arXiv subjects

André LeClair

Publications and source records attributed to André LeClair.

At least 37 records · Page 2Linked to original sources

arXiv2017

Some Riemann Hypotheses from Random Walks over Primes

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is $O(\sqrt{N})$, then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this $\sqrt{N}$ growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results we obtain an equation which relates every individual non-trivial zero of the $L$-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet $L$-functions due to the existence of the pole at $s=1$, in which the Riemann $ζ$-function is a particular case.

math.NT↗

arXiv2017

On large gaps between zeros of $L$-functions from branches

It is commonly believed that the normalized gaps between consecutive ordinates $t_n$ of the zeros of the Riemann zeta function on the critical line can be arbitrarily large. In particular, drawing on analogies with random matrix theory, it has been conjectured that $$λ' ={ lim ~ sup} ~( t_{n+1} - t_n ) \frac{ \log( t_n /2 πe)}{2π}$$ equals $\infty$. In this article we provide arguments, although not a rigorous proof, that $λ'$ is finite. Conditional on the Riemann Hypothesis, we show that if there are no changes of branch between consecutive zeros then $λ' \leq 3$, otherwise $λ'$ is allowed to be greater than $3$. Additional arguments lead us to propose $λ'\leq 5$. This proposal is consistent with numerous calculations that place lower bounds on $λ'$. We present the generalization of this result to all Dirichlet $L$-functions and those based on cusp forms.

math.NT↗

arXiv2016

Central Limit Theorems for series of Dirichlet characters

For a given Dirichlet character $χ(n) = e^{i θ_n}$, we prove central limit theorems for the series $\sum_{p'} \cos θ_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$ are integers based on a variant of Cramér's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\Re (s) = t \to \infty$.

math.NT↗

arXiv2015

An asymptotic upper bound on prime gaps

The Cramér-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n < \sum_{n=1}^N \log^2 p_n$. In the remainder of the paper we investigate which properties of the fluctuations $\fluc (x) = π(x) - \Li(x)$ would imply the Cramér-Granville conjecture is true and present two such properties, one of which assumes the Riemann Hypothesis; however we are unable to prove these properties are indeed satisfied. We argue that the conjecture is related to the enormity of the Skewes number.

math.NT↗

arXiv2015

On the validity of the Euler product inside the critical strip

The Euler product formula relates Dirichlet $L(s,χ)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re (s) > 1/2$ in a specific sense. Namely, the logarithm of the Euler product, although formally divergent, is meaningful because it is Cesàro summable, and its Cesàro average converges to $\log L (s,χ)$. Our argument relies on the prime number theorem, an Abel transform, and a central limit theorem for the Random Walk of the Primes, the series $\sum_{n=1}^N \cos\left(t\log p_n\right)$, and its generalization to other Dirichlet $L$-functions. The significance of ${\Re(s) > 1/2}$ arises from the $\sqrt{N}$ growth of this series, since it satisfies a central limit theorem. $L$-functions based on principal Dirichlet characters, such as the Riemann $ζ$-function, are exceptional due to the pole at $s=1$, and require $\Im (s) \neq 0$ and a truncation of the Euler product. Compelling numerical evidence of this surprising result is presented, and some of its consequences are discussed.

math.NT↗

arXiv2015

Transcendental equations satisfied by the individual zeros of Riemann $ζ$, Dirichlet and modular $L$-functions

We consider the non-trivial zeros of the Riemann $ζ$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $ζ$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function.

math.NT↗

arXiv2014

Statistical and other properties of Riemann zeros based on an explicit equation for the $n$-th zero on the critical line

We show that there are an infinite number of Riemann zeros on the critical line, enumerated by the positive integers $n=1,2,\dotsc$, whose ordinates can be obtained as the solution of a new transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of such zeros already saturates the counting formula for the numbers of zeros on the entire critical strip. These results thus constitute a concrete proposal toward verifying the Riemann hypothesis. We perform numerical analyses of the exact equation, and its asymptotic limit of large ordinate. The starting point is an explicit analytical formula for an approximate solution to the exact equation in terms of the Lambert $W$ function. In this way, we neither have to use Gram points or deal with violations of Gram's law. Our numerical approach thus constitutes a novel method to compute the zeros. Employing these numerical solutions, we verify that solutions of the asymptotic version are accurate enough to confirm Montgomery's and Odlyzko's pair correlation conjectures and also to reconstruct the prime number counting function.

