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(Following question 2269073. See also respective lower bounds.)

Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ r+3q, \ldots \tag{P} $$

Dirichlet proved that there are infinitely many primes in progression (P).

Let $R(n,q,r)$ be the $n$th record gap between primes in progression (P). For example, with $q=6$ and $r=1$, we have $R(n,6,1)=\mbox{A268925}(n)$; see http://oeis.org/A268925.

Conjecture (see arXiv:1709.05508): Almost all record gaps satisfy $$ R(n,q,r) < \varphi(q) n^2 + (n+2)q\log^2 q. \tag{1} $$

Question 1: Find a counterexample to inequality $(1)$. (You will likely need to write a program and run it long enough. No counterexamples exist for $r<q\le2000$ and $n\le14$.)

For comparison, here is a tighter conjectural bound (also for almost all record gaps): $$ R(n,q,r) < \varphi(q) n^2 + (n+2)\varphi(q)\log^2 q. \tag{2} $$ Here $\varphi(q)$ is Euler's totient function.

A few counterexamples to $(2)$ are known; e.g. for $q=20$, $r=17$.

Question 2: Find more counterexamples to inequality $(2)$.

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Take $q=23$ and $r=4$. The record gaps between primes $p\equiv 4$ (mod $23$) are $$ 138 = 211 - 73,$$ $$322 = 809 - 487, $$ $$1150 = 4259 - 3109, $$ $$1380 = 144973 -143593, $$ $$2070 = 459337 - 457267 \ldots $$

The 5th record gap occurring between primes 457267 and 459337 in residue class 4 (mod 23) is $$ 2070 = 459337-457267 > 22\cdot5^2 + (5+2)\cdot22\cdot(\log23)^2 \approx2064.02. $$ This is a counterexample to inequality $(2)$ for $n=5$. (This only answers question 2.)

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