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A Quantum Model of the Distribution of Prime Numbers and the Riemann Hypothesis

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Abstract

After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed.

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Notes

  1. The story of Hilbert-Polya conjecture is documented on Odlyzko’s personal website. See: http://www.dtc.umn.edu/~odlyzko/

  2. The logarithmic integral function is defined here for x > 1 as \( \mathrm{li}\left(\mathrm{x}\right)=\mathrm{p}.\mathrm{v}.{\int}_0^{\mathrm{x}}\frac{\mathrm{dt}}{\ln \left(\mathrm{t}\right)} \), p.v. stands for Cauchy principal value, instead of the more usual function\( \mathrm{Li}\left(\mathrm{x}\right)={\int}_2^{\mathrm{x}}\frac{\mathrm{dt}}{\ln \left(\mathrm{t}\right)} \). Obviously when x > 2 the two functions differ only for a constant li(x) = li(2) + Li(x), li(2) = 1,045… see A069284 of On-line Encyclopedia of Integer Sequences (OEIS).

  3. I adopt here, for the quantum counterpart of prime numbers, the same name used in previous works dealing with free gas models.

  4. I use here the same notation as in the paper of De Reyna and Toulisse, (2013), to indicate the inverse function of li(x).

  5. w.p. stands for “with probability”

  6. It would be impossible to mention here all the great work of Odlyzko on the zeros of the zeta function, see for example [31].

  7. This step requires some clarification both from a dimensional point of view and because of discrete charge distribution. In order for the replacement to be correct, we need to assume the spatial distribution of charges in [s1, sn − 1] in terms of mean charge value per space unit,

    (n − 1)/(sn − 1 − s1), is equal to that in [sn + 1, ∞), which is an acceptable assumption when n ≫ 1.

  8. Here and in the rest of the paper potential energy and energy levels will be denoted by their corresponding mathematical function and number sequence, since the correct physical dimensions can always be restored by appropriate constant dimensional factors.

  9. This “physical” assumption is common throughout the scientific literature on the subject (e.g., [37]), and it can be related with the “observation” of a minimal spacing among primes of 2 units [10].

  10. Potential and energy eigenvalues, which are assumed as axioms in this paper, can be derived from a combinatorial model of the distribution of primes, as I said in the Introduction. Giving an account of combinatorial theory here would take too many pages. This paper is mainly dedicated to study the physical model and its applications.

  11. The exact expression contains the term \( -{\left|{\uppsi}_{\mathrm{n}}^{\ast}\left(\mathrm{x}\right){\mathrm{H}}_{\mathrm{n}}\left(\mathrm{x}\right){\uppsi}_{\mathrm{n}}\left(\mathrm{x}\right)\right|}_{\mathrm{x}=\mathrm{n}}=-{\mathrm{E}}_{\mathrm{n}}{\upvarphi}_{\mathrm{n}}\left(\mathrm{n}\right) \), to be added to the above equation. I omit this term since it decreases as \( \mathcal{O}\left(\exp \left(-2{e}^{\frac{r(n)}{n}-1}\right)\right) \) as n → ∞.

  12. A slightly modified condition is needed in order that rn is an increasing function of n:

    \( \mathrm{n}<{\mathrm{n}}_0{\mathrm{e}}^{\left(2/\mathrm{e}\right){\mathrm{e}}^{\left({\mathrm{r}}_0/{\mathrm{n}}_0\right)}-1/\mathrm{W}\left(\mathrm{e}/2\right)} \); W is the Lambert function (e.g., [14]),

    W(e/2) ≅ 0,685.

  13. I keep the symbol r instead of x to remember the meaning of n(r) as stationary point-counting function.

  14. More precisely there are positive constants C and D such that for infinitely many x it is

    \( \uppi \left(\mathrm{x}\right)-\mathrm{li}\left(\mathrm{x}\right)>\mathrm{C}\left(\frac{{\mathrm{x}}^{1/2}\cdotp \mathrm{lnlnln}\left(\mathrm{x}\right)}{\ln \left(\mathrm{x}\right)}\right) \) and \( \uppi \left(\mathrm{x}\right)-\mathrm{li}\left(\mathrm{x}\right)<-\mathrm{D}\left(\frac{{\mathrm{x}}^{1/2}\cdotp \mathrm{lnlnln}\left(\mathrm{x}\right)}{\ln \left(\mathrm{x}\right)}\right) \), as x → ∞.

  15. Se note 4 in Introduction about the choice of the symbol ali().

  16. Brackets indicate events in their respective probability spaces. I give here an intuitive definition of the concepts, avoiding any formalization.

  17. In the case studied in Sections 35 where En is given by (13), it is \( \frac{\sqrt{\updelta \left(\mathrm{n}\right)}}{\mathrm{n}}=\frac{\sqrt{3}}{2}<1 \).

