---
address: '32 Avenue de l’Observatoire, 25044 Besançon Cedex'
author:
- Michel PLANAT
title: |
     Class numbers in the imaginary quadratic field\
    and the $1/f$ noise of an electron gas 
---

Introduction to Noise\
in Electrical Circuits {#Electrical}
======================

Noise in electrical circuits is found in many forms, some of them have been well explained: thermal noise, shot noise, partition noise, burst noise... and one which is still subject to much debate due to its universality and the lack of a general accepted model: 1/f noise.

Let us first review briefly our understanding of electrical thermal noise. Due to thermal agitation free electrons in a metallic conductor are moving around continuously causing collisions with the atoms and a continuous exchange of energy between the modes. This was first investigated experimentally by Johnson [@Joh.28] and theoretically explained by Nyquist [@Nyq.28]. The noise in any circuit kept at uniform temperature $T$ can be described by a noise voltage$(\overline{v^2})^{1/2}$ in series with a resistor R of the circuit such that for a small frequency interval $df$ $$\overline{v^2}~=~4~kRTp(f)~df, \label{thermal}$$ where $p(f)=\frac{hf}{kT}(e^{hf/kT}-1)^{-1}$ is the Planck factor, $k=1.38~10^{-23}$ [J/K]{} is Boltzmann’s constant and $h=6.62~10^{-34}$ [J.s]{} is Planck’s constant. For room temperature and not too high frequency $hf/kT \ll 1$, so that $p(f)$ is equal to unity. These equations are verified from experiments. For example with a resistor of $R=1~K\Omega$, at room temperature $T=300~K$, if one uses a filter of bandwith $df=1~MHz$, the expected voltage fluctuation is of the order $4~\mu V$ easily detectable on an oscilloscope.

Our second example is shot noise in a diode. Such a noise is produced by random emission of electrons from the cathode to the anode. Schottky theorem [@Sch.18] states that the mean current $I$ undergoes a fluctuation $$\overline{i^2}~=~2~eI~df, \label{shot}$$ where $e=1.6~10^{-19}$ C is the charge of one electron. Using a typical value $I=1~mA$ and the same bandwith as above the expected current fluctuation is on the order $18~nA$.

Our third exemple is partition noise which occurs any time a current is distributed randomly between two electrodes. Its net effect is an extra multiplicative factor in the relation for the shot noise current [@Van.70]. A modern variant of shot noise and partition noise is quantum partition noise. Eq. (\[shot\]) was used as an extra proof that the electrical charge is indivisible and the measurement of current fluctuations allowed the determination of the unit $e$. Recently it has been argued that low dimensional systems like quantum wires or quantum dots may produce a fractional charge $e\frac{p}{q}$ ($p$ and $q$ integers) associated to a quasi particle tunneling state. The corresponding fractional shot noise current has been observed [@Sam.97].

In most small size transistors a noise of the low frequency type is observed: burst noise also called random telegraph signal (or RTS noise). The noise typically consists of random discrete-switching events in the current flowing through a transistor. It was first seen in reverse-biased p-n junctions and bipolar transistors [@Buc.83]. It was then related to random single electron capture events into localized defects of very small transistors such as MOSFET’s [@Kir.89]. Assuming that to the probability of an electron making a transition from one state of amplitude $0$ to another state of amplitude I there is associated a switching time $\tau_1$. The power spectral density is found to be of the Lorentzian type $$S_I(f)=\frac{2~I^2 \tau_1}{4+(2\pi f \tau_1)^2}, \label{RTS}$$ that is a flat spectrum up to the turn over frequency and a $1/f^2$ dependance above.

