The parity of the number of prime factors among integers with restrictions

When/Where:

April 2, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

This talk will provide the background for my second talk next week on the parity of the generalized divisor function, which is recent joint work with my PhD student Ankush Goswami.

 

The celebrated Prime Number Theorem is equivalent to the statement that the number of prime factors function takes odd and even values with asymptotically equal frequency. More refined versions of this statement are equivalent to the Riemann Hypothesis. In this talk, as a background, I will present my results dating back to 1977-87 on the parity of the number of prime factors in which we consider
(i) only integers all whose prime factors are >y, or
(ii) integers for which all prime factors are <y, or
(iii) all integers, but their prime factors are <y.

Here the integers are from an interval [1,x], and the emphasis is on uniform estimates as y varies with x. These questions provide striking variations of the classical theme. The problems are attacked by using a combination of classic analytic techniques, sieve methods, and difference-differential equations