B\'{a}ez-Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals

Abstract

Criterion for the Riemann hypothesis found by B\'{a}ez-Duarte involves certain real coefficients $c_{k\text{}}$defined as alternating binomial sums. These coefficients can be effectively investigated using N\"{o}% rlund-Rice's integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of $c_{k\text{}}$for large values of $k$, at least to $k=4\cdot 10^{8}$. We give explicit expressions both for the trend and for the oscillations. The first tends to zero and is therefore, in view of the criterion, irrelevant for the Riemann hypothesis. The oscillations can be further decomposed into a series of harmonics with amplitudes diminishing quickly. Possible violation of the Riemann hypothesis would indicate that the amplitude of some high harmonic increases.