Problems & Puzzles: Conjectures

Conjecture 77. Gaps between primes of the form p=qn + r

Alexei Kourbatov sent the following conjecture for primes p=qn+r, gcd(q,r)=1.

Conjecture:
Gaps between primes p = qn + r up to x are less than phi(q)*log^2(x)
Here phi(q) is Euler's totient function, the number of positive integers <= q coprime to q. (This is a generalization of Cramer's conjecture. We get Cramer's case if q=2, r=1.)
 
No counterexamples exist for any 1<=r < q<=50, gcd(q,r)=1; x<10^10.
 
Example: q=10 r=1
The primes 180666691 and 180667801 both have the form 10n+1.
In between, there are no other primes of the form 10n+1.
The gap 1110 = 180667801 - 180666691 is less than phi(10)*log^2(180667801) = 4*(19.012)^2 = 1445.85
(NOTE: the logarithm is taken of the larger end of the gap.)
 
Other examples can be found in the OEIS:
A268925  Record (maximal) gaps between primes of the form 6k + 1.
A268928  Record (maximal) gaps between primes of the form 6k + 5.
A268799  Record (maximal) gaps between primes of the form 4k + 3.
 
A quick update: still no counterexamples. More data in the OEIS now:
 
A268984  Record (maximal) gaps between primes of the form 10k + 1.
A269234  Record (maximal) gaps between primes of the form 10k + 3.
A269238  Record (maximal) gaps between primes of the form 10k + 7.
A269261  Record (maximal) gaps between primes of the form 10k + 9.
A268925  Record (maximal) gaps between primes of the form 6k + 1.
A268928  Record (maximal) gaps between primes of the form 6k + 5.
A084162  Record (maximal) gaps between primes p = {1, 2} modulo 4.
A268799  Record (maximal) gaps between primes of the form 4k + 3.
 
See a convenient query to see the relevant OEIS sequences:

Weaker conjectures:

(I) Almost all maximal gaps between primes p = qn + r below x are less than phi(q)*log^2(x)
(II) Gaps between primes p = qn + r below x are O(phi(q)*log^2(x)).
 
These conjectures are based on the following "ingredients":
- the prime number theorem;
- Dirichlet's theorem on arithmetic progressions;
- a heuristic application of extreme value theory.

Q. Prove these conjectures or find counterexamples.


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