Analogues of the Robin–Lagarias criteria for the Riemann hypothesis

Robin’s criterion states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the sum of divisors of [Formula: see text] and [Formula: see text] is the Euler–Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that [Formula: see text] for all odd numbers [Formula: see text]. Lagarias’s criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the [Formula: see text]th harmonic number. We establish an analogue to Lagarias’s criterion for the Riemann hypothesis by creating a new harmonic series [Formula: see text] and demonstrating that the Riemann hypothesis is equivalent to [Formula: see text] for all odd [Formula: see text]. We prove stronger analogues to Robin’s inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of [Formula: see text] and its behavior in Robin’s inequality.

Some properties of harmonic numbers

Let Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer . In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.

Optomechanical microwave oscillator and frequency comb generation in a full phononic bandgap 1D optomechanical crystal cavity

In this work we show that a silicon optomechanical crystal cavity can be used as an optomechanical oscillator when driven to the phonon lasing condition with a blue-detuned laser. The optomechanical cavity is designed to have a breathing like mode vibrating at Ωm/2π =3.897 GHz in a full phononic bandgap. Our measurements show that the first harmonic displays a phase noise of -100 dBc/Hz at 100 kHz. Stronger bluedetuned driving leads eventually to the formation of an optomechanical frequency comb, with lines spaced by the mechanical frequency. The measured phase noise grows up with the harmonic number, as in classical harmonic mixing. We present real-time measurements of the comb waveform and show that it can be adjusted to a theoretical model recently presented. Our results suggest that silicon optomechanical cavities can play a role in integrated microwave photonics.

Asymptotic inequalities for alternating harmonics

For [Formula: see text] the [Formula: see text]th alternating harmonic number [Formula: see text] is given in the form [Formula: see text] where [Formula: see text] is a parameter controlling the magnitude of the error term [Formula: see text] estimated as [Formula: see text]