Zeta Zeros Research Articles

Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field

For a function field k over a finite field with Fq as the field of constants, and a finite abelian group G whose exponent divides q−1, we study the distribution of zeta zeroes for a random G-extension of k, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.

Riemann zeta zeros and zero-point energy

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

A CENTRAL LIMIT THEOREM FOR THE ZEROES OF THE ZETA FUNCTION

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Spohn and Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.

On the exact location of the non-trivial zeros of Riemann's zeta function

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.

Ergodicity of the Action of K* on

Connes gave a spectral interpretation of the critical zeros of zeta- and L-functions for a global field K using a space of square integrable functions on the space A_K/K* of adele classes. It is known that for K=Q the space A_K/K* cannot be understood classically, or in other words, the action of Q* on A_Q is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.

On the zeros of Epstein zeta functions

Abstract We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

О законе Грама в теории дзета-функции Римана

Some statements concerning the distribution of imaginary parts of zeros of the Riemann zeta\,-function are established. These assertions are connected with so\,-called `Gram's law' or `Gram's rule'. In particular, we give a proof of several Selberg's formulae stated him without proof in his paper `The Zeta Function and the Riemann Hypothesis' (1946), and some of their equivalents.

Distribution of zeta zeroes of Artin–Schreier covers

We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier curves over Fq when q is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed interval of [−π, π) has a standard Gaussian distribution (when properly normalized).

Generalizing Riemann: from the L -functions to the Birch/Swinnerton-Dyer conjecture

To prove: that the generalized Riemann hypothesis is true, namely, that all the zeta zeros of L -functions are balanced on the one-half number line, using the scale of the harmonic zeta function. Now that the Riemann hypothesis has been proven (in “Summa Characteristica and the Riemann Hypothesis: Scaling Riemann’s Mountain”, Journal of Interdisciplinary Mathematics, Vol. 11 (6) (December 2008)), it can be extended by analytical continuation to prove the generalized hypothesis concerning L-functions. The initial “gene” or generating (one half) principle that proves the Riemann hypothesis also proves that no L-series has a zero with real part on the complex plane other than one half. As a result of this proof, the Birch/Swinnerton-Dyer conjecture is also resolved. Finally, what impact, if any, do these proofs have on security codes based upon prime numbers?

ON THE RIEMANN HYPOTHESIS, AREA QUANTIZATION, DIRAC OPERATORS, MODULARITY, AND RENORMALIZATION GROUP

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = ½ + iρn. A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation [Formula: see text], where the functional form of [Formula: see text] is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the [Formula: see text] terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large [Formula: see text] has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

The first-digit frequencies of prime numbers and Riemann zeta zeros

Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, we show that a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digitdin datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.

Summa characteristica and the Riemann hypothesis: scaling Riemann’s mountain

To prove: Riemann’s hypothesis that all zeta zeros have real part one half. Premise one : the (characteristic) harmonic triangle of Pascal is characterized by the harmonic series. Premise two: when s equals one in the zeta function, the function is identical with the harmonic series. Three : when s equals any complex number, the zeta function is the same as the complex number plane. Four: since the harmonic pattern of the series in the triangle of Pascal completely carries over into the zeta function, and the zeta function charts the complete complex plane (premises one through three combined), whatever places restrictions on the Pascal harmonic pattern also places constraints on the number pattern of the complex plane. Five : the defining (characteristic) restriction responsible for the generation of the whole triangle-plane is the number of both the binomial theorem and the geometrical expansion, which is one half. Because this fraction-interval is recursively present throughout the entire harmonic series, it ties all of the zeta zeros to the real line one half. By recursion we can follow one half as the real part of the zeta zeros forward throughout the entire harmonic series and onto the complex plane. Conclusion: all zeta zeros have real part one half. Since the premises are all true, the conclusion must also be true. Therefore, the Riemann hypothesis is correct. Nature is the realization of the simplest conceivable mathematical ideas. A physicist’s greatest challenge is to uncover the fundamental and universal laws from which a process of simple deduction can lead to a picture of the world. [.] This is what Leibniz felicitously called a ‘re-established’ harmony. Albert Einstein, 1918

Quantum mechanical potentials related to the prime numbers and Riemann zeros

Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions. Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the Riemann zeta(s) function. According to the Hilbert-Pólya conjecture, there exists a Hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of zeta(s) . This idea has encouraged physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marchenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the zeta(s) function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress.

