Zeros of Riemann’s Zeta Functions in the Line z=1/2+it_{0}
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Abstract
It was found that, in addition to trivial zeros in points (z = − 2N, N = 1, 2…, natural numbers), the Riemann’s zeta function ζ(z) has zeros only on the line {\( z=\frac{1}{2}+\mathrm{i}{\mathrm{t}}_0 \), t_{0} is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function η(z), leading to the possibility fix the zeros of the zeta function ζ(z), was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation).
Keywords
Superconductivity phase-slip events time-dependent Ginzburg-Landau equation Riemann's zeta-function1 Introduction
Investigations of Josephson effect, current flow in narrow superconducting strips [1], and dynamical states [2] in superconductors with use of TDGLE (time-dependent Ginzburg-Landau equation) lead to the necessity to deal with an important phenomenon: phase slip events. It is interesting that the study of the distribution of zeros for Riemann’s ζ function (see below Eqs. (15–17)) also requires an analysis of the same phenomenon. It means that there exists a deep internal connection between all these problems. It turns out that the Euler Γ function is also the essential ingredient of the scenario.
Riemann’s ζ function and Euler Γ function appear in many physical problems. For example, the study of the spin-orbit interaction in inhomogeneous superconductors state reveals the existence of the infinite set of nontrivial exact relations for Euler {ψ} function [3]; these relations are essential for the evaluation of the critical temperature. These equations contain the Bernoulli numbers B_{n}. In general, these problems appear as the consequence of mentioned internal ties and arise while the temperature technique and analytical continuation are used in the theory of superconductivity. We are trying to analyze these ties. We hope that the investigation of phase-slip events with the use of the distribution of zeros for Riemann’s ζ function will lead to new results, which are important not only in mathematics. They will provide a deep understanding of the phenomena mentioned above, especially for the case of strong suppression of superconductivity by an external field.
2 Main Equations
Note that in the points {t_{0} ln 2 = 2πN, N ≠ 0, ν = 1/2}, we have D_{1} = D_{2} = 0 [5].
Then on the Stock’s line, we obtain the following:
Equations (8, 11, 13, 14) allow us to estimate the location of the interval, where the zero point with number N is placed.
Note that there exists a large parameter t_{0} ≫ 1 in the limit ln(t_{0}). It leads to the drastic increase of the first term in Eq. (10) relative the second term with an increase in ν; as a result, the existence of a pair of zeros (ν, −ν) with ν ≠ 0 is impossible. The ratio of these two terms is exp(ν ln(t_{0}/2π)). Equations (6, 13) allow us get the exact expression for the zeros of the Ξ function with a given number N and an asymptotic exact expression for \( {\eta}_1\left({t}_0^{(N)}\right) \) and \( {\eta}_2\left({t}_0^{(N)}\right) \) on a Stock’s line.
The Table contains the values of 29 first zeros of the zeta function from Ref. [7] and additional seven points with numbers 30, 40, 41, 42, 269, 270, and 271, which follow from Eqs. (10, 13, 14). For all these points, we added the values of functions \( {\eta}_1\left({t}_0^{(N)}\right) \) and \( {\eta}_2\left({t}_0^{(N)}\right) \).
The existence of the large parameter, lnt_{0}, allows to analyze Eqs. (10, 13, 14) with the use of the perturbation theory and estimate the functions {\( {\eta}_1\left({t}_0^{(N)}\right),{\eta}_2\left({t}_0^{(N)}\right) \)}.
with the use of the perturbation theory relative to the second term on the right side of Eq. (17) and Eq. (15). Here f^{′}(t_{0}) = ∂f(t_{0})/∂t_{0}.
