A multiplicative analogue of Schnirelmann's theorem

Abstract

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group ( Z / q Z ) × and prove a similar result. For all suitably large primes q we define P η to be the set of primes less than η q , viewed naturally as a subset of ( Z / q Z ) × . Considering the k-fold product set P η ( k ) = { p 1 p 2 ⋯ p k : p i ∈ P η } , we show that, for η ≫ q − 1 / 4 + ϵ , there exists a constant k depending only on ϵ such that P η ( k ) = ( Z / q Z ) × . Erdös conjectured that, for η = 1 , the value k = 2 should suffice: although we have not been able to prove this conjecture, we do establish that P 1 ( 2 ) has density at least 1 64 ( 1 + o ( 1 ) ) . We also formulate a similar theorem in almost-primes, improving on existing results.

Locations

• Bulletin of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View