Some heuristics on the gaps between consecutive primes

Abstract

We propose the formula for the number of pairs of consecutive primes $p_n,p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi (x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log (x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.

new computer data, updated literature

References in corpus (3)

• Estimates of Some Functions Over Primes without R.H.
• Mertens' Proof of Mertens' Theorem
• There Are Infinitely Many Prime Twins

Cited by in corpus (9)

• Maximal gaps between prime k-tuples: a statistical approach
• The distribution of maximal prime gaps in Cramer's probabilistic model of primes
• On the nth record gap between primes in an arithmetic progression
• Jumping champions and prime gaps using information-theoretic tools
• Studies of entropy measures concerning the gaps of prime numbers
• On primes amongst squares of naturals
• On the moments of the gaps between consecutive primes
• On the first occurrences of gaps between primes in a residue class
• On the distribution of primes in the alternating sums of concecutive primes