New Upper and Lower Bound Sifting Iterations

with fκ(s) as large as possible (resp. Fκ(s) as small as possible) given that the above inequality holds for all choices of A satisfying (1). Selberg [3] has shown (in a much more general context) that the functions fκ(s), Fκ(s) are continuous, monotone, and computable for s > 1, and that they tend to 1 exponentially as s goes to infinity. Let β = β(κ) be the infimum of s such that fκ(s) > 0. Selberg [3] has shown that we have κ e(1− 1 κ )2 < β < 2κ+ 0.4454,