Arithmetic function: Difference between revisions

Revision as of 19:48, 21 August 2023

Revision as of 21:08, 21 August 2023

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Let <math>H_n = 1 + \frac 1 2 + \frac 1 3 + \cdots +\frac{1}{n}</math> &nbsp; be the ''n''th [[harmonic number]]. Then

:<math> \sigma(n) \le H_n + e^{H_n}\log H_n</math> &nbsp; is true for every natural number ''n'' if and only if the [[Riemann hypothesis]] is true. &nbsp; &nbsp;<ref>See [[Divisor_functionDivisor function#Growth_rateGrowth rate|Divisor function]].</ref>

The Riemann hypothesis is also equivalent to the statement that, for all ''n'' > 5040,

<math display="block">\sigma(n) < e^\gamma n \log \log n </math> (where γ is the [[Euler–Mascheroni constant]]). This is [[Divisor_functionDivisor function#Growth_rateGrowth rate|Robin's theorem]].

:<math>\sum_{p}\nu_p(n) = \Omega(n).</math>