Euler–Riemann–Dirichlet Lattices: Applications of η(s) Function in Physics

Abstract

We discuss applications of the Dirichlet $\eta \left(s\right)$ function in physics. To this end, we provide an introductory description of one-dimensional (1D) ionic crystals, which are well-known in the condensed matter physics literature, to illustrate the central issue of the paper: A generalization of the Coulomb interaction between alternating charges in such crystalline structures. The physical meaning of the proposed form, characterized by complex (in the mathematical sense) ion–ion interactions, is argued to have emerged in many-body systems, which may include effects from vacuum energy fluctuations. We first consider modifications to the bare Coulomb interaction by adding an imaginary component to the exponent of the Coulomb law of the form $s=1+ib$, where b is a real number. We then extend the results to slower-decaying interactions, where the exponent becomes $s=a+ib$, presenting numerical results for values $1/2\le a\le 2$, which include the critical strip relevant to the Riemann hypothesis scenario.

1. Introduction

The Riemann hypothesis, which conjectures that the non-trivial zeros of the Riemann $\zeta \left(s\right)$ function, $\zeta (1/2\pm i{\mu }_n)=0$, occur solely for real values of ${\mu }_n$, has sparked considerable interest not just among mathematicians specializing in number theory but also in a wide range of scientific disciplines, including physics, which is the focus of this paper. The applications of the $\zeta \left(s\right)$ function in physics are numerous and have been thoroughly reviewed in [1]. To provide an overview of the current status of the field, we briefly review recent works that are relevant to our discussions. We begin with the physical applications.

The connections between the properties of $\zeta \left(s\right)$, including the Riemann hypothesis, and scattering amplitudes in quantum field theory are explored in [2]. A relationship between $\zeta \left(s\right)$ and a physical system, known as the Hilbert–Pólya conjecture, was proposed in the early 1900s, suggesting that ${\mu }_n$ could be interpreted as the eigenvalues of some quantum mechanical Hamiltonian. Since then, significant efforts have been devoted to identifying such an operator or uncovering other links to physics. In this context, the author constructs a closed-form scattering amplitude, revealing the remarkable result that real ‘masses’ ${\mu }_n$ are consistent with the Riemann hypothesis.

Various physical problems related to the Riemann hypothesis have been reviewed in [3], including the Hilbert–Pólya conjecture, connections to random matrix theory, the relationship with the Lee–Yang theorem on the zeros of the partition function, and phase transitions, as well as topics such as random walks, billiards, and more.

Numerical series of prime numbers can be generated as sequences of points in a 1D lattice occupied by electric charges [4]. Based on this mapping, a single-particle Hamiltonian model is proposed for the distribution of primes $\rho \left(n\right)$ along the x-axis, which are treated as quantum particles located at stationary points, $x_n$, of a Lennard-Jones-like potential. A particle-counting function, ${\pi }_Q\left(x\right)$, analogous to the prime-counting function, $\pi \left(x\right)$, is defined and obtained with the help of the Hellmann–Feynman theorem. The conditions on the sequence of energy eigenvalues, $E\left(n\right)$, for the system to ensure that the Riemann hypothesis holds for ${\pi }_Q\left(x\right)$, are derived.

An alternative approach to the prime-counting function $\pi \left(x\right)$ involves a connection between the distribution of the zeros of $\zeta \left(s\right)$ and the poles of the scattering matrix S of a quantum system. Arguments are presented in [5]. They are aimed at explaining the still unresolved question of why the poles and zeros of the S-matrix of an ideal system, which could potentially satisfy the Riemann hypothesis, always occur in pairs and are related by complex conjugation. This result thus suggests a potential pathway toward a proof of the Riemann conjecture.

A novel formulation of the zeta function represented as the sum of an infinite series of delta and cosine functions is studied in [6]. The formula exhibits duality properties, establishing a link between the Riemann hypothesis and the new formulation. Furthermore, the quantum behaviors of the energy and eigenstates for this new mathematical model are examined, offering valuable insights into several key physical phenomena, including the collapse of the wavefunction during measurement, quantum entanglement, and double-slit experiments.

An introduction to the Riemann hypothesis using familiar mathematical techniques is presented in [7], providing useful heuristic arguments in support of the hypothesis. The latter is shown to be connected to quantum mechanics by interpreting the Dirichlet series as a superposition of probability amplitudes, which leads to a unique potential with a logarithmic energy spectrum.

