Sieve weights and their smoothings

Journal article

We obtain asymptotic formulas for the 2kth moments of partially smoothed divisor sums of the Möbius function. When 2k is small compared with A, the level of smoothing, then the main contribution to the moments comes from integers with only large prime factors, as one would hope for in sieve weights. However if 2k is any larger, compared with A, then the main contribution to the moments comes from integers with quite a few prime factors, which is not the intention when designing sieve weights. The threshold for "small'' occurs when A=12k(2kk)−1.

One can ask analogous questions for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, with even less cancelation in the divisor sums. We give, we hope, a plausible explanation for this phenomenon, by studying the analogous sums for Dirichlet characters, and obtaining each type of behavior depending on whether or not the character is "exceptional''.

Authors: Granville, A; Koukoulopoulos, D; Maynard, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Société Mathématique de France
• Journal: Annales Scientifiques de l'Ecole Normale Superieure
• Volume: 54
• Issue: 5
• Pages: 1089-1177
• Publication date: 2021-12-15
• Acceptance date: 2020-04-08
• DOI: 10.24033/asens.2478
• EISSN: 0012-9593
• ISSN: 0012-9593
• Language: English
• Keywords: FFR