2014/08/02 by Kolbjørn Tunstrøm, Tunstrøm, Kolbjørn
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1408.0420
Minor revisions in preparation for submission
arxiv created 2014/08/12 · arxiv updated 2014/08/13
The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval sk:=\pk2, …,pk+12-1\ is fully sieved by the k first primes. Here we take advantage of this essential characteristic and present evidence for the conjecture that πk ∼ |sk|/ log pk+12, where πk is the number of primes in sk; or even stricter, that y=x1/2 is both necessary and sufficient for the prime number theorem to be valid in intervals of length y. In addition, we propose and substantiate that the prime counting function π(x) is best understood as a sum of correlated random variables πk. Under this assumption, we derive the theoretical variance of π(pk+12)=∑j=1k πj, from which we are led to conjecture that |π(x)-\textrmli(x)| =O(√\textrmli(x)). Emerging from our investigations is the view that the intervals between consecutive primes squared hold the key to a furthered understanding of the distribution of primes; as evidenced, this perspective also builds strong support in favour of the Riemann hypothesis.