Department of

# Mathematics

Seminar Calendar
for Graduate Student Number Theory events the year of Tuesday, January 1, 2019.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2018           January 2019          February 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1          1  2  3  4  5                   1  2
2  3  4  5  6  7  8    6  7  8  9 10 11 12    3  4  5  6  7  8  9
9 10 11 12 13 14 15   13 14 15 16 17 18 19   10 11 12 13 14 15 16
16 17 18 19 20 21 22   20 21 22 23 24 25 26   17 18 19 20 21 22 23
23 24 25 26 27 28 29   27 28 29 30 31         24 25 26 27 28
30 31


Thursday, February 21, 2019

2:00 pm in 241 Altgeld Hall,Thursday, February 21, 2019

#### A note on the Liouville function in short intervals

Abstract: We will begin discussing a note of Kaisa Matomaki and Maksym Radziwill on the Liouville function in short intervals. Come prepared to discuss and participate. You can find the note here: https://arxiv.org/abs/1502.02374

Thursday, April 11, 2019

2:00 pm in 241 Altgeld Hall,Thursday, April 11, 2019

#### Conversations on the exceptional character

Abstract: We will spend the last few weeks of the semester discussing Landau-Siegel zeros. In particular, we will be discussing Henryk Iwaniec's survey article "Conversations on the exceptional character."

Friday, August 30, 2019

3:00 pm in 343 Altgeld Hall,Friday, August 30, 2019

#### Organizational Meeting

###### Aubrey Laskowski (UIUC)

Abstract: This will be the organizational meeting for the graduate student number theory seminar. We will discuss the schedule for weekly meetings, as well as begin sign-up for speakers.

Friday, September 6, 2019

3:00 pm in Illini Hall 1,Friday, September 6, 2019

#### Series and Polytopes

###### Vivek Kaushik (Illinois Math)

Abstract: Consider the series $S(k)=\sum_{n \geq 0} \frac{(-1)^{nk}}{(2n+1)^k}$ for $k \in \mathbb{N}.$ It is well-known that $S(k)$ is a rational multiple of $\pi^k$ using standard techniques from either Fourier Analysis or Complex Variables. But in this talk, we evaluate $S(k)$ through multiple integration. On one hand, we start with a $k$-dimensional integral that is equal to the series in question. On the other hand, a trigonometric change of variables shows the series is equal to the volume of a convex polytope in $\mathbb{R}^k.$ This volume is proportional to a probability involving certain pairwise sums of $k$ independent uniform random variables on $(0,1).$ We obtain this probability using combinatorial analysis and multiple integration, which ultimately leads to us finding an alternative, novel closed formula of $S(k).$

Friday, September 20, 2019

3:00 pm in 1 Illini Hall,Friday, September 20, 2019

#### The prime number theorem through the Ingham-Karamata Tauberian theorem

###### Gregory Debruyne (Illinois Math)

Abstract: It is well-known that the prime number theorem can be deduced from certain Tauberian theorems. In this talk, we shall present a Tauberian approach that is perhaps not that well-known through the Ingham-Karamata theorem. Moreover, we will give a recently discovered "simple" proof of a so-called one-sided version of this theorem. We will also discuss some recent developments related to the Ingham-Karamata theorem. The talk is based on work in collaboration with Jasson Vindas.

Friday, November 8, 2019

3:00 pm in Illini Hall 1,Friday, November 8, 2019

#### On two central binomial series related to $\zeta(4)$

###### Vivek Kaushik (UIUC)

Abstract: In this expository talk, we prove two related central binomial series identities: $\sum_{n \geq 0} \frac{{2n}\choose{n}}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $\sum_{n \in \mathbb{N}} \frac{1}{{n^4}{{2n}\choose{n}}}=\frac{17 \pi^4}{3240}.$ These series resist all standard approaches used to evaluate other well-known series such as the Dirichlet $L$ series. Our method to prove these central binomial series identities in question will be to evaluate two log-sine integrals that are equal to the series representations. The evaluation of these log-sine integrals will lead to computing closed forms of polylogarithms evaluated at certain complex exponentials. After proving our main identities, we discuss some polylogarithmic integrals that can be readily evaluated using the knowledge of these central binomial series.