Other aspects of interest for our discussions include the question of non-Hermitian Hamiltonians, quantum dissipative systems, and the case of complex interactions in many-body systems. We briefly illustrate these different issues with some examples.

A nonlinear field theory, which describes frictional effects in dissipative systems through a Schrödinger-type field equation with a logarithmic nonlinearity, has become a paradigm in the field [8]. The nonlinear field equation for the damped harmonic oscillator can be solved exactly, showing that wavefunctions and a wavepacket-like solution with a Gaussian shape display reasonable physical properties that differ from those of the undamped case. Properties of the nonlinear friction term are examined and compared to similar terms used by other authors. Finally, a nonlinear friction term equivalent to the logarithmic nonlinearity is obtained but expressed in terms of combinations of position and linear momentum operators, along with their mean values.

A purely field-based description of matter, represented by well-defined elementary densities rather than by classical fields and point particles, allows for overcoming the present nonphysical divergences in gravitational and Coulomb self-energy [9]. In this approach, the gravitational mass is linked to the real part of the integral of the complex densities of matter, while the electric charge is related to the imaginary part of the same spatial integral in such a way that the field laws would not break down at small distances.

A detailed review of the foundations and applications of non-Hermitian classical and quantum physics has recently become available [10]. Key theorems and fundamental concepts in non-Hermitian linear algebra, such as the Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry, are presented in an accessible and mathematically rigorous way.

It is well known that the Hermitian condition for a Hamiltonian, $H=H^{†}$, is sufficient to ensure a real energy spectrum and unitary time evolution. However, the requirement of Hermiticity is not essential to achieve these properties in general [11]. Actually, it is possible to describe natural processes using non-Hermitian Hamiltonians where the Hermiticity condition can be replaced by a more physically intuitive condition of parity-time (PT) symmetry, $H=H^{PT}$, without violating any fundamental principles of quantum mechanics. When a Hamiltonian satisfies this relation, it is referred to as PT-symmetric.

From the early days of quantum mechanics, numerous heuristic attempts to model dissipative phenomena in nuclear, atomic, and molecular physics have relied on effective non-Hermitian Hamiltonians. In the past two decades, the introduction of parity-time (PT) symmetry has opened the door for a more systematic exploration of non-Hermitian physics. It is now understood that non-Hermitian Hamiltonians can be rigorously justified, as in the case in which the system dynamics is confined either to a subspace or within the framework of quantum measurement theory, by conditioning quantum trajectories on specific measurement outcomes [12].

To conclude this brief review, we draw the attention of the reader to recent works dealing with the mathematical aspects of the Riemann hypothesis. In [13], new sums over the primes of non-principal Dirichlet characters, which exhibit a conjectured random walk behavior, have been derived. It is shown that the Euler product formula for their L-functions holds to the right of the critical line ($\Re \left(s\right)>1/2$), which implies that the Riemann hypothesis is valid for this class of L-functions. From these findings, a new algorithm is proposed for calculating very high Riemann zeros and computing the googol-th (${10}^{100}$-th) zero to over 100 digits, which surpasses current computational limits.

The approach discussed in [14] involves an analytical study of the functional equation of $\zeta \left(s\right)$ based on a complex function, which is treated as a real function of two real variables, and its modulus, alongside extensive numerical analysis to support the Riemann hypothesis within the computational resources. These findings offer a new mathematical perspective on assessing the validity of Riemann’s hypothesis. Finally, a novel approach to the Riemann hypothesis, emphasizing the relationship between prime gaps and the non-trivial zeros of the Riemann zeta function, has been discussed recently [15]. The key findings are that past non-trivial zeros may predict future prime gaps; nonlinear interactions between prime gaps and non-trivial zeros are present; and intricate feedback loops between the two may exist. These insights open new avenues for future mathematical exploration.

In this work, we investigate potential applications of the Riemann $\zeta \left(s\right)$ function formalism, particularly in relation to the Dirichlet $\eta \left(s\right)$ function, within condensed matter physics. While a similar problem is briefly discussed in [1], our model introduces, to the best of our knowledge, several novel features making it a valuable contribution for readers interested in ongoing discussions surrounding the Riemann hypothesis in a broader context.

