Department of

# Mathematics

Seminar Calendar
for Graduate Student Number Theory Seminar events the year of Sunday, May 19, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      April 2019              May 2019              June 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6             1  2  3  4                      1
7  8  9 10 11 12 13    5  6  7  8  9 10 11    2  3  4  5  6  7  8
14 15 16 17 18 19 20   12 13 14 15 16 17 18    9 10 11 12 13 14 15
21 22 23 24 25 26 27   19 20 21 22 23 24 25   16 17 18 19 20 21 22
28 29 30               26 27 28 29 30 31      23 24 25 26 27 28 29
30


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

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 $S(k).$