Nontrivial Zeros Research Articles

Non-trivial zeros of riemann zeta function and riemann hypothesis

Non-trivial zeros of riemann zeta function and riemann hypothesis

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function , Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended to the complex plane z and conjectured that all nontrivial zeros are in the axis. The nonadditive entropy , where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function . It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as , which recover the number x for . The q-prime numbers are then defined as the q-natural numbers , where n is a prime number We show that, for any value of q, infinitely many q-prime numbers exist; for they diverge for increasing prime number, whereas they converge for ; the standard prime numbers are recovered for . For , we generalize the function as follows: (). We show that this function appears to diverge at , . Also, we alternatively define, for , and , which, for , generically satisfy , in variance with the case, where of course .

The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős–Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive ‘experimental’ confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.

Approximation of Analytic Functions by Shifts of Certain Compositions

In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function.

Solubility of additive sextic forms over $\mathbb{Q}_2(\sqrt{-1})$ and $\mathbb{Q}_2(\sqrt{-5})$

Michael Knapp, in a previous work, conjectured that every additive sextic form over $\mathbb{Q}_2(\sqrt{-1})$ and $\mathbb{Q}_2(\sqrt{-5})$ in seven variables has a nontrivial zero. In this paper, we show that this conjecture is true, establishing that $\Gamma^*(6, \mathbb{Q}_2(\sqrt{-1})) = \Gamma^*(6, \mathbb{Q}_2(\sqrt{-5})) = 7 $.

The zeros of the Lerch zeta-function are uniformly distributed modulo one

UDC 511.311We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one.

Joint discrete universality for periodic zeta-functions. IV

In the paper, a theorem on the approximation of collections of a wide class of analytic functions by collections of shifts of zeta-functions with periodic co-efficients involving imaginary parts of non-trivial zeros of the Riemann zeta-function is obtained. For this, a version of the Montgomery pair correlation conjecture is required.

Generation of off-critical zeros for hypercubic Epstein zeta functions

We study the Epstein zeta-function formulated on the d-dimensional hypercubic lattice ζ(d)(s)=12∑′n1,…,nd(n12+…+nd2)−s/2 where the real part ℜ(s)>d and the summation runs over all integers except of the origin (0,0,…,0). An analytical continuation of the Epstein zeta-function to the whole complex s-plane is constructed for the spatial dimension d being a continuous variable ranging from 0 to ∞. Zeros of the Epstein zeta-function ρ=ρx+iρy are defined by ζ(d)(ρ)=0. The nontrivial zeros split into the “critical” zeros (on the critical line) with ρx=d2 and the “off-critical” zeros (off the critical line) with ρx≠d2. Numerical calculations reveal that the critical zeros form closed or semi-open curves ρy(d) which enclose disjunctive regions of the plane (ρx=d2,ρy). Each curve involves a number of left/right edge points ρ*=(d*2,ρy*), defined by a divergent tangent dρy/dd|ρ*. Every edge point gives rise to two conjugate tails of off-critical zeros with continuously varying dimension d which exhibit a singular expansion around the edge point, in analogy with critical phenomena for second-order phase transitions. For each dimension d>9.24555… there exists a conjugate pair of real off-critical zeros which tend to the boundaries 0 and d of the critical strip in the limit d→∞. As a by-product of the formalism, we derive an exact formula for limd→0ζ(d)(s)/d. An equidistant distribution of critical zeros along the imaginary axis is obtained for large d, with spacing between the nearest-neighbour zeros vanishing as 2π/lnd in the limit d→∞.

More exact values of the function Γ⁎(k)

Consider a homogeneous additive polynomial of the form F(x)=a1x1k+a2x2k+⋯+asxsk, where the coefficients are integers. The function Γ⁎(k) is defined as the smallest number s of variables such that F is guaranteed to have a nontrivial zero in every p-adic field Qp regardless of the coefficients. In this article, we calculate the value of Γ⁎(k) for all 33≤k≤64 for which this value is not already known.

Counting zeros of the Riemann zeta function

In this article, we show that|N(T)−T2πlog⁡(T2πe)|≤0.1038log⁡T+0.2573log⁡log⁡T+9.3675 where N(T) denotes the number of non-trivial zeros ρ, with 0<Im(ρ)≤T, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large T. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett et al. on counting zeros of Dirichlet L-functions.

Counting zeros of Dedekind zeta functions

Given a number field K K of degree n K n_K and with absolute discriminant d K d_K , we obtain an explicit bound for the number N K ( T ) N_K(T) of non-trivial zeros (counted with multiplicity), with height at most T T , of the Dedekind zeta function ζ K ( s ) \zeta _K(s) of K K . More precisely, we show that for T ≥ 1 T \geq 1 , | N K ( T ) − T π log ⁡ ( d K ( T 2 π e ) n K ) | ≤ 0.228 ( log ⁡ d K + n K log ⁡ T ) + 23.108 n K + 4.520 , \begin{equation*} \Big | N_K (T) - \frac {T}{\pi } \log \Big ( d_K \Big ( \frac {T}{2\pi e}\Big )^{n_K}\Big )\Big | \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, \end{equation*} which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet L L -functions.

Riemann zeros from Floquet engineering a trapped-ion qubit

The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in random matrix theory and quantum chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped-ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.

Prime zeta function statistics and Riemann zero-difference repulsion

We present a derivation of the numerical phenomenon in which differences between the Riemann zeta function’s nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ‘repulsion’ between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function’s magnitude on the one-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.

Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function. II

Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function. II

Accurate estimation of sums over zeros of the Riemann zeta-function

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.

Multidimensional scaling and visualization of patterns in distribution of nontrivial zeros of the zeta-function

In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then we employ a variety of different metrics, such as the Jeffreys and Lorentzian ones, to calculate the distances between the objects. The set of the calculated distances is then processed by the MDS algorithm that produces the loci, organized according to the objects features. Then they are analyzed from the perspective of the emerging patterns. Surprisingly, in the case of the Lorentzian metric, this procedure leads to the very clear periodical structures both in the case of the objects in form of the single nontrivial zeros of the Riemann zeta-function and in the case of the vectors with a given number of neighboring zeros. The other tested metrics do not produce such periodical structures, but rather chaotic ones. In this paper, we restrict ourselves to numerical experiments and the visualization of the produced results. An analytical explanation of the obtained periodical structures is an open problem worth for investigation by the experts in the analytical number theory.

Continuous crop circles drawn by Riemann's zeta function

Let η(s)=∑n=1∞(−1)n+1n−s be the alternating zeta function. For a real number τ we define certain complex numbers bM,m(τ) and consider finite Dirichlet series υM(τ,s)=∑m=1MbM,m(τ)m−s and ηN(τ,s)=∑M=1NυM(τ,s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far.First, numerical data show that ηN(τ,s) approximates η(s) with high accuracy for s in the vicinity of 1/2+iτ; this allows one to surmise that(*)η(s)=∑M=1∞υM(τ,s). Moreover, it looks thatlimM→∞⁡mυM(τ,1−σ+it)‾mυM(τ,σ+it)=η(σ+it)mη(1−σ+it)‾m; in other words, the individual summands in expected expansion (⁎) satisfy with an increasing accuracy a counterpart of the classical functional equation.Let ϒM(τ,σ+it)=υM(τ,σ+it)/η(σ+it). When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒM(τ,σ+it) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

On the logarithm of the Riemann zeta-function near the nontrivial zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|\zeta(\rho+z)|)$ and $(\arg\zeta(\rho+z)).$ Here $\rho=\frac12+i\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\gamma \leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|\zeta^\prime(\rho)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.