Our work builds on well-established results from condensed matter physics on the cohesive energy in ionic crystals [16,17,18,19]. We focus on the one-dimensional case of an infinite ionic chain with alternating electric charges. In addition to the Coulomb interaction, we use the Born exponential form for the repulsive core between neighboring ions, which prevents the lattice from collapsing, as opposed to a Lennard-Jones type, since in ionic systems, such as alkali halides, the charge transfer from the alkali atom to the halide atom is incomplete, which leads to a partially closed shell of outer electrons.

This work is motivated by the following physical considerations concerning many-body interactions in strongly coupled electron-ion systems. In real materials, it is anticipated that interactions beyond the fundamental Coulomb force between charges may become significant. Specifically, the repulsive short-range interaction arises from an interplay of electrostatic and quantum mechanical effects, such as the Pauli exclusion principle. At medium-range distances, induced dipole–dipole (van der Waals) interactions can become significant, while electron–phonon couplings may also contribute to anomalies in the structural properties of materials. In addition to these interactions, phenomena related to vacuum polarization and self-energy effects on electrons may also play a crucial role in determining the system’s stability in certain conditions.

To account for these complex scenarios in a straightforward manner, we propose introducing an effective Coulomb (EC) interaction, $V_{\mathrm{EC}}\left(r\right)$, between charges separated by a distance r, which leads to the following

$$\mathrm{CONJECTURE}:V_{\mathrm{EC}}\left(r\right)=\pm V_0{\displaystyle \frac{1}{r^s}},\mathrm{with}s=a+ib.$$

(1)

Here, both a and b are real numbers. To illustrate how the effective interaction $V_{\mathrm{EC}}\left(r\right)$ actually works, we apply it to the ionic chain described above because it admits an exact solution. Applications to real systems go beyond the scope of the present work. We then show that the total energy per site of the chain can be written as,

$$E_{\mathrm{T}}(a_0,s)=-A{\displaystyle \frac{\hslash c}{a_0\left(s\right)}}\Re \left[\eta \left(s\right)\right],$$

(2)

where A is a dimensionless quantity and $a_0\left(s\right)$ is the lattice constant at equilibrium. Here, we first consider the case $a=1$ and $b>0$, which reduces to the bare Coulomb limit when $b=0$, and then present some results for the broader case, $1/2\le a\le 2$, which covers the critical strip.

The paper is organized as follows. In Section 2, we review the connection of $\eta \left(s\right)$ to $\zeta \left(s\right)$. In Section 3, we define our lattice of charges and revisit the main results for a standard one-dimensional ionic crystal in Section 3.1. Section 3.2 introduces the Euler–Riemann–Dirichlet (ERD) lattice model, which uses $\eta \left(s\right)$ to generalize the Coulomb law. We discuss the implications of this new interaction in detail, providing plausibility arguments for the physical mechanism behind the complex interaction, which includes imaginary components. In Section 4, we present ERD results at and near the critical strip, comparing them with those from the previous section. In addition, we include three appendices: The first, in Appendix A, discusses the properties of the $\eta \left(s\right)$ function. The second, in Appendix B, covers prime numbers, presenting both numerical and analytical results that extend previous literature. The third, in Appendix C, considers the critical strip of the Riemann zeta function, where the behavior of the non-trivial zeros of $\zeta \left(s\right)$ is examined. Finally, our concluding remarks are presented in Section 5.

2. The Function $\mathsf{\eta }\left(\mathit{s}\right)$ and Its Relation to $\mathsf{\zeta }\left(\mathit{s}\right)$

Leonhard Euler was fascinated by infinite numerical series [20]. This culminated in 1735 while in Saint Petersburg, when he published the solution to the famous Basel problem, ${\sum }_{n=1}^{\infty }{n}^{-2}={\pi }^2/6$. One hundred years later, Bernhard Riemann generalized this class of series to the complex plane by defining the function [21]

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}},\mathrm{with}s=a+ib,$$

(3)

where both $(a,b)\in R$. This is now known as the Riemann zeta function. We note that, in standard notation, $a=\sigma$ and $b=t$. The complex function $\zeta \left(s\right)$ in Equation (3) is absolutely convergent when $a>1$ with a single pole at $s=1$ and residue 1 (see below). In [21], Riemann showed that $\zeta \left(s\right)$ satisfies the functional equation (see the discussion in [20], and also the detailed monograph in [22]),

$$\zeta \left(s\right)={\pi }^{s-1/2}{\displaystyle \frac{\mathsf{\Gamma }\left(\right\{1-s\}/2)}{\mathsf{\Gamma }(s/2)}}\zeta (1-s).$$