N | α(J_{a}. E) | α_{ cal} | η_{1} | η_{2} |
---|---|---|---|---|
1 | 14.134725 | 14.134725 | 0.157875 | − 0.67088 |
2 | 21.022040 | 21.022047 | − 0.2205789 | − 0.63322 |
3 | 25.010856 | 25.010856 | 0.3342749 | − 0.68525 |
4 | 30.424878 | 30.424878 | − 0.5374147 | − 0.50314 |
5 | 32.935057 | 32.935065 | 0.5740572 | − 0.51096 |
6 | 37.586176 | 37.586175 | − 0.2939078 | − 0.77282 |
7 | 40.918720 | 40.918725 | − 0.2046136 | − 0.46588 |
8 | 43.327073 | 43.327073 | 0.6459731 | − 0.6406 |
9 | 48.005150 | 48.005155 | − 0.8508028 | − 0.4339 |
10 | 49.773832 | 49.773825 | 0.4764723 | − 0.31471 |
11 | 52.970 | 52.97031 | 0.2600957 | − 0.82039 |
12 | 56.446 | 56.446238 | − 0.3591892 | − .076845 |
13 | 59.347 | 59.34704 | − 0.4384483 | − 0.21453 |
14 | 60.833 | 60.83178 | 1.026932 | − 0.37641 |
15 | 65.113 | 65.11254 | − 0.7642258 | − 0.67143 |
16 | 67.080 | 67.079783 | 6.272686 ⋅ 10^{−2} | − 0.41039 |
17 | 69.546 | 69.54639 | 0.2614599 | − 0.59757 |
18 | 72.067 | 72.06715 | 0.3503908 | − 0.88480 |
19 | 75.705 | 75.7042 | − 0.9904536 | − 0.3584 |
20 | 77.145 | 77.14482 | 0.3521394 | − 0.1489 |
21 | 79.337 | 79.337345 | 0.7290875 | − 0.73164 |
22 | 82.910 | 82.91038 | − 0.6992546 | − 0.71322 |
23 | 84.734 | 84.7355 | 7.80726 ⋅ 10^{−2} | − 0.50791 |
24 | 87.426 | 87.425253 | − 0.3003618 | − 0.31313 |
25 | 88.809 | 88.809105 | 1.014014 | − 0.49443 |
26 | 92.494 | 92.49189 | − 0.7591807 | − 0.803829 |
27 | 94.651 | 94.65131 | − 0.5337027 | − 8.5412 ⋅ 10^{−2} |
28 | 95.871 | 95.87063 | 0.9503964 | − 0.21802 |
29 | 98.831 | 98.83123 | 3.5379 ⋅ 10^{−2} | − 0.9336 |
30 | 101.31786 | − 0.2645489 | − 0.81867 | |
40 | 122.94674 | 3.15446 ⋅ 10^{−3} | − 4.6178 ⋅ 10^{−2} | |
41 | 124.2568 | 1.193301 | − 0.4390 | |
42 | 127.51671 | − 0.5508639 | − 0.9550723 | |
269 | 498.5809 | − 0.30956 | − 1.25627 | |
270 | 500.30905 | − 0.948835 | − 0.666625 | |
271 | 501.6045 | − 0.64338 | 0.116255 |
There is a very interesting interval {122.4 ≤ t_{0} ≤ 123.3} with very small values of both functions {η_{1}, η_{2}}. Inside this interval, a zeta function zero with number N = 40 is located at the point {t_{0} = 122.9467}. The value of functions {η_{1}, η_{2}} at this point is η_{1}(122.9467) = 3.154 ⋅ 10^{−3}; η_{2}(122.9467) = − 4.618 ⋅ 10^{−2}.
The values of the function {η_{1}, η_{2}} in the interval 122.4 ≤ t_{0} ≤ 123.3 are presented in Fig. 4 on a large scale.
The similar situation takes place in the third interval in the vicinity of a zero of the zeta function with the number {N = 269, t_{0} = 498.5809} 498.2 < t < 499.2. Functions {η_{1}, η_{2}} display fast oscillations with small amplitude in this interval near the values {− 0.31; − 1.28}. It looks as some special “zeros” are “attractive resonant centers.” Definitely, there exists the final concentration of such “centers.”
3 Conclusions
The key results of the paper are presented by the equations (10), (13), and (14). Equation (13) represents a strong improvement of the well-known result described in the book [5]. Equation (10) is the proof of the Riemann’s hypothesis. We introduce two analytical functions {ϕ and η}. The exact asymptotical expression for function ϕ on the stripe 0 < Re z < 1 for t_{0} ≫ 1 is obtained. With the use of the functions {ϕ, η}, one can create additional relation between the functions {Γ(z), ζ(z)}, and this allows us to produce a numeration of zeros of the Riemann’s ζ function. Equation (10) was solved for the large values of the parameter \( \ln \left(\frac{t_0}{2\pi}\right) \) for functions {η_{1},η_{2}; η = η_{1} + iη_{2}}. We also obtained the solution (in the first order of the perturbation theory) for three intervals 76.8 ≤ t_{0} ≤ 83.2, 122.4 ≤ t_{0} ≤ 127.9, and 498 ≤ t_{0} ≤ 502.
The condition (14) connected with symmetry of the Riemann’s zeta function and corresponding simultaneous equality to zero of both terms in Eq. (7) appears to be very essential.
The study of the phase-slip events is connected with the zeros, which are discussed above. Unlike all previous studies based on the use of the perturbation theory, the present analysis allows us to obtain an exact solution, without invoking any small parameter.
The connection between the numbers N of zeros of the Riemann’s ζ function inside of the abovementioned “resonant centers” and the primes is rather obvious.
Notes
Acknowledgments
I am grateful to Prof. Peter Fulde for his fruitful discussions of the problem, and Prof. Roderich Moessner for the hospitality in LIFW, Dresden.
Funding Information
Open access funding provided by Max Planck Society.
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