Powers in arithmetic progressions

We investigate the function $$P_{a,b;N}(\ell )$$ describing the number of $$\ell $$ -th powers among the first N terms of an arithmetic progression $$ax+b$$ . We completely describe the arithmetic progressions containing the most $$\ell $$ -th powers asymptotically. Based on these results we formulate problems concerning the maximum of $$P_{a,b;N}(\ell )$$ , and we give affirmative answers to these questions for certain small values of $$\ell $$ and N.