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 $0.2844$. 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.

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. 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.