Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2015 Talks are at noon on Monday in room W565 of University Hall For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: June 1 at 11:00am in C630 Adam Felix (KTH Royal Institute of Technology, Sweden) How close is the order of $\pmb{a}$ mod $\pmb{p}$ to $\pmb{p-1}$? Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let $f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ is prime. There are many results that suggest $p-1$ and $f_{a}(p)$ are close. For example, Artin's conjecture and Hooley's subsequent proof upon the Generalized Riemann Hypothesis. We will examine questions related to the relationship between $p-1$ and $f_{a}(p)$.

 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Jan 12 everyone Open problem session at noon in UHall W565 Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Jan 19 Nathan Ng A subconvexity bound for modular $\pmb{L}$-functions at noon in UHall W565 Let $L(s)$ be the $L$-function associated to a modular form of weight $k$ for the full modular group. Using spectral theory, Anton Good proved that $L(k/2+it) \ll t^{1/3} (\log t)^{5/6}$. Matti Jutila discovered a simpler proof which makes use of the Voronoi summation formula, exponential integrals, and Farey fractions. We shall present Jutila's argument. Jan 26 Nathan Ng A subconvexity bound for modular $\pmb{L}$-functions, part 2 at noon in UHall W565 This is a continuation of last week's talk. I will sketch Matti Jutila's proof of a subconvex bound for a modular $L$-function on the critical line. The main ideas are an approximate functional equation, the use of Farey fractions, Voronoi's summation formula, and exponential integral and sum estimates. Feb 2 no seminar (PIMS Distinguished Visitor will speak at noon) Feb 9 Nathan Ng Gaps between the zeros of the Riemann zeta function at noon in UHall W565 In this talk we will show how to exhibit large and small gaps between the zeros of the Riemann zeta function, assuming the Riemann hypothesis. This is based on a technique of Montgomery and Odlyzko. The problem of finding small gaps between the zeros leads to a very interesting optimization problem. Feb 23 Gabriel Verret Automorphism groups of vertex transitive graphs at 9am in UHall D632 (University of Western Australia) no abstract available Mar 11 Peter J. Cho Zeros of $\pmb{L}$-functions Wednesday at 9am in UHall D632 (University of Buffalo) In 20th century, one of the most striking discoveries in number theory is Montgomery's pair-correlation. It says that pair-correlation of zeros of the Riemann zeta function is the same with that of eigenvalues of unitary matrices. In 1990's, Rudnick, Katz and Sarnak studied the zeros of $L$-functions more systematically. Moreover, Katz and Sarnak proposed the $n$-level density conjecture which claims that distributions of low-lying zeros of $L$-functions in a family is predicted by one of compact matrix groups, which are $U(N)$, $SO(\text{even})$, $SO(\text{odd})$, $O(N)$, and $Sp(2N)$. At the end of the talk, I will state an $n$-level density theorem for some families of Artin $L$-functions and talk about counting number fields with local conditions. I will start with a friendly definition of $L$-functions and give some examples. No background or knowledge for $L$-functions are required for this talk. Mar 13 Daniel Vallieres Abelian Artin $\pmb{L}$-functions at zero Friday at 9am in UHall C620 (Binghamton University) In the early 1970s, Harold Stark formulated a conjecture about the first non-vanishing Taylor coefficient at zero of Artin $L$-functions. About 10 years later, he refined his conjecture for abelian $L$-functions having order of vanishing one at zero, under certain hypotheses. In 1996, Karl Rubin extended this last refinement of Stark to the higher order of vanishing setting. In this expository talk for a general audience, we will give a survey of this area of research and present a more general conjecture, which we formulated in the past few years. At the end, we will present evidence for our conjecture and indicate one possible direction for further research. Mar 23 Tristan Freiberg Square totients at noon in UHall W565 (University of Missouri) A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p-1)(q-1)$ is a perfect square. This is joint work with Carl Pomerance (Dartmouth College). May 8 Ram Murty Consecutive squarefull numbers Friday at 10:30am in UHall C640 (Queen's University) A number $n$ is called squarefull if for every prime $p$ dividing $n$, we have $p^2$ also dividing $n$. Erdos conjectured that the number of pairs of consecutive squarefull numbers $(n, n+1)$ with $n < N$ is at most $(log N)^A$ for some $A >0$. This conjecture is still open. We will show that the abc conjecture implies this number is at most $N^e$ for any $e>0$. We will also discuss a related conjecture of Ankeny, Artin and Chowla on fundamental units of certain real quadratic fields and discuss its connection with the Erdos conjecture. This is joint work with Kevser Aktas. June 1 Adam Felix How close is the order of $\pmb{a}$ mod $\pmb{p}$ to $\pmb{p-1}$? at 11:00am in UHall C630 (KTH Royal Institute of Technology, Sweden) Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let $f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ is prime. There are many results that suggest $p-1$ and $f_{a}(p)$ are close. For example, Artin's conjecture and Hooley's subsequent proof upon the Generalized Riemann Hypothesis. We will examine questions related to the relationship between $p-1$ and $f_{a}(p)$.
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