Prime Values of Quadratic Polynomials

Abstract

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq x^{1/2},\, n \text{ odd}}\Lambda(qn^2+a)\gg qx^{1/2}/2\varphi(q)$ is achieved for $q\ll (\log x)^b$, where $ b\geq 0 $ is a constant.