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.**

A. Kourbatov wrote on Set 19, 2017

Regarding
Conjecture 77 - I have found an exceptional case:

For q=1605, r=341
(also q=3210, r=341), we have the primes

3415781 =
3624431 = 341 (mod 1605), phi(1605)=phi(3210)=848,

and the gap G_{q,r} between
these primes is 208650 > phi(q) log^2 (3624431) = 193434.64...

The "almost always"
version of conjecture 77 seems very plausible; exceptions like
the above are extremely rare (I do not know of any other
exceptions at the moment).

This and related
conjectures are included in arXiv:1610:03340

(The paper is not
updated since January 2017, so it does not currently tell about
this particular exception.)

...

By now I have found
these counterexamples to Conjecture 77:

gap
prime1 prime2

208650
3415781 3624431 q=3210 r=341

316790
726611 1043401 q=4010 r=801

229350
1409633 1638983 q=4170 r=173

I do believe that
the "almost always" version of conjecture 77 is true. (Last year
I already updated the corresponding OEIS entries to state
"almost always" in conjectures.)

***