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

For any function D(x) that is real for real x, positivity of Tuck's function Q(x) ≡ D′2(x)/(D′2(x) − D(x)D″(x)) is a condition for the absence of complex zeros close to the real axis. Study of the probability distribution PN(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. PN(Q) has singularities at Q = 0, Q = 1 and Q = N. The moments (averages of Qm) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of PN(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q = 1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of PN(Q) for the GUE seems difficult.

ON STRATEGIES TOWARDS THE RIEMANN HYPOTHESIS: FRACTAL SUPERSYMMETRIC QM AND A TRACE FORMULA

The Riemann hypothesis (RH) states that the non-trivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert–Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu–Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions ∑p W(x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu–Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss–Jacobi theta series. An explicit completion relation ("trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied, this could be another test of the Riemann Hypothesis. In an appendix, we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of [Formula: see text]-invariant Quantum Mechanics, because [Formula: see text] where s are the complex eigenvalues of the scaling-like operators. We show why [Formula: see text] invariance requires that s(1 - s) = real , which implies that s is real and/or it lies in the critical Riemann line.

The distribution of zeros of Epstein zeta functions over GLn

The distribution of zeros of Epstein zeta functions over GL_n

ON THE RIEMANN HYPOTHESIS AND TACHYONS IN DUAL STRING SCATTERING AMPLITUDES

It is the purpose of this work to pursue a novel physical interpretation of the nontrivial Riemann zeta zeros and prove why the location of these zeros zn = 1/2+iλn corresponds physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Realz = 1/2 (but inside the critical strip), these putative zeros do not correspond to any poles of the bosonic open string scattering (Veneziano) amplitude A(s, t, u). The physical relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes an important connection between string theory and the Riemann Hypothesis. In addition, one has also a geometrical interpretation of the zeta zeros in the critical line in terms of very special (degenerate) triangular configurations in the upper-part of the complex plane.

A New Unconditional Result about Large Spaces Between Zeta Zeros

A New Unconditional Result about Large Spaces Between Zeta Zeros

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

Each simple zero 1 2 + i γ n \frac {1}{2}+i\gamma _n of the Riemann zeta function on the critical line with γ n > 0 \gamma _n > 0 is a center for the flow s ˙ = ξ ( s ) \dot {s}=\xi (s) of the Riemann xi function with an associated period T n T_n . It is shown that, as γ n → ∞ \gamma _n \rightarrow \infty , \[ log ⁡ T n ≥ π 4 γ n + O ( log ⁡ γ n ) . \log T_n\ge \frac {\pi }{4}\gamma _n+O(\log \gamma _n). \] Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γ n + 1 − γ n ≫ γ n − θ \gamma _{n+1}-\gamma _n \gg \gamma _n^{-\theta } for some exponent θ > 0 \theta >0 , we obtain the upper bound log ⁡ T n ≪ γ n 2 + θ \log T_n \ll \gamma ^{2+\theta }_n . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log ⁡ T n = π 4 γ n + O ( log ⁡ γ n ) \log T_n = \frac {\pi }{4}\gamma _n+O(\log \gamma _n) . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

Zero spacing distributions for differenced L-functions

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 2i (ξ(s + h) − ξ(s − h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line ℜ(s) = 1 and are simple zeros. The number of zeros of these functions to height T has asymptotically the same density as the Riemann zeta zeros. For fixed |h| ≥ 1 the distribution of normalized zero spacings of these functions up to height T converge as T → ∞ to a limiting distribution, which consists of equal spacings of size 1. That is, these zeros are asymptotically regularly spaced. Assuming the Riemann hypothesis, the same properties hold for all nonzero h. In particular, these averaging and differencing operations destroy the (conjectured) GUE distribution of the zeros of the ξ-function, which should hold at h = 0. Analogous results hold for all completed Dirichlet L-functions ξχ(s) having χ a primitive character.