(4)

Equation (4) allows one to obtain $\zeta \left(s\right)$ for $a<0$, from $\zeta (1-s)$ with $1-s=1+\left|a\right|-ib$. In particular, note that when $a=-2m$ for integer $m>0$, Equation (4) gives the so-called trivial zeros of the Riemann zeta function, $\zeta (-2m)=0$, due to the simple poles of $\mathsf{\Gamma }(s/2)$ at $s/2=-m$, with residues ${(-1)}^m/m!$. The next question is how to calculate $\zeta \left(s\right)$ within the range $0\le a\le 1$, also known as the critical strip, which brings us to the central issue of this paper.

For this purpose, the alternating Riemann zeta function, commonly known as the Dirichlet eta function, is emphasized,

$$\eta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^{n-1}}{n^s}},$$

(5)

which converges for $\Re \left(s\right)=a>0$. It can be shown that $\eta \left(s\right)$ is proportional to $\zeta \left(s\right)$,

$$\eta \left(s\right)=\left(1-2^{1-s}\right)\zeta \left(s\right),$$

(6)

thus allowing us to evaluate $\zeta \left(s\right)$ over the entire range $0\le a\le 1$, except at $s=1$ ($a=1$, $b=0$) where the simple pole is located, while the value $\eta \left(0\right)=1/2$ can be derived using the functional relation Equation (4) (see Appendix A and [23] for details). As a result, Equation (6) yields $\zeta \left(0\right)=-1/2$, and using the value $\eta \left(1\right)=log2$, one finds $\zeta \left(s\right)=1/(s-1)$ for $s\to 1$.

Another important result derived from Equation (6) is that $\eta \left(s\right)$ should have simple zeros, $s_m$, when the factor $1-2^{1-s_m}=0$, which is consistent with the single simple pole of $\zeta \left(s\right)$ at $s=1$. This yields

$$\eta \left(s_m\right)=0,\mathrm{with}{s}_m=1+i{\displaystyle \frac{2\pi m}{log2}},$$

(7)

for integer $m\ne 0$. Efficient algorithms have been designed to calculate $\zeta \left(s\right)$ for generic s (see, for instance, [24]).

3. Euler–Riemann–Dirichlet Lattices

In Section 3.1, we provide an overview of the known results for standard 1D ionic lattices, offering the reader the necessary background for developing a more general model in Section 3.2. The latter is proposed to describe lattices with many-body interactions between ions, referred to as Euler–Riemann–Dirichlet (ERD) lattices.

3.1. Standard Ionic Lattices

To illustrate applications of the functions defined in Section 1, we begin by considering a one-dimensional ionic crystal, where the ions are arranged alternately as shown in Figure 1. In addition to the Coulomb interaction, we assume the presence of a short-range repulsive force that prevents the lattice from collapsing [16,19]. At equilibrium, the total Coulomb energy acting on a reference ion is exactly balanced by the short-range forces between the ion and its two nearest neighbors. Let us specify these forces in detail.

We consider the reference ion (index 0 and positive charge $q_{\mathrm{I}}$) to be located at the origin of coordinates, $x_0=0$, and the remaining ions at $x_j=j{a}_0$, with $\left|j\right|\ge 1$, carrying charges ${(-1)}^j{q}_{\mathrm{I}}$. The Coulomb interaction energy between the reference ion and the one at $x_j$ is given by

$$E_{0j}^{\mathrm{C}}=-{\displaystyle \frac{q_{\mathrm{I}}^2}{4\pi {ϵ}_0}}{\displaystyle \frac{{(-1)}^{j-1}}{|x_j|}},\left|j\right|\ge 1.$$

(8)

For convenience, we define, $q_{\mathrm{I}}=\pm qe$, where q is a positive real number, e is the unit of electric charge, $n=\left|j\right|$, $|x_j|=n{a}_0$, and the fine-structure constant ${\alpha }_0=e^2/(4\pi {ϵ}_0\hslash c)\simeq 1/137$. Therefore, denoting $E_{\mathrm{C}}\left(n\right)\equiv E_{0j}^{\mathrm{C}}$, Equation (8) becomes,

$$E_{\mathrm{C}}\left(n\right)=-{\alpha }_0{q}^2{\displaystyle \frac{\hslash c}{a_0}}{\displaystyle \frac{{(-1)}^{n-1}}{n}},n\ge 1.$$

