Sohail Farhangi: On certain aspects of the Möbius randomness principle

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.

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