun Kim, TBA

Alex Kontorovich, TBA

Florian Luca, TBA

Jens Marklof, TBA

Kaisa Matomäki, TBA

Ram Murty, TBA

Cristian Popescu, TBA

Zeev Rudnick, TBA

Peter Sarnak, TBA

Igor Shparlinski, Weyl Sums: Large, Small and Typical
While Vinogradov's Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments of the Weyl sums $S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+\ldots+u_dn^d))$ where $$u = (u_1,\ldots,u_d) \in [0,1)^d,$$ very little is known about the distribution, or even existence, of $$u \in [0,1)^d$$, for which these sums are very large, or small, or close to their average value $$N^{1/2}$$. In this talk, we describe recent progress towards these and some related questions.
In particular, we give new results on the spectrum of possible sizes of Weyl sums. We also present some new bounds on `slices' of $$S(u;N)$$, that is, bounds which hold for almost all $$(u_i)_{i\in I}$$ and all $$(u_j)_{j\in J}$$, where $$I \cup J$$ is a partition of $${1,\ldots,,d}$$, which improve a previous result of T. Wooley (2015).
Finally, we discuss lower bounds on slices of Weyl sums, generalising that of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020).
Joint work with Julia Brandes, Changhao Chen, Bryce Kerr, James Maynard.

Kannan Soundararajan, TBA

Maosheng Xiong, TBA

Ae Ja Yee, TBA