Friday, April 22, 4-5 pm, MW 154

Abstract: I will discuss a criterion due to Daboussi and Kátai for checking that a bounded sequence a : ℕ → ℂ is asymptotically orthogonal to multiplicative functions (such as the Möbius or Liouville function). This allows for alternative proofs of (and generalizations of) the orthogonality results of Davenport and Green and Tao and provides motivation for a result about the structure of multiplicative functions due to Bergelson, Kułaga-Przymus, Lemańczyk, and Richter.

Friday, April 8, 4-5 pm, MW 154

Abstract: We will discuss few equivalent formulations of the Prime Number Theorem (PNT). Our focus will be on statements that can be obtained by elementary methods and pertain to the growth rate of the First (resp. second) Chebyshev’s function, sum of Mobius function and von Mangoldt function (with and without weights). The presentation will be self-contained and the proofs elementary (as expected).

Friday, April 1, 4-5 pm, MW 154

Abstract: We will talk about the recent work of Davit Karagulyan mentioned in the title. In order to better understand the relationship between Chowla’s conjecture, Sarnak’s conjecture, and the Riemann Hypothesis we will formulate properties (Chw), (S), and (R) for sequences taking values in {-1,0,1} such that the Möbius function satisfying property (Chw) is equivalent to Chowla’s conjecture, satisfying property (S) is equivalent to Sarnak’s conjecture, and satisfying property (R) is equivalent to the Riemann hypothesis. It will be the case that property (Chw) implies property (S), but we will show that properties (Chw) and (R) are independent, properties (S) and (R) are independent, and that properties (S) and (R) together need not imply property (Chw).  These results emphasize the importance of the multiplicative properties of the Möbius function when trying to derive relationships between Chowla’s conjecture, Sarnak’s conjecture, and the Riemann Hypothesis.