(9)

The repulsive interaction energy is modeled using the Born exponential form [16],

$$E_{\mathrm{R}}\left(n\right)=E_0{e}^{-n{a}_0/\rho },\mathrm{for}n=1,$$

(10)

and for the remaining terms, $n>1$, we assume $E_{\mathrm{R}}\left(n\right)=0$, since their contribution is negligible because in most physical cases, $\rho \ll a_0$. There are other forms for $E_{\mathrm{R}}$ known in the literature involving short-range power-law repulsions say, $E_{\mathrm{R}}\simeq E_0{(\rho /a_0)}^{m_{\mathrm{R}}}$, with $m_{\mathrm{R}}\approx 12$ for a Lennard-Jones type of interaction energy typical of noble gases. We stick at the Born shape in this work for simplicity (see also the discussion on this matter in [19]).

The total lattice energy per site, $E_{\mathrm{T}}$, is just the sum over all ion positions, which takes the form

$$E_{\mathrm{T}}=2\sum _{n=1}^{\infty }[E_{\mathrm{R}}\left(n\right)+E_{\mathrm{C}}\left(n\right)]=2E_0{e}^{-a_0/\rho }-2{\alpha }_0{q}^2{\displaystyle \frac{\hslash c}{a_0}}\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^{n-1}}{n}},$$

(11)

where we recognize the Dirichlet function $\eta \left(1\right)=log2$ in the last term (Equation (5)).

Now, the condition of equilibrium requires a stable minimum, i.e., $\partial E_{\mathrm{T}}/\partial a^{\prime }{|}_{a_0}=0$, and ${\partial }^2{E}_{\mathrm{T}}/\partial a^{\prime 2}{|}_{a_0}>0$, yielding

$$\partial E_{\mathrm{T}}/\partial a^{\prime }{|}_{a_0}=0\to a_0^2{e}^{-a_0/\rho }={\alpha }_0{q}^2\eta \left(1\right){\displaystyle \frac{\hslash c\rho }{E_0}},$$

(12)

and

$${\partial }^2{E}_{\mathrm{T}}/\partial a^{\prime 2}{|}_{a_0}=2{\alpha }_0{q}^2\eta \left(1\right){\displaystyle \frac{\hslash c}{\rho a_0^2}}\left(1-2{\displaystyle \frac{\rho }{a_0}}\right)>0,$$

(13)

while the total energy per site at the minimum becomes,

$$E_{\mathrm{T}}\left(a_0\right)=-2{\alpha }_0{q}^2\eta \left(1\right){\displaystyle \frac{\hslash c}{a_0}}\left(1-{\displaystyle \frac{\rho }{a_0}}\right)<0.$$

(14)

To obtain $a_0$, Equation (12) must be solved numerically. It is convenient to consider an example to get an idea of the order of magnitude of the quantities involved, as illustrated in Figure 2. To be noted is that the interaction energies between nearest-neighbor ions give already the right order of magnitude for the total lattice energy. For instance, according to Equations (8) and (10) we find, $E_{\mathrm{C}}\left(1\right)\simeq -13.868$ eV and $E_{\mathrm{R}}\left(1\right)\simeq +0.922$ eV, respectively.

3.2. ERD Complex Lattices: Beyond the Standard Picture

In real systems, there are interactions beyond the Coulomb force. Specifically, the repulsive short-range interaction, given by Equation (10), arises from a non-trivial combination of both electrostatic and quantum mechanical effects, such as those due to the Pauli exclusion principle. These short-range interactions are challenging to calculate, but significant progress has been made by employing density functional theory for the electron states, coupled with molecular dynamics for the ions [25].

For medium-range distances, the van der Waals (VDW) interaction can play an important role, particularly between neutral closed-shell atoms. It can be understood as an induced attractive dipole–dipole interaction (see [19] for a simple derivation of its origin). In the case of ions, the VDW interaction also contributes to the total energy, introducing specific power-law terms that decay with the typical sixth-power exponent. These additional terms can be analyzed in the same way as the Coulomb interaction, leading to a lattice sum (see, e.g., [16]). A review of different types of lattice sums is provided in [26].

