Weyl's and Browder's theorem for an elementary operator

F. Lombarkia; A. Bachir

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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $σ (n) : = \sum_{d | n} d < e^{γ} n log log n$ is satisfied for $n \geq 5041$ , where $γ$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n \geq 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that $n$ must be divisible by a fifth power $> 1$ . As consequence we obtain that RH holds true iff every natural number divisible by...