| Date: | October 6, 2022 |
| Time: | 2:30pm |
| Location: | BLOC 302 |
| Speaker: | Louis Gaudet, Rutgers University |
| Title: | The least Euler prime via a sieve approach |
| Abstract: | Euler primes are primes of the form $p = x^2+Dy^2$ with $D>0$. In analogy with Linnik’s theorem, we can ask if it is possible to show that $p(D)$, the least prime of this form, satisfies $p(D) \ll D^A$ for some constant $A>0$. Indeed Fogels showed this in 1962, but it wasn’t until 2016 that an explicit value for $A$ was determined by Thorner and Zaman, who showed one can take $A=694$. Their work follows the same outline as the traditional approach to proving Linnik’s theorem, relying on log-free zero-density estimates for Hecke L-functions and a quantitative Deuring-Heilbronn phenomenon. In an ongoing work (as part of my PhD thesis) we propose an alternative approach to the problem via sieve methods that avoids the use of the above technical results on zeros of the Hecke L-functions. We hope that such simplifications may result in a better value for the exponent $A$. |
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| Date: | October 13, 2022 |
| Time: | 2:30pm |
| Location: | BLOC 302 |
| Speaker: | Nathan Green, Louisiana Tech University |
| Title: | A motivic pairing and applications |
| Abstract: | We describe our recent result on several pairings between the t-motive and the dual t-motive associated to a t-module. Specializations of these pairings allow us to give explicit formulas for the exponential and logarithmic functions of the t-module. We also explain a few applications of variations of these pairings, such as a Mellin transform-like formula for Carlitz zeta values. We describe possible generalizations of this formula to higher rank Drinfeld modules, which is a current work in progress. Time permitting, we will also explain the connection between these pairings and a noncommutative factorization of the Carlitz exponential which gives multiple zeta values. |
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| Date: | October 27, 2022 |
| Time: | 2:30pm |
| Location: | BLOC 302 |
| Speaker: | Matt Papanikolas, Texas A&M University |
| Title: | Convolutions of Goss and Pellarin L-series |
| Abstract: | Function field L-series in positive characteristic were introduced by Carlitz in the 1930's, where he studied special values of what is now called the Carlitz zeta function. Later in the 1970's and 1980's, Goss developed a general theory of function field valued L-functions associated with a Drinfeld module, defined through an Euler product using the characteristic polynomial of Frobenius acting on its Tate module. More recently, Pellarin has defined L-functions that are deformations of Carlitz's zeta function and that exhibit a number of special value properties. In this talk we will investigate convolutions of Goss and Pellarin L-series, which occur as a kind of Rankin-Selberg convolution, and present results on special values that arise from class module formulas of Demeslay and Taelman. Joint with W.-C. Huang. |
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| Date: | November 10, 2022 |
| Time: | 2:30pm |
| Location: | BLOC 302 |
| Speaker: | Hung Viet Chu, University of Illinois at Urbana-Champaign |
| Title: | Divots in the distribution of missing sums in sumsets & more-sum-than-difference sets |
| Abstract: | We begin by talking about missing sums in sumsets. For a finite set $A$ of integers, define the sumset $A + A := \{a+b: a, b\in A\}$ and the difference set $A - A := \{a-b: a, b\in A\}$. Consider the probability model where $A$ is formed by choosing each integer in $\{0, 1, \ldots, n-1\}$ with probability $p$. Note that if $A = \{0,1,\ldots, n-1\}$, then $|A + A| = 2n-1$. Hence, the random variable $X := 2n-1 - |A+A|$ counts the number of missing sums in the sumset of $A$. Let $\mathbb{P}_{p, n}(|X| = k)$ denote the probability that we observe $k$ missing sums. This probability depends on both $p$ and $n$. In 2011, Zhao proved that $\mathbb{P}_{p}(|X| = k) := \lim_{n\rightarrow\infty}\mathbb{P}_{p, n}(|X|= k)$ exists. In 2013, Lazarev, Miller, and O’Bryant showed an interesting behavior in the distribution of missing sums in the uniform model:
$$\mathbb{P}_{1/2}(|X| = 6) \ >\ \mathbb{P}_{1/2}(|X| = 7) \ <\ \mathbb{P}_{1/2}(|X| = 8).$$
This result says that in the uniform model, it is less likely to miss $7$ sums than both to miss $6$ sums and to miss $8$ sums. We call $7$ a divot when $p = 1/2$ and investigate whether a divot can happen earlier when $p$ varies. We proved
$$\mathbb{P}_{p}(|X| = 0) \ >\ \mathbb{P}_{p}(|X| = 1) \ <\ \mathbb{P}_{p}(|X| = 2),\forall p \geqslant 0.68.$$
Next, we move on to talk about more-sum-than-difference (MSTD) sets, which are sets $A$ satisfying $|A + A| > |A - A|$. In 2007, Martin and O’Bryant proved a surprising result that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{0, 1, \ldots, n-1\}$ as $n\rightarrow\infty$. We proved that for any $k\geqslant 2$, it is possible to partition $\{0, 1, \ldots, n-1\}$ into $k$ MSTD subsets and provide bounds on the smallest $n$ for such partitions to take place. This answered a question by Asada, Manski, Miller, and Suh in 2017.
The key idea in both of the above results is fringe analysis, which centers around the observation that it is much more likely to miss a sum in the two end |
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| Date: | November 15, 2022 |
| Time: | 2:00pm |
| Location: | BLOC 302 |
| Speaker: | Soumendra Ganguly, Texas A&M University |
| Title: | Subconvexity for twisted L-functions on GL(3) x GL(2) and GL(3) |
| Abstract: | In 2012, Blomer showed q-aspect subconvexity bounds
for quadratic Dirichlet character twists of degree 6 and degree 3 L-functions
for q prime. In this talk, we discuss a broad generalization of this
to cube-free q and non-quadratic twists. |
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| Date: | December 8, 2022 |
| Time: | 1:00pm |
| Location: | BLOC 302 |
| Speaker: | Yen-Tsung Chen, Penn State University |
| Title: | On Thakur's basis conjecture for multiple zeta values in positive characteristic |
| Abstract: | Classical multiple zeta values (MZV's) are generalizations of special values of the Riemann zeta function at positive integers. The machinery of regularized double shuffle relations produces many linear relations among MZV's. There is a famous conjecture of Zagier predicting the dimension of the vector space spanned by MZV's of the same weight. Furthermore, due to the numerical computation, Hoffman proposed a conjectural basis for the space in question. In the parallel but quite different world, namely the global function fields, Thakur introduced an analogue of classical MZV's in the function field setting. In this talk, we give affirmative answers to the conjectures of Todd and Thakur which are analogues of the conjectures of Zagier and Hoffman respectively for MZV's in positive characteristic. This is joint work with Chieh-Yu Chang and Yoshinori Mishiba. |
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