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.**