An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis
The Jensen moment of Balazard, Saias and Yor expresses the zeros of the Riemann zeta function outside the critical line through the mean value of the logarithm of its absolute value on that line. Jensen's inequality converts any mollifier into an upper bound for this contribution. For Dirichlet-polynomial mollifiers, the bound cannot fall below 0.0386, a barrier arising from the pole at s = 1. We instead use the reciprocal of a smoothed Euler factor, denoted E-star(X,s), constructed from prime powers up to a constant multiple of X together with an explicit pole term. The quotient of the zeta…
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