H. Halberstam

A comparison of two sieves, I

Number theoretic applications of the Diamond-Halberstam-Richert (DHR) sieves have yielded results that are in most cases better than those of Ankeny-Onishi (AO) and in no case worse. For each value of the sieve parameter к, the DHR upper and lower estimation functions agree with those of AO for small arguments. It is likely that the DHR estimates are better than those of AO from some point onward, but it appears difficult to show this theoretically. Here we show the interrelation between some measures of the quality of the sieves, show that the DHR upper sieve is at most of the size of that of AO at the point where the upper estimates first can deviate, and obtain a new lower bound for the sieving limit of the DHR sieve for large values of к.