International journal of physics research and applications, Jun 5, 2023
How to cite this article: Merlini D, Rusconi L, Sala M, Sala N. Spin ½ model in statistical mecha... more How to cite this article: Merlini D, Rusconi L, Sala M, Sala N. Spin ½ model in statistical mechanics and relation to a truncation of the Riemann ξ function in the Riemann Hypothesis.
Using the first discrete derivatives for the expansion in z=0 of the oscillating part λ tiny (n) ... more Using the first discrete derivatives for the expansion in z=0 of the oscillating part λ tiny (n) =λ n * of the "tiny" Li-Keiper coefficients , we analyse two series in the variable z=1-1/s ~0 for the first low values and compare them with the exact series. The numerical results suggest interesting more "sophisticated" approximations.
Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first sp... more Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first specify a lower bound emerging from the infinite set and give a characterization of it. Then, we propose a possible new upper and lower bound for the coefficients in few of the partitions occurring in the cluster functions furnishing in a nonlinear way the coefficients. A numerical experiment up to n=15 confirms the proposed bounds and an experiment, i.e. the counting of the zeros in the binary representation of an integer for a constant related to the Glaisher-Kinkelin constant is also given up to n=32.
In the first part of this short work (in the form of a comment) we add the plots of two more valu... more In the first part of this short work (in the form of 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".
Our purpose is to describe some recent progress in applying fractal concepts to systems of releva... more Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material, Furthermore, we discuss the finding that the exponent a quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.
We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe f... more 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 Li-Keiper 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 numerical finding which support our stability conjecture that the tiny part lambda-tiny(n) (the fluctuations around the trend) are bounded in absolute values by gammaxn, where gamma is the Euler-Mascheroni constant.
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functi... more Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented.
We present a calculation involving a function related to the Riemann Zeta function and suggested ... more We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi, while that at r = 0 is given by (-4 + gamma + 3*log(2) + Pi/2)*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m(r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1...
The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypoth... more The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true.
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functi... more Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented.
We present a calculation involving a function related to the Riemann Zeta function and suggested ... more We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for
New expansions for some functions related to the Zeta function in terms of the Pochhammer's polyn... more New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coecients b k , d k ,d k andd k). In some formal limit our expansion b k obtained via the alternating series gives the regularized expansion of Maslanka for the Zeta function. The real and the imaginary part of the function on the critical line is obtained with a good accuracy up to I(s) = t < 35. Then, we give the expansion (coecientd k) for the derivative of ln((s − 1)ζ(s)). The critical function of the derivative, whose bounded values for R(s) > 1 2 at large values of k should ensure the truth of the Riemann Hypothesis (RH), is obtained either by means of the primes or by means of the zeros (trivial and non-trivial) of the Zeta function. In a numerical experiment performed up to high values of k i.e. up to k = 10 13 we obtain a very good agreement between the two functions, with the emergence of twelve oscillations with stable amplitude. For a special case of values of the two parameters entering in the general Pochhammer's expansion it is argued that the bound on the critical function should be given by the Euler constant gamma.
[math.NT], where a number of integral equalities involving integrals of the logarithm of the Riem... more [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented. Our integral equalities, together with the equalities given in the previous paper, include all earlier known criteria of this kind, viz. Wang, Volchkov and Balazard-Saias-Yor criteria, which are certain particular cases of the general approach proposed.
International journal of physics research and applications, Jun 5, 2023
How to cite this article: Merlini D, Rusconi L, Sala M, Sala N. Spin ½ model in statistical mecha... more How to cite this article: Merlini D, Rusconi L, Sala M, Sala N. Spin ½ model in statistical mechanics and relation to a truncation of the Riemann ξ function in the Riemann Hypothesis.
Using the first discrete derivatives for the expansion in z=0 of the oscillating part λ tiny (n) ... more Using the first discrete derivatives for the expansion in z=0 of the oscillating part λ tiny (n) =λ n * of the "tiny" Li-Keiper coefficients , we analyse two series in the variable z=1-1/s ~0 for the first low values and compare them with the exact series. The numerical results suggest interesting more "sophisticated" approximations.
Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first sp... more Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first specify a lower bound emerging from the infinite set and give a characterization of it. Then, we propose a possible new upper and lower bound for the coefficients in few of the partitions occurring in the cluster functions furnishing in a nonlinear way the coefficients. A numerical experiment up to n=15 confirms the proposed bounds and an experiment, i.e. the counting of the zeros in the binary representation of an integer for a constant related to the Glaisher-Kinkelin constant is also given up to n=32.
In the first part of this short work (in the form of a comment) we add the plots of two more valu... more In the first part of this short work (in the form of 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".
Our purpose is to describe some recent progress in applying fractal concepts to systems of releva... more Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material, Furthermore, we discuss the finding that the exponent a quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.
We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe f... more 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 Li-Keiper 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 numerical finding which support our stability conjecture that the tiny part lambda-tiny(n) (the fluctuations around the trend) are bounded in absolute values by gammaxn, where gamma is the Euler-Mascheroni constant.
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functi... more Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented.
We present a calculation involving a function related to the Riemann Zeta function and suggested ... more We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi, while that at r = 0 is given by (-4 + gamma + 3*log(2) + Pi/2)*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m(r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1...
The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypoth... more The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true.
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functi... more Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented.
We present a calculation involving a function related to the Riemann Zeta function and suggested ... more We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for
New expansions for some functions related to the Zeta function in terms of the Pochhammer's polyn... more New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coecients b k , d k ,d k andd k). In some formal limit our expansion b k obtained via the alternating series gives the regularized expansion of Maslanka for the Zeta function. The real and the imaginary part of the function on the critical line is obtained with a good accuracy up to I(s) = t < 35. Then, we give the expansion (coecientd k) for the derivative of ln((s − 1)ζ(s)). The critical function of the derivative, whose bounded values for R(s) > 1 2 at large values of k should ensure the truth of the Riemann Hypothesis (RH), is obtained either by means of the primes or by means of the zeros (trivial and non-trivial) of the Zeta function. In a numerical experiment performed up to high values of k i.e. up to k = 10 13 we obtain a very good agreement between the two functions, with the emergence of twelve oscillations with stable amplitude. For a special case of values of the two parameters entering in the general Pochhammer's expansion it is argued that the bound on the critical function should be given by the Euler constant gamma.
[math.NT], where a number of integral equalities involving integrals of the logarithm of the Riem... more [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented. Our integral equalities, together with the equalities given in the previous paper, include all earlier known criteria of this kind, viz. Wang, Volchkov and Balazard-Saias-Yor criteria, which are certain particular cases of the general approach proposed.
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