math.NT↗

arXiv2014

On the zeros of L-functions

We generalize our recent construction of the zeros of the Riemann $ζ$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an infinite number of zeros on the critical line which are in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. We obtain an exact equation on the critical line that determines the $n$-th zero of these $L$-functions. We show that the counting formula on the critical line derived from such an equation agrees with the known counting formula on the entire critical strip. We provide numerical evidence supporting our statements, by computing numerical solutions of this equation, yielding $L$-zeros to high accuracy. We study in detail the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function. The same analysis for a more general class of $L$-functions is also considered.

math.NT↗

arXiv2013

An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N

We construct a vector field E from the real and imaginary parts of an entire function xi (z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Gamma and Riemann zeta functions and is known to satisfy the functional identity xi (z) = xi (1-z). E satisfies the conditions for a static electric field. The structure of E in the critical strip is determined by its behavior near the Riemann zeros on the critical line Re (z) = 1/2, where each zero can be assigned a + or - vorticity of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of this "electric" field. We formulate our argument more precisely in terms of the potential Phi satisfying E = - gradient Phi, and construct Phi explicitly. One outcome of our analysis is a formula for the n-th zero on the critical line for large n expressed as the solution of a simple transcendental equation. Riemann's counting formula for the number of zeros on the entire critical strip can be derived from this formula. Our result is much stronger than Riemann's counting formula, since it provides an estimate of the n-th zero along the critical line. This provides a simple way to estimate very high zeros to very good accuracy, and we estimate the 10^{10^6}-th one.

math-ph↗

arXiv2013

Scrutinizing the Cosmological Constant Problem and a possible resolution

We suggest a new perspective on the Cosmological Constant Problem by scrutinizing its standard formulation. In classical and quantum mechanics without gravity, there is no definition of the zero point of energy. Furthermore, the Casimir effect only measures how the vacuum energy {\it changes} as one varies a geometric modulus. This leads us to propose that the physical vacuum energy in a Friedman-Lemaître-Robertson-Walker expanding universe only depends on the time variation of the scale factor $a(t)$. Equivalently, requiring that empty Minkowski space is gravitationally stable is a principle that fixes the ambiguity in the zero point energy. On the other hand, if there is a meaningful bare cosmological constant, this prescription should be viewed as a fine-tuning. We describe two different choices of vacuum, one of which is consistent with the current universe consisting only of matter and vacuum energy. The resulting vacuum energy density $\rhovac$ is constant in time and approximately $k_c^2 H_0^2$, where $k_c$ is a momentum cut-off and $H_0$ is the current Hubble constant; for a cut-off close to the Planck scale, values of $\rhovac$ in agreement with astrophysical measurements are obtained. Another choice of vacuum is more relevant to the early universe consisting of only radiation and vacuum energy, and we suggest it as a possible model of inflation.

hep-th↗

arXiv2012

Possible Cooper instabilities in pair Green functions of the two-dimensional Hubbard model

In analogy with ordinary BCS superconductivity, we identify possible pairing instabilities with poles in a certain class of Green functions of the two-dimensional repulsive Hubbard model, which normally signify bound states. Relating the gap to the location of these poles, we find that these instabilities exist in the range of hole doping between 0.03 and 0.24. The magnitude of the gap can be calculated without introducing an explicit cut-off, and for reasonably large coupling U/t = 10, the maximum gap is on the order of 0.08 in units of the hopping parameter t.

cond-mat.str-el↗

arXiv2012

3D Ising and other models from symplectic fermions

We study a model of N component symplectic fermions in D spacetime dimensions. It has an infra-red stable fixed point in 2<D<4 dimensions referred to as Sp{2N}{D}. Based on the comparison of exponents, we conjecture that the critical exponents for the 3D Wilson-Fisher fixed point for an O(N) invariant N-component bosonic field can be computed in the Sp{-2N}{3} theory. The 3D Ising model corresponds to Sp{-2}{3}. The exponent beta agrees with known results to 1 part in 1000 and is within current error bars. The nu exponent agrees to 1% and we suggest this because we only went to 1-loop for this exponent.

cond-mat.stat-mech↗

arXiv2012

Edge states for topological insulators in two dimensions and their Luttinger-like liquids

Topological insulators in three spatial dimensions are known to possess a precise bulk/boundary correspondence, in that there is a one-to-one correspondence between the 5 classes characterized by bulk topological invariants and Dirac hamiltonians on the boundary with symmetry protected zero modes. This holographic characterization of topological insulators is studied in two dimensions. Dirac hamiltonians on the one dimensional edge are classified according to the discrete symmetries of time-reversal, particle-hole, and chirality, extending a previous classification in two dimensions. We find 17 inequivalent classes, of which 11 have protected zero modes. Although bulk topological invariants are thus far known for only 5 of these classes, we conjecture that the additional 6 describe edge states of new classes of topological insulators. The effects of interactions in two dimensions are also studied. We show that all interactions that preserve the symmetries are exactly marginal, i.e. preserve the gaplessness. This leads to a description of the distinct variations of Luttinger liquids that can be realized on the edge.

