Primes representable by polynomials and the lower bound of the least primes in arithmetic progressions

Letf (m) be an irreducible quadratic polynomial with integral coefficients and positive leading coefficient. Under the assumption of Extended Riemann Hypothesis, we obtain new remainder terms in the upper bounds on primes represented byf(m) orf(p) which greatly improve Bantle's recent results. As an application, we obtain, in the second part of the paper, a new result on the lower bound of the least primes in arithmetic progressions with some difference.