A note on trigonometric polynomials for lower bounds of $\hat{I}\P (s)$

Abstract

Non-negative trigonometric polynomials satisfying certain properties are employed when studying a number of aspects of the Riemann zeta function. When establishing zero-free regions in the critical strip, the classical polynomial $3+4\cos (\hat{I}¸)+\cos (2\hat{I}¸)$ used by de la Vallée Poussin has since been replaced by more beneficial polynomials with larger degree. The classical polynomial was also employed by Titchmarsh to provide a lower bound on $|\hat{I}\P (\ddot{I}ƒ+it)|$ when $\ddot{I}ƒ>1$. We show that this polynomial is optimal for this purpose.

8 pages