cond-mat.str-el↗

arXiv2012

Holographic classification of Topological Insulators and its 8-fold periodicity

Using generic properties of Clifford algebras in any spatial dimension, we explicitly classify Dirac hamiltonians with zero modes protected by the discrete symmetries of time-reversal, particle-hole symmetry, and chirality. Assuming the boundary states of topological insulators are Dirac fermions, we thereby holographically reproduce the Periodic Table of topological insulators found by Kitaev and Ryu. et. al, without using topological invariants nor K-theory. In addition we find candidate Z_2 topological insulators in classes AI, AII in dimensions 0,4 mod 8 and in classes C, D in dimensions 2,6 mod 8.

cond-mat.str-el↗

arXiv2011

On the viscosity to entropy density ratio for unitary Bose and Fermi Gases

We calculate the ratio of the viscosity to the entropy density for both Bose and Fermi gases in the unitary limit using a new approach to the quantum statistical mechanics of gases based on the S-matrix. In the unitary limit the scattering length diverges and the S-matrix equals -1. For the fermion case we obtain eta/s > 4.7 times the proposed lower bound of \hbar/4 pi k_B which came from the AdS/CFT for gauge theories, consistent with the most recent experiments. For the bosonic case we present evidence that the gas undergoes a phase transition to a strongly interacting Bose-Einstein condensate, and is a more perfect fluid, with eta/s > 1.3 times the bound.

cond-mat.quant-gas↗

arXiv2010

Superconductivity in the two-dimensional Hubbard model based on the exact pair potential

We analyze solutions to a superconducting gap equation based on the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping. The Cooper pair potential can be calculated exactly and expressed in terms of elliptic functions. The Fermi surfaces at finite temperature and chemical potential are calculated based on the exact two-body S-matrix of the Hubbard model using the formalism we recently developed, which allows variation of hole doping. The resulting solutions to the gap equation are strongly anisotropic, namely largest in the anti-nodal direction, and zero in the nodal directions of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate to BSCO, and a physically natural choice for the cut-off, our self-contained analytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06 and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits to the Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 at optimal doping, both in good agreement with experiments. The superconducting gap is non-zero for all hole-doping h < 0.35, and increases all the way down to zero doping, suggesting that it evolves into the pseudogap.

cond-mat.supr-con↗

arXiv2010

S-matrix approach to quantum gases in the unitary limit II: the three-dimensional case

A new analytic treatment of three-dimensional homogeneous Bose and Fermi gases in the unitary limit of negative infinite scattering length is presented, based on the S-matrix approach to statistical mechanics we recently developed. The unitary limit occurs at a fixed point of the renormalization group with dynamical exponent z=2 where the S-matrix equals -1. For fermions we find T_c /T_F is approximately 0.1. For bosons we present evidence that the gas does not collapse, but rather has a critical point that is a strongly interacting form of Bose-Einstein condensation. This bosonic critical point occurs at n lambda^3 approximately 1.3 where n is the density and lambda the thermal wavelength, which is lower than the ideal gas value of 2.61.

cond-mat.quant-gas↗

arXiv2009

Non-Fermi liquid properties of 2d symplectic fermions: the role of a dynamically generated (pseudo)-gap

The interacting symplectic fermion model in two spatial dimensions is further analyzed. As an effective low energy theory, the model is unitary. We show that a relativistic mass m is dynamically generated and derive a gap equation for it. By incorporating a finite temperature we study some fundamental properties of the model, such as the specific heat and spin response, which clearly show non-Fermi liquid properties. We find that various physical properties are suppressed at temperatures T< T* where the cross-over scale is T* = m. As a simplified, toy model of high Tc superconductivity, we thus identify the pseudogap energy scale with the zero temperature relativistic mass m, and show that this reproduces some qualitative aspects of the observed phenomenology of the pseudogap. The effects of the pseudogap and finite temperature on the d-wave gap equation are analyzed. In this model, the pseudogap is a distinct phenomenon from superconductivity and in fact competes with it. Our analysis of Tc suggests that the quantum critical point of our model, where the pseudogap vanishes, occurs inside the superconducting dome near optimal doping. For an antiferromagnetic exchange energy of J/k_B ~ 1350K, solutions of the d-wave gap equation give a maximum Tc of about 110K.

cond-mat.str-el↗