Number Theory

A Smoothed Functional Defined by the Zeros of the Riemann Zeta Function

Authors: Yufei Liu

We define a function L(x)=∑_ρ(1-e^{-x/ρ}) for x>0, where the sum runs over all non-trivial zeros ρ of the Riemann zeta function ζ(s), taken in the symmetric pairing ρ and 1-ρ to ensure absolute convergence. We prove that L(x) converges absolutely for every x>0, that it is differentiable, and that its derivative is given by L'(x)=∑_ρ (1/ρ) e^{-x/ρ} (with the same pairing). We also show that L(x) is real-valued. This functional serves as a continuous analogue of the Li coefficients. We state the conjecture that the Riemann Hypothesis is equivalent to the strict positivity L'(x)>0 for all x>0. All results are unconditional and rely only on standard zero-density estimates.

Comments: 4 Pages.

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Submission history

[v1] 2026-04-12 07:40:47
[v2] 2026-04-16 12:49:23

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