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Evaluate the real logarithmic derivative of Riemann xi at real part two, where every summand is positive and comparable to this kernel. The gamma logarithmic derivative and the bounded zeta logarithmic derivative give the logarithmic bound. It implies the local unit-interval count, and conversely such local counts imply this smoothed bound by summing the decaying tails.

Riemann hypothesis is equivalent to monotonicity of $\sigma \mapsto |\xi (\sigma +it)|$ for every fixed $t$ on $\sigma \ge 1/2$. Under Riemann hypothesis the real logarithmic derivative of Riemann xi is positive for $\sigma >1/2$. Conversely a zero right of that line would force the nonnegative monotone modulus to vanish on an interval, contrary to the identity theorem. Functional symmetry rules out zeros to the left.