Prime Values of Quadratic Polynomials

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.