We now turn to $1/f$ or flicker noise. Examples above refer to a constant power spectral density $S_V(f)=4kRT$ for thermal noise and $S_I(f)=2eI$ for shot noise. Flicker effect is the large ammount of noise generated in any solid state device (vacuum tube, diode, transistor...) at low frequencies which was discovered by Johnson [@Joh.28]. It is generally described phenomenologically as $$S_V(f)~=~K\frac{V^a}{f^b}, \label{1/f noiseshot}$$ with $a$ close to $2$ and $b$ close to $1$. By integrating between the low frequency cut-off $1/\tau$ and the high frequency cut-off $f_c$, we get $$\overline{v^2}~=~\int_{1/\tau} ^{f_c} S_V(f~)~df \sim K V^2
ln(f_c\tau), \label{power}$$ that is the variability in flicker noise depends logarithmically on the integration time $\tau$, as was observed experimentally [@Kle.88]. Until now $1/f$ noise in solid state systems eluded attempts to find its universel origin. Fundamental experiments[@Vos.76] found flicker noise associated to thermal noise both in metals and semiconductors without a clear explanation on its microscopic origin.

From many experiments it was found that the noise intensity is inversely proportional to the total number of carriers $N_0$ in the sample that is $K\sim\gamma/N_0$ with $\gamma$ the Hooge factor [@Hoo.69]. Values of $\gamma$ in the range $10^{-3}$ to $10^{-8}$ have been found. They were generally attributed to different scattering mechanisms, by the lattice or the impurities, leading to mobility fluctuations of the electrons. This claims for a non linear origin of the $1/f$ noise. Fine structures revealing the interaction of electrons with bulk and surface phonons were observed in different solid-state physical systems by Mihaila [@Mih.99]. On the theoretical side a quantum electrodynamical theory was developed by P. Handel [@Han.80] based on infrared divergent coupling of the electrons to the electromagnetic field in the scattering process. The basic result for the $\gamma$ parameter is as follows $$\gamma = \frac{4\alpha}{3\pi}~\frac{\overline{(\delta w)^2}}{c^2},
\label{Handel}$$ where $\delta w$ is the change of the velocity of the accelerated charge in the collision processes governing the carrier mobility and the carrier diffusion constant, $c= 3.10^8$ [m/s]{} is the velocity of light, $\alpha=\frac{e^2}{2~hc\epsilon_0} \simeq
1/137$ is the fine structure constant and $\epsilon_0 =
8.85~10^{-12}~F/m$ is the permittivity of free space. Thirty years after its introduction the theory is still under debate [@Kiss],[@Weiss].

In large devices such as MOSFET’s $1/f$ noise is often explained as arising from a summation of RTS’s due to the defects in the insulator whenever a large range of time constants $\tau_1$ is involved.

$1/f$ noise is also observed in oscillators. One way to characterize it is from the Allan variance $$\sigma_y^2(\tau)=\frac{1}{2}{\overline{ \delta y(\tau)^2}},
\label{Allan}$$ where $\delta y(\tau)=y_{i+1}(\tau)-y_i(\tau)$ is the deviation between two consecutive samples of index $i$ and $i+1$ counted over an averaging time $\tau$ and $y_i=f_i/f_0$ refers to the ratio between the instantaneous frequency and the mean frequency of the oscillator. It can be shown [@All.87] that $1/f$ noise of the power spectral density $S_y(f)=C/f$, with C a constant related to the physical set-up, is associated to a constant value of the Allan deviation $\sigma_y(\tau)\sim2~ln2~C$. This so-called flicker floor is the limit to the stability of oscillators. Values in the range $10^{-12}$ for quartz oscillators to $10^{-15}$ for Hydrogen maser based oscillators have been obtained [@Audoin98].

