| Date: | January 22, 2020 |
| Time: | 1:45pm |
| Location: | BLOC 220 |
| Speaker: | Alex Dunn, UIUC |
| Title: | Moments of half integral weight modular L-functions, bilinear forms and applications |
| Abstract: | Given a half-integral weight holomorphic newform f, we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan-Petersson conjecture for the form f. This gives a very sharp Lindelöf on average result for L-series attached
to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salié sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski-Zaharescu. |
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| Date: | February 12, 2020 |
| Time: | 1:45pm |
| Location: | BLOC 220 |
| Speaker: | Sheng-Chi Liu, Washington State University |
| Title: | A GL_3 Analog of Selberg's Result on S(t) |
| Abstract: | The function S(t) appears in an asymptotic formula for counting the number of nontrivial zeros of the Riemann zeta function with imaginary part less than t. It was shown by Littlewood that the function has a lot of cancellation on the average over t. Later Selberg studied the moments of S(t) and the moments of the analog function associated with a Dirichlet L-function for a primitive Dirichlet character. A GL_2 analog of Selberg's result was proved by Hejhal and Luo. In this talk we will discuss a GL_3 analog of such results. This is joint work with Shenhui Liu. |
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| Date: | February 20, 2020 |
| Time: | 10:30am |
| Location: | BLOC 220 |
| Speaker: | Nahid Walji, American University of Paris |
| Title: | On the distribution of Hecke eigenvalues in the complex plane |
| Abstract: | Let r be a cuspidal automorphic representation of non-solvable polyhedral type for GL(2) over a number field. We establish the existence of sets of primes with positive upper Dirichlet density for which the associated Hecke eigenvalues satisfy prescribed bounds on their argument and/or size. For example, if r is not self-dual we show that there is a positive upper density of Hecke eigenvalues in any sector of size 2.64 radians. |
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