If the Laplace-weighted Möbius sum Σ μ(n)e^{-nx} is O(x^{-1/2}) as x→0+, the Riemann hypothesis holds; an explicit formula with new double-pole terms is proved under a simplicity assumption.
The Riemann hypothesis is true up to $3\cdot 10^{12}$
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abstract
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $\beta + i\gamma$ of the Riemann zeta-function with $0<\gamma\leq 3\cdot 10^{12}$ have $\beta = 1/2$.
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2026 1verdicts
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Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis
If the Laplace-weighted Möbius sum Σ μ(n)e^{-nx} is O(x^{-1/2}) as x→0+, the Riemann hypothesis holds; an explicit formula with new double-pole terms is proved under a simplicity assumption.