We recently discovered a possible relationship between $1/f$ frequency noise in oscillator measurements and prime number theory [@Planat1], [@PRE02]. In this paper we pursue this quest by relating exact statistical mechanics of electrons of mass $m$ in a box of size $L$ to the quadratic field $Q(\sqrt{-p})$ of negative discriminant, with $p$ the number of energy quanta in units of $\delta E=h^2/8mL^2$. We find numerically a $1/f$ noise of the fractional density of states about the average classical value. Sec. 2 shows that analytical number theory (and the Riemann zeta function) underlies the statistical mechanics of the perfect non degenerate gas. Sec. 3 shows that the degeneracy of energy levels needs the extended frame of algebraic number theory (and the field of quadratic forms). To let the text accessible to a large audience of readers we do not enter into the mathematical details that can be found elsewhere [@Edwards],[@Grosswald].

The Non Degenerate Perfect Gas {#nondegenerate}
==============================

The classical approach\
to the perfect gas
-----------------------

In statistical thermodynamics the partition function allows to establish all thermodynamical properties [@Kestin]. It is called generating function in the context of number theory. For a perfect classical gas the partition function per degree of freedom and per particle is $$Z_0=\sum_{n=1}^{n_{\rm{max}}} \exp{(\frac{-E_n}{kT})}.$$ The energy levels are obtained by solving Schrödinger equation with free boundary conditions $$\begin{aligned}
&&\frac{d^2\psi}{dx^2}+\frac{2mE}{\hbar^2}\psi =0,\nonumber \\ &&
\psi=0~~\rm{at}~x=0~\rm{and}~x=L.\end{aligned}$$ The solution is $$\psi=(\frac{2}{L})^{1/2}\sin(\frac{n\pi x}{L}),$$ with $E_n=\delta E~ n^2$, $(n=1,2,\cdots)$ so that the partition function becomes $$Z_0=\sum_{n=1}^{n_{\rm{max}}} \exp{(-\pi n^2 x) },$$ with $x=\frac{\delta E}{kT}$, $\delta E=\frac{h^2}{8mL^2}$.Using the first order Mac-Laurin expansion and if $x$ is small $$Z_0(x)\simeq \frac{1}{2\sqrt x}\rm{erf}(n_{\rm{max}}\sqrt {\pi x})
-\frac{1}{2}\simeq \frac{1}{2\sqrt x}. \label{MacL}$$ For the three dimensional gas of $N$ particles we need the indistinguishability parameter $N!$ so that

$$Z =\frac{Z_0^{3N}}{N!}.$$

The gas pressure is defined from $$P(V,T)=3NkT(\frac{\partial \ln Z}{\partial
V})_{T,N}~~~~\rm{with}~V=L^3,$$ The law of perfect gas follows $$PV=RT~~~~\rm{with}~R=kN_A,$$ where $N_A$ is the Avogadro constant.

Others useful state equations are derived such as the mean energy per mole of the gas $$E=-\frac{\partial \ln Z}{\partial
\beta}=\frac{3}{2}RT~~~~\rm{with}~\beta=\frac{1}{kT},$$ and the free energy $F$ and the entropy $S$ as $$S=-\frac{\partial F}{\partial T},~~~~F=-kT \ln Z.$$ The Sackur-Tetrode equation follows $$\begin{aligned}
&S=S(E,V,N)\nonumber \\&=Nk\left[ \frac{5}{2}+\ln(\frac
{V}{N})+\frac{3}{2}\ln(\frac{m}{3\pi\hbar^2}\frac{E}{N})\right].\end{aligned}$$ Thanks to the inclusion of $N!$ term in the definition of partition function the entropy equation satisfies the property of extensivity.

The classical law of the perfect gas\
and the Riemann zeta function
-------------------------------------

The partition function $Z_0$ appears in Riemann’s pioneering work on prime number theory. It starts from the integral representation of gamma function $$\Gamma (s)=\int_0^{\infty}\exp(-x) x^{s-1} dx~~~~\rm{with}~
\Re(s)>0, \label{gamm}$$ which generalizes the factorial function into the complex plane $s$. In particular $\Gamma(s+1)=s\Gamma(s)$ and as result $\Gamma(n+1)=n!$ if $n$ is an integer.