There are also phenomena related to vacuum polarization and self-energy effects on electrons, which are generally expected to be quite small. However, much less is known about these effects in confined systems, such as molecules or chains of atoms. Last but not least, electrons can interact with each other through the vibrational modes of the lattice, known as phonons, which play a crucial role in superconductivity (see, e.g., the book on finite fermionic systems that discusses these issues in detail [27]).

In order to take into account these complex scenarios in a simple way, we suggest to introduce an effective Coulomb (EC) interaction between charges separated by a distance r, of the form (Conjecture Equation (1), with $a=1$),

$$V_{\mathrm{EC}}\left(x\right)=\pm V_0{\displaystyle \frac{1}{x^s}},\mathrm{with}s=1+ib,$$

(15)

where $x=r/a_0$, and, for convenience, we define $V_{\mathrm{EC}}\left(x\right)=V_0{F}_{\mathrm{EC}}\left(x\right)$, with $V_0>0$, where

$$F_{\mathrm{EC}}\left(x\right)=\pm {\displaystyle \frac{1}{x}}\left[cos(blogx)-isin(blogx)\right],$$

(16)

is a complex scaling function. The latter is illustrated in Figure 3 for opposite charges (i.e., corresponding to the negative sign in Equation (15)), and few values of b, for $x_{\min }<x<2$. The lower limit is arbitrary (we use $x_{\min }=0.1$ in the plot), and it has been introduced in order to avoid the non-physical oscillations of $F_{\mathrm{EC}}\left(x\right)$ when $x\to 0$.

It is interesting to note that, due to the cosine function, the real part of the scaling function becomes less attractive and even repulsive, compared to the bare Coulomb decay, $-1/x$, suggesting some form of screening effect between charges. A plot for the case $b=1$ is shown in Figure 4 over a longer range, $0.1<x<10$.

In the following, we apply this model to the ionic lattice and examine its effects on the total lattice energy and the stability criteria of the chain. To this end, we rewrite Equation (8) so that it is of the form

$$E_{0j}^{\mathrm{EC}}\left(s\right)=-{\displaystyle \frac{q_{\mathrm{I}}^2}{4\pi {ϵ}_0}}{\displaystyle \frac{1}{a_0}}{\displaystyle \frac{{(-1)}^{j-1}}{\left(\right|x_j{|/a_0)}^s}},\left|j\right|\ge 1,s=1+ib,$$

(17)

in which we retain the presence of alternating charges, $q_{\mathrm{I}}=\pm q_{b}e$, where $q_b$ is still a positive real number obeying $q_0=q$. Therefore, denoting $E_{\mathrm{EC}}(n,s)\equiv E_{0j}^{\mathrm{EC}}\left(s\right)$, Equation (17) becomes,

$$E_{\mathrm{EC}}(n,s)=-{\alpha }_0{q}_b^2{\displaystyle \frac{\hslash c}{a_0}}{\displaystyle \frac{{(-1)}^{n-1}}{n^s}},n\ge 1,$$

(18)

such that Equations (17) and (18) reduce to Equations (8) and (9) when $b=0$. To keep the model as simple as possible, we retain the Born exponential form for the repulsive interaction, as in Equation (10), so that the total lattice energy per site, $E_{\mathrm{T}}(a^{\prime },s)$, can be written as

$$E_{\mathrm{T}}(a^{\prime },s)=2E_0{e}^{-a^{\prime }/\rho }-2{\alpha }_0{q}_b^2{\displaystyle \frac{\hslash c}{a^{\prime }}}\eta \left(s\right),s=1+ib.$$

(19)

Due to the presence of $\eta \left(s\right)$, this model brings us close to the issue of non-Hermiticity mentioned in the introduction. Further analytical work would be required to implement the effective Coulomb model for applications in quantum systems [10,11,12]. Since we are dealing with a complex expression for the total energy, Equation (19), we stipulate that the physical value is given by $\Re \left[E_{\mathrm{T}}(a^{\prime },s)\right]$, which corresponds to taking $\Re \left[\eta \right(s\left)\right]$ in (19).

