Let $P(n)$ denote the largest prime factor of $n$. I wanted to know if an analogue of Baker-Harman estimate for smooth shifted primes in arithmetic progressions with fixed moduli is there in literature.


Baker-Harman proved that there exist infinitely many primes $p$ such that $P(p-1)<p^{0.2961}$ and the exponent was recently improved to $0.2844$ by Lichtman [recently](https://arxiv.org/abs/2211.09641). Has someone considered the problem of obtaining an inifnitude of such primes which are of form $1\pmod{a}$ for some fixed modulus $a$?


Thanks in advance for any help.



Infinitude of smooth shifted primes in arithmetic progression with fixed moduli
The only relevant result I found is [a paper](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-52/issue-3/Smooth-values-of-shifted-primes-in-arithmetic-progressions/10.1307/mmj/1100623415.full) by Banks, Harcharras, and Shparlinski. They did not establish existence of such smooth numbers but instead gave an upper bound.


Let


$$
\pi\_h(x,y;q,a)=\#\{p\le x:P(p-h)<y\wedge p\equiv a\pmod q\}.
$$


Then uniformly for $x\ge y\ge2$ and $q\ge1$, they proved that


$$
\pi\_h(x,y;q,a)\ll{u\rho(u)\over\varphi(q)}\prod\_{\substack{p|h\\p>2}}{p-1\over p-2}\pi(x).
$$