Riemann’s zeta function is defined as $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}~~~~\rm{with}~\Re(s)>1.$$ Substituting $\pi n^2 x$ to $x$ in the integral (\[gamm\]) and summing for all integers $n$ one gets $$\pi^{-s} \Gamma(s) \zeta(2s)=\int_0^{\infty} Z_0(x) x^{s-1} dx,$$ which is the desired relation between $\zeta(s)$ and the Mellin transform of $Z_0(x)$. Here we assumed $n_{\rm{max}} \rightarrow
\infty$ in the definition of the partition function. The expression $$\xi(s)=\pi^{-s/2} \Gamma(s/2)\zeta(s),$$ is the so-called completed Riemann zeta function. This is because $\xi(s)$ admits an analytic continuation to the whole complex plane $s$ except at the simple poles $s=0$ and $s=1$ of residus $-1$ and $1$ respectively. It satisfies the functional relation $\xi(s)=\xi(s-1)$.

The approximation of $Z_0(x)$ may be obtained by taking the inverse Mellin transform and by applying the residue theorem to the two poles of $\xi(2s)$ at $s=0$ and $s=1/2$ so that $$Z_0(x) \simeq \frac{1}{2 \sqrt x}-\frac{1}{2},$$ which is similar to the previous Mac-Laurin approximation (\[MacL\]). The step from the discrete to the continuous is condensed into the contribution of poles of the Riemann zeta function.

The remaining poles at $s=-m$ ($m$ integer $\ge 1$) of $\Gamma(s)$ are absorbed by the trivial zeros at $s=-l$ ($l$ a positive integer) of $\zeta(2s)$. The unproven Riemann hypothesis [@Planat1] is that all remaining zeros are located on the critical line $s=1/2$. The modulating action of the zeros is expected to be significant within a box at the nanoscopic scale $x=\frac{\delta E}{kT}\ge 1$.

The Degenerate Perfect Gas
==========================

First form of quantum degeneracy\
due to wave-particle aspects
---------------------------------

In the quantum gas, the first form of degeneracy is due to the indiscernability of (wave)particles of size $\lambda_{\rm{th}}$ (the de Broglie wavelength) larger than the typical spacing $d$. The degeneracy parameter is [@Kestin] $$\frac{\lambda_{\rm{th}}}{d}=\frac{h/(mv)}{V^{1/3}},~~\rm{with}~v=(\frac{2
\pi k T}{m})^{1/2}.$$ (For conduction electrons in silver at room temperature the ratio is about $40$).

The partition function for electrons is taken to satisfy Fermi-Dirac statistics with chemical potential $\alpha$ $$\ln Z=\sum_s \ln \left[ 1+ \exp(-\alpha -\beta E_s) \right] +N
\alpha.$$ In the conventional approach of statistical mechanics, the discretness of quantum states is neglected and one goes from the discrete to the continuous by taking the number of states $\Omega_0(E)$ as $\frac{1}{8}$, times the volume of the Fermi sphere of radius $R_F$, times $2$ (for the two spin states of the electron). The density of states $\omega_0(E)$ per unit volume $V=L^3$ follows from $$\omega_0(E)=\frac{1}{V}\frac{d \Omega_0(E)}{dE}=2 \pi
 (\frac{2m}{h^2})^{3/2} E^{1/2}.
\label{equation2}$$

Second quantum aspect\
due to the degeneracy of energy levels
--------------------------------------

Here the number of possible quantum states is calculated exactly from number theory. $$\Omega(E)=\sum_{n_1^2+n_2^2+n_3^2 \le R_F^2}
1~~\rm{with}~~R_F^2=\frac{E_F}{\delta E}, \label{equation1}$$ where $n_1$, $n_2$ and $n_3$ are positive integers, $E_F$ is the Fermi energy and $\delta E$ is the energy quantum as above. In quantum dots the energy is quantized in units of $\delta E$: this may be observed when the Coulomb blockade energy $E_B=\frac{e^2}{
C}$ is such that $E_B \simeq \delta E \gg kT$, that is at very low temperatures or for small samples with large capacity $C$. The partition function is $$Z=\sum_{p=1}^{p_{\rm{max}}} g_p\exp
(-\frac{E_p}{kT})~~\rm{with}~E_p=\delta E ~ p,$$ where the degeneracy parameter $g_p$ is the number of states of energy $E_p$.

