A note on trigonometric polynomials for lower bounds of $\zeta(s)$

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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the critical strip, the classical polynomial $3+4\cos(\theta)+\cos(2\theta)$ used by de la Vall\'ee Poussin has since been replaced by more beneficial polynomials with larger degree.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 $|\zeta(\sigma+it)|$ when $\sigma>1$. We show that this polynomial is optimal for this purpose.