5 papers · 1 filter
On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors
Ethan S. Lee, Rowan O'Clarey
Every natural number greater than can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-fr…
Minimal zero-free regions for results on primes between consecutive perfect th powers
Ethan Simpson Lee
We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect th powers. Our computa…
On the error in the prime number theorem in short intervals
Ethan Simpson Lee
Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem using a new smoothing argument. Our results i…
Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis
Ethan Simpson Lee, PaweÅ Nosal
In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that …
Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias
Shehzad Hathi, Ethan S. Lee
The first result of our article is another proof of Mertens' third theorem in the number field setting, which generalises a method of Hardy. The second result concerns the sign of…