Parity questions concerning the generalized divisor function with restrictions on the prime factors

When/Where:

April 9, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

Let \(\nu_y(n)\) denote the number of distinct prime factors of n which are \(<y\). We discuss the asymptotic behavior of the sum

\( S_{-k}(x,y)=\sum_{n\le x}(-k)^{\nu_y(n)}.\)

 

The emphasis is on uniform estimates as y varies in the interval \([2,x]\). We study this sum using a combination of analytic techniques, sieve methods, and difference- differential equations. It turns out that there is a difference in behavior when \(k=p-1\), and \(k\ne p-1\), where p is a prime, which is clearly understood by considering certain Dirichlet series and representing them in terms of the Riemann zeta function.

 

This is joint work with my PhD student Ankush Goswami.