On the random Chowla conjecture

• Published in: Geometric and functional analysis Vol. 33; no. 3; pp. 749 - 777
• Main Authors: Klurman, Oleksiy, Shkredov, Ilya D., Xu, Max Wenqiang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2023; Springer Nature B.V
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Normal distribution; Polynomials
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-023-00641-y

We show that for a Steinhaus random multiplicative function f : N → D and any polynomial P ( x ) ∈ Z [ x ] of deg P ≥ 2 which is not of the form w ( x + c ) d for some w ∈ Z , c ∈ Q , we have 1 N ∑ n ≤ N f ( P ( n ) ) → d C N ( 0 , 1 ) , where C N ( 0 , 1 ) is the standard complex Gaussian distribut...