A sharp estimate of the discrepancy of a certain numerical sequence

TL;DR

This paper provides a precise estimate of how quickly a specific sequence of rational numbers, derived from primes, converges to uniform distribution, using advanced discrepancy and prime number techniques.

Contribution

It offers the first sharp convergence rate estimate for the discrepancy of a prime-based rational sequence, combining discrepancy bounds with prime number asymptotics.

Findings

Established the exact rate of discrepancy convergence for the sequence

Connected discrepancy estimates with prime number asymptotics

Demonstrated the sequence's equidistribution properties

Abstract

We consider a certain equidistributed sequence of rational numbers constructed from the primes. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.