[Math] Divergence of a series similar to $\sum\frac{1}{p}$

nt.number-theoryprime numberssequences-and-series

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic progressions $r_i+m\cdot p_i$ as $r_{k+1}$ which is going to be the new residue class for a (random) prime $p_{k+1}$.
We proceed in this way "covering" the natural numbers (without changing the $r_i$'s.)
Question:Is it true that $\sum\limits_{n=1}^\infty\frac{1}{r_n}=+\infty$?

Motivation:
This would directly imply Dirichlet's theorem on primes in arithmetic progressions if we make use of this
Lemma:
Let $a_n$ be a sequence of natural numbers, strictly increasing with $\gcd(a_i,a_j)=1$.
Then if $\sum\limits_{n=1}^\infty\frac{1}{a_n}=+\infty$ then the sequence contains infinitely many prime numbers.

I tried to modify some proofs which show the divergence of $\sum\frac{1}{p}$ but without much success.
Thank you very much in advance!

EDIT: The primes are distinct and the residue classes are not reduced modulo $p$.Suppose we start with the primes $2,3,7$ and their residue classes are $(1)_2 , (2)_3 , (4)_7$ which means $r_1=1$, $r_2=2$ and $r_3=4$. The least number not covered by the progressions $1+2k , 2+3k , 4+7k$ is $6$.We define then $r_4=6$ and $6$ is going to be a new residue class for a new prime (random choice) let's say $p_4=17$.Then $r_5=10$ and we choose a new prime (Let's say $p_5=5$) and continue in this direction.
The $r_i$'s could be much greater than the $p_i$'s

Best Answer

The question makes sense if the $r$'s are all positive. The answer is likely to be yes for the folllowing reason: if it were no, there would be an explicit sequence where the $r$'s grow faster than $O(1/f'(n))$, where $f(n)$ is smaller than any iterated $\log$ function, and thus smaller than $\log \log\log\log\log n$, call this $\log_5 n$ or $g(n)$. Current lower bounds of the conceivable maximal growth rate of the $r$'s are like $$\dfrac1{g'(n)(\log_3 n)^2},$$ see recent work of Ford, Konyagin, Green, Maynard, and Tao. Having such a sequence of $r$'s and $p$'s would allow one to radically improve on this bound, for one could leverage this to find really large gaps between primes.

Of related interest may be Kanolds work in 1963-65 on such sequences of $r$'s. If the $r$'s grow slowly enough (something like $n^{2-c}$), one can get Linnik's theorem on the least prime in arithmetic progressions as well as a non-quantitative version of Dirichlet's Theorem through elementary means.

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