Problems & Puzzles: Conjectures

Conjecture 84.  Conjecture related to primes in A.P.

Alexei Kourbatov wrote on Novemebre, 2019

Let q>r be fixed coprime positive integers. Consider the arithmetic progression:
r, r+q, r+2q, r+3q, ...

Let R(n,q,r) be the n-th record gap between primes in this progression.
For example, with q=6 and r=5 we have R(n,6,5)=A268928(n) in OEIS.

The n-th record prime gap R(n,q,r) almost always satisfies the double inequality 1/6 phi(q) n^2  <  R(n,q,r)  <  phi(q) n^2 + (n+2)q ln^2 q

Q. find counterexamples to the above inequalities.


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