The stability conditions are then the same as those obtained in Equations (12) and (13), with $\eta \left(1\right)$ replaced by $\Re \left[\eta \right(s\left)\right]$ and q by $q_b$, where now $a_0\left(b\right)$ is a function of b. The final expression for the total energy is then given by (Equation (2) with $s=1+ib$)

$$E_{\mathrm{T}}(a_0,s)=-2{\alpha }_0{q}_b^2\Re \left[\eta (1+ib)\right]{\displaystyle \frac{\hslash c}{a_0\left(b\right)}}\left(1-{\displaystyle \frac{\rho }{a_0\left(b\right)}}\right).$$

(20)

To illustrate the behavior of the total lattice energy in the presence of a complex Coulomb exponent, $s=1+ib$, we have plotted the real values of $E_{\mathrm{T}}(a^{\prime },s)$ in Figure 5a, using Equation (19). The values of b chosen for the figure correspond to $b=0$ as the reference case, $b\simeq 5.58$, yielded the lowest possible lattice energy per site and a case with a larger value of $E_{\mathrm{T}}(a^{\prime },s)$, namely $b\simeq 8.19$.

In Figure 5b, we display the lattice constant, $a_0\left(b\right)$, as a function of b, showing that the most stable situation occurs for $b\simeq 5.58$, where $a_0\simeq 0.93425$, being about 10% smaller than in the standard case, $b=0$. For $b>5.58$, the energy increases, and so does the lattice constant. This behavior is related to the fact that $\eta (1+ib)=0$ when $b=2\pi /log2\simeq 9.065$, as discussed in Equation (7). Clearly, since the generalized Coulomb term vanishes, the lattice becomes unstable, and no bound state can exist.

It is interesting to observe that the vanishing of $\Im \left[\eta (1+i{b}_1)\right]$ occurs at $b_1\simeq 5.27185$ (see the position of the vertical line in Figure 5b), which is close to $b_0\simeq 5.58$, where the minimum of the total lattice energy occurs. This difference can be easily eliminated by adjusting the parameters of the model. For example, we fit the ‘renormalized’ charge, $q_{b_1}$, so that the minimum of $a_0$ occurs at $b_1\simeq 5.27185$ where $\Im \left[\eta (1+i{b}_1)\right]=0$. This is achieved by imposing the condition, $q_{b_1}^2\Re \left[\eta (1+i{b}_1)\right]=\Re \left[\eta (1+i{b}_0)\right]$, yielding,

$$q_{b_1}=\sqrt{\Re \left[\eta (1+i{b}_0)\right]/\Re \left[\eta (1+i{b}_1)\right]}\simeq 1.0056\gtrsim 1,$$

(21)

which is remarkably close to $q=1$.

In Figure 6a, we show the results for $E_{\mathrm{T}}(a_0,s)$ as a function of b, which are consistent with the previous ones. In Figure 6b, we consider the lattice ‘stiffness’, represented by the second derivative of the energy with respect to the lattice constant, as given in Equation (13), with $\eta \left(1\right)$ replaced by $\Re \left[\eta \right(s\left)\right]$, and $q_b=1$. The phenomenon of stiffness may also have consequences for the electron–phonon couplings, possibly leading to enhanced effective electron–electron exchange, which is important for the emergence of superconductivity properties in materials (see e.g., [28]). We can argue that finite values of the imaginary component $b>0$ may result from the interaction of matter waves with vacuum energy fluctuations, thus having a purely quantum nature. Therefore, the small excess charge appearing in Equation (21) can be interpreted as being supplied by the vacuum.

4. The ERD Complex Lattices Within the Range: $\mathbf{1}/\mathbf{2}\le \mathit{a}\le \mathbf{2}$

In the following, we consider the general case $s=a+ib$ in the expression of the total lattice energy per site (cf. Equation (20)),

$$E_{\mathrm{T}}(a_0,s)=-2{\alpha }_0{q}_b^2\Re \left[\eta (a+ib)\right]{\displaystyle \frac{\hslash c}{a_0\left(b\right)}}\left(1-{\displaystyle \frac{\rho }{a_0\left(b\right)}}\right),$$

(22)

to complement the results discussed in Section 3.2. We do not attempt to relate the ERD lattice model to the question of the non-trivial zeros of $\eta \left(s\right)$. Instead, we study Equation (22) in the range $1/2\le a\le 2$, which includes the critical strip $0<a<1$. Some illustrative results are reported in Figure 7.

