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We construct the partition function of small spin ½ systems defined on Ci, (Ci is a circle where are sitting the 2N spin variables. The spins interact with two body ferromagnetic interactions and are immersed in a magnetic field a one body interaction) and compare them with a polynomial truncation of the Riemann ξ function, the idea being to use some rigorous results on the zeros of the spin systems to obtain some information about the possible \"non negativity\" property of the Li-Keiper coefficients. The analysis is carried out up to 2N = 12 spin systems and a numerical experiment is performed up to 2N = 56. (For simplicity, only for the case where all the two body interaction strengths are equal is considered, i.e., the case where X(i, j)= X= e(-2K) for all (i, j); the magnetic field variable is given by z = e(-2h)). The zeros of the truncated ξ function are then on the unit circle but the associated values of the first few Li-Keiper coefficients so obtained holds only for a maximum value of N. A non zero lower bound to the values of the coefficients in the form of a conjecture is also presented. Possible developments are also indicated.

The aim of this work is to describe a new formula relating the nontrivial zeros of the Riemann Zeta function to the energy levels of the harmonic oscillator, which we call the \"Riemann wave,\" whose nodes are located at the height of the non-trivial zeros (on the Riemann Hypothesis, RH). We illustrate the formula by means of various Figures, and we present a calculation up to relatively \"high\" heights. Then, we propose formally an \"operator\" in agreement to the Polya's idea, which involves here the Lambert W function. We call it the \"Quantum Riemann Wave.\" Some approximations of the implicit equation for this operator, as well as a special interesting approximation (with the use of the Montgomery bound on the fluctuations S(t) and an additional factor), are also discussed and illustrated with some numerical experiments and Figures.

We introduce a kind of \"perturbation\" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the LiKeiper coefficients. We then check the correctness of some of the many possible solutions offered by the system, related to the discrete derivative of order n of a function. We also report a numerical finding which support our stability conjecture, i.e., that the tiny part λtiny(n) (the fluctuations around the trend) are bounded in absolute values by γ.n, where γ is the Euler-Mascheroni constant.

We develop an expression for the Li-Keiper coefficients λn in terms of k-blocks partitions, to begin with, for low values of n. The k-blocks partitions are given in terms of our cluster functions φn and the main point of this work lies in the emergence of an alternating sequence of values converging toward values of λn near the true values, i.e., increasing the index k of the blocks one obtains an increasing range of positivity of the Li-Keiper coefficients. With the contribution of k = 1 and k = 2 blocks, positivity of the λn is reached already until n = 26-27. The treatment is given here until k = 4 blocks up to n = 30. λn are all found to be positive.

In connection with the binomial transform, we establish a concrete asymptotic expression for the (trend, tiny part and complete) Li-Keiper coefficients in terms of elementary functions. The tiny fluctuations are computed up to n=5000 and a careful numerical analysis allows the derivation of an asymptotic formula using the extreme values of such fluctuations which is correct up to n=100000. Such an asymptotic expression - do not assume - but it is in agreement with the assumption of the truth of the Riemann Hypothesis.

Starting with a rigorous integral Equality, i.e., an integral over the logarithm of the modulus of the zeta function (an Equivalent of the Riemann Zeta Function having no zeros for Re(s) > 0.9) with a density (reciprocal of an hyperbolic function of parameter a, at Re(s) = 0.9), we define the Primitive Riemann Wave and compute some of its zeros (nodes). Alternatively, we study such a sequence of zeros by means of the Gauss -Lucas Theorem: attractive as well as repulsive points of the associated map are given up to t = 50, where t is the height in s = 0.9+i.t. We also report some numerical results for the case of vanishing parameter a, i.e., a→0 in the reciprocal hyperbolic measure. The numerical results suggest that the Primitive Riemann Wave has zeros extending to infinity i.e., that its convex hull contain in fact, the infinity of zeros of the Integrand, i.e., that no symmetry breaking occurs, supporting that the node of the Primitive Riemann Wave extends to infinity (with a vanishing amplitude if a > 0).

In the first part of this work (realized as a comment), we add the plots of two more values of the Li-Keiper coefficients λ5 and λ6, computed as in our recent work where the first four values were in particular given. This for the trend as well as for the oscillating path (\"tiny,\" the term coined by Maslanka in his pioneering work). Then, in the second part looking at the tiny oscillations, we propose a \"numerical conjecture\" in a more strong form, i.e., with a logarithmic behaviour and carry out a short numerical experiment on the new \"numerical conjecture.\"

In this note, we consider a constant c related to the Glaisher-Kinkelin constant, obtained by a shift of the infinite system of Equations for the Li-Keiper coefficients and related to the Euler formula at s = 2, i.e. ζ(2) = π2/6 for the Zeta function. The aim of the experiment we are carrying out is to relate the Primes with the tiny part (the fluctuations) of the Li-Keiper coefficients as well as with the trend part of them. The contribution of the series for the tiny part or for the trend part are thus related to a series for the primes and we obtain in this ways a upper and a lower bound of the constant c, which will be given with some decimals. Other relations may be obtained along the same lines and additional examples are given.

Integral equalities involving integrals of the logarithm of the Riemann ς -function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp( − 2 π t ). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.

Integral equalities involving integrals of the logarithm of the Riemann theta -function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp( -2 pi t ). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.