The number of quantum states $g_p$ also equals $\frac{1}{8}$, times 2, times the number $r_3(p)$ of solutions of the three squares diophantine equation $$n_1^2+n_2^2+n_3^2=p ~~~\rm{with}~~ p\le R_F^2,
\label{equation3}$$

and $n_1,~n_2,~n_3~,p$ integers. This is a central problem of number theory. Gauss proved that for $p>4$ and squarefree $$r_3(p)= \left\{\begin{array}{ll} 24h(-p) &~~\mbox{for}~ p\equiv
3~(mod~8),\\ 12  h (-4p)&~~ \mbox{for}~p \equiv 1,2,5~\rm{or}~6
~(mod~8),
\\ 0&~~ \mbox{for}~ p\equiv 7~ (mod~8),
\end{array}\right.
\label{equation4}$$ where $h(-p)$ is the class number in the field $Q(\sqrt{-p})$ (that is the number of primitive quadratic forms with given negative discriminant -p). The general case is $r_3(p)=\sum _{d^2|p}R_3(\frac{p}{d^2})$, where $R_3(p)$ holds for square free values of $p$. More explicit solutions can be found [@Grosswald]. They are available in the Mathematica package [@Wolfram].

The class number $h(-p)$, and thus $r_3(p)$ is a highly discontinuous function. We defined the (exact) density of states as

$$\begin{aligned}
&\omega(p)=\frac{1}{\delta p}\sum_{n=p}^{n=p+\delta p}
r_3(n)\nonumber \\ &\simeq \omega_0(p)=2\pi
p^{1/2}~~\rm{with}~~~\delta p \ll p. \label{equation5}\end{aligned}$$

Relative fluctuations of the density of states around the mean value $\omega_0(p)$ were calculated as $\epsilon(p)=\frac{\omega(p)-\omega_0(p)}{\omega_0(p)}$ (see Fig. 1). As a result the power spectral density of low frequency fluctuations versus Fourier frequency $f$ was found as $FFT
(f)\sim \frac{A~ \delta p}{f^{\alpha}}$, with $\alpha$ close to $1$ and $A\sim 0.1$, that is a $1/f$ spectrum as found in Fig. 2.

Physically one can take $\delta p=[\frac{kT}{\delta E}]$, with $[~]$ the integer part and $\delta E$ the granularity parameter, since $kT$ is the width of the Fermi surface at finite temperature T. One can conclude that the $1/f$ spectrum arising in the arithmetical approach connects to the different regimes one gets for the electron gas in quantum dots, versus the typical size $L$ of the dot, the temperature $T$ and the effective mass $m$.

Due to the low frequency noise in the density of states all thermodynamical quantities are perturbed, as well as the conductivity of electronic devices. This will be developed in a future paper.

Concluding Remarks
==================

Thermodynamical quantities attached to the quantum description of an electronic gas in a cubic box are not smooth versus energy or time, contrary to the classical viewpoint: this is because the summatory function of an arithmetical function such as $r_3(n)$ is discontinuous. The fractional deviation with respect to the asymptotic average very often leads to a low frequency power spectrum of the fractal type $1/f^b$, $b$ close to $1$. Number theory thus is a good candidate to explain many low frequency noises encountered in solid state physics and engineering as well as in others fields.

Acknowledgements {#acknowledgements .unnumbered}
================

I acknowledge Rémi Barrère for his help in the programming of arithmetical functions used in this paper.

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