In Figure 7a, we show the behavior of the equilibrium lattice constant, $a_0\left(s\right)$, $s=a+ib$, as a function of a for selected values of b. We observe that $a_0\left(s\right)$ decreases with increasing a, as expected, but for $b\lesssim 3$, the opposite trend occurs, with $a_0\left(s\right)$ increasing with a. This suggests that the physics may depend crucially on the imaginary component, b. Similar results are shown in Figure 7b, where $a_0\left(s\right)$ is plotted versus b for some values of a, showing a crossover of the curves near $b_c\simeq 3.2$. In this case, the values $a_0(1/2,b)$ are the largest for $b<b_c$ and the smallest for $b>b_c$. Thus, for the latter, the case $a=1/2$ is more stable than for larger values of a.

The results for the total lattice energy per site, $E_{\mathrm{T}}(a_0,s)$ (Figure 7c), and for the lattice stiffness (Figure 7d), plotted as a function of b, are consistent with those shown in Figure 7b. These results suggest a quite rich scenario of the physics of strongly interacting systems modeled by the simple ERD lattice, which deserves further consideration, both theoretically and experimentally.

For the interested reader, we have also prepared two appendices dealing with the Riemann hypothesis: The first, in Appendix B, reviews the connection between $\zeta \left(s\right)$ and prime numbers, which contains a wealth of numerical results that complement those in the literature [29,30,31,32,33,34,35,36,37]. The second, in Appendix C, discusses simple arguments related to the Riemann hypothesis.

5. Concluding Remarks

We have presented an application of the Dirichlet $\eta \left(s\right)$ function to describe 1D ionic systems with alternating charges characterized by a complex Coulomb exponent $s=1+ib$. This leads to an effective interaction, $V_{\mathrm{EC}}\left(r\right)\sim \pm q^2/r^s$, between charges $\pm q$ separated by a distance r, where the case $b=0$ corresponds to the standard Coulomb law. The effects of a finite imaginary component, $b>0$, were studied by calculating the cohesive energy of a 1D lattice, $E_{\mathrm{T}}(a_0,b)\sim \Re \left[\eta \left(s\right)\right]$, from which a new lattice constant, $a_0\left(b\right)$, is obtained as a function of b. For small values of b, the lattice shrinks, i.e., $a_0\left(b\right)<a_0\left(0\right)$, and its cohesive energy decreases, $E_{\mathrm{T}}(a_0,b)<E_{\mathrm{T}}(a_0,0)$. Both the lattice constant and the cohesive energy reach a minimum at $b\simeq 5.58$ and then start increasing again for larger values of b. Correspondingly, the lattice stiffness, represented by the second derivative ${\partial }^2{E}_{\mathrm{T}}/\partial a^{\prime 2}{|}_{a_0}$, reaches a maximum at $b\simeq 5.58$ and begins to decrease for $b>5.58$. Finally, the lattice becomes unstable when $\eta (1+i{b}_u)=0$, with $b_u=2\pi /log2\simeq 9.065$.

We argue that the imaginary component $b>0$ could result from effective many-body interactions between localized electrons due to their coupling with phonons and vacuum energy fluctuations. Further theoretical work is needed to quantitatively validate these ideas by comparing the model with experimental observations in various physical contexts. Specifically, applications to real three-dimensional structures require the use of generalized forms of the $\eta \left(s\right)$ function to deal with more elaborated lattice sums (see, e.g., [16,26]).

Finally, the connection between the Riemann zeta function and prime numbers has also been explored in light of the ongoing interest in the still unproven Riemann hypothesis. Numerical calculations on the prime numbers gaps distribution, as well as an evaluation of their correlations using a fluctuation analysis method, are discussed. The former follows an exponentially decaying distribution function, which is derived numerically from the first two million primes. The fluctuation analysis reveals that hidden correlations exist between gaps, which tend to disappear as the number of primes increases. This gradual approach to the asymptotic behavior is typical of other quantities, such as the prime number value as a function of the number of primes considered. Finally, we demonstrate that the non-trivial zeros of the Riemann zeta function, $\rho =a+ib$, exhibit distinct behaviors depending on whether the real part satisfies $a=1/2$ or $a\ne 1/2$. This difference is significant and may suggest that actual zeros can exist only when $a=1/2$.