  18. I use the same symbol usually adopted in mathematical papers on the subject after [22]

  19. The model leading to Conjecture 1 does not depend on RH, as we saw in Section 5. It can be shown that other models can be defined analogously, in and outside the validity of RH, all having the same value (74) for the probability α.

  20. The statistical procedure proposed above can be easily converted to measure the fluctuations of π(x)-li(x), through the counting of the crossover points between the two functions.

  21. For any reference to statistical terms used in this section, see [25], chapter 3.

  22. This expression defines the “logarithmic density” of the positive integers k such that π(k) > li(k).

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs and mathematical tables. Nat. Bureau of Standards, Washington (1964)

  2. Bays, C., Hudson, R.H.: A new bound for the smallest x with π(x) > li(x). Mathematics of Computation. 69(231), 1285–1289 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  3. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications, Wiley (2004)

  4. Bender, C.M., Brody, D.C., Müller, M.P.: Hamiltonian for the Zeros of the Riemann zeta function. Phys. Rev. Lett. 118, 130201 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  5. Berry, M., Keating, J.: H = Xp and the Riemann Zeros. In: Lerner, et al. (eds.) Supersymmery and trace formulae: chaos and disorder, p. 355. Kluwer Academic Publishers, New York (1999a)

    Chapter  Google Scholar 

  6. Berry, M., Keating, J.: The Riemann Zeros and eigenvalue Asymptotics. SIAM Rev. 41(2), 236–266 (1999b)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bombieri, E.: Problems of the Millennium: the Riemann Hypothesis. Clay Math Institute Online at http://www.claymath.org/sites/default/files/official_problem_description.pdf (2000)

  8. Born, M.: Atomic Physics. Blackie & Son Ltd, Glasgow (1969)

    Google Scholar 

  9. Carfì, D.: The pointwise Hellmann-Feynman theorem. AAPP Classe di Scienze Fisiche, Matematiche e Naturali 88(1) (2010)

  10. Castro, C.: On strategies towards the Riemann hypothesis: fractal Supersymmetric QM and a trace formula. International Journal of Geometric Methods in Modern Physics. 04(05), 861–880 (2007)

    Article  MathSciNet  Google Scholar 

  11. Chao, K.F., Plymen, R.: A new bound for the smallest x with π(x) > li(x). International Journal of Number Theory. 06(03), 681–690 (2010)

    Article  MathSciNet  Google Scholar 

  12. Connes, A.: Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Sel. Math. N. Ser. 5, 29–106 (1999)

    Article  MathSciNet  Google Scholar 

  13. Conrey, J.B.: The Riemann hypothesis. Notices of the AMS (2003)

  14. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert function. Adv. Comput. Math. 5(1), 329–359 (1996)

    Article  MathSciNet  Google Scholar 

  15. De Reyna, J.A., Toulisse, J.: The n-th prime asymptotically. Journal de theorie des nombres de Bordeaux. 25(3), 521–555 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Feynman, R.P.: Forces in Molecules. Phys. Rev. 56, 340 (1939)

    Article  ADS  Google Scholar 

  17. Feynman, R.P.: The Feynman Lectures on Physics (Vol. 2, Chap. 5). Addison-Wesley Publishing, Inc. Reading, Massachusetts (1964)

  18. Gatteschi, L.: Funzioni Speciali. UTET, Torino (1973)

    MATH  Google Scholar 

  19. Hellmann, H.: Einführung in Die Quantenchemie. Frank Deuticke, Leipzig (1937)

    Google Scholar 

  20. Ismail, M.E.H., Zhang, R.: On the Hellmann- Feynman theorem and the variations of Zeros of certain special functions. Adv. Appl. Math. 9, 439–446 (1988)

    Article  MathSciNet  Google Scholar 

  21. Julia, B. (1990). Statistical Theory of Numbers. In: Luck, J.M., Moussa, P., Waldschmidt, M. (Eds) Number Theory and Physics. Springer Proceedings in Physics (47) 276. Springer-Verlag, Berlin, Statistical Theory of Numbers

  22. Kotnik, T.: The prime-counting function and its analytic approximations. Adv. Comput. Math. 29, 55–70 (2008)

    Article  MathSciNet  Google Scholar 

  23. Latorre, J.I., Sierra, G.: Quantum Computation of Prime Number Functions. Quant. Inf. and Comp. 14, 0577 (2014)

    MathSciNet  Google Scholar 

  24. Latorre, J.I., Sierra, G.: There is entanglement in the primes. Quant. Inf. and Comp. 15, 0622 (2015)

    MathSciNet  Google Scholar 

  25. Lehmann, E.L.: Testing Statistical Hypotheses, 2nd edn. John Wiley and Sons Inc., New York (1986)

    Book  Google Scholar 

  26. Lennard-Jones, J.E.: On the determination of molecular fields. Proc. R. Soc. Lond. A. 106(738), 463–477 (1924)

    Article  ADS  Google Scholar 

  27. Littlewood, J.E.: Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris. 158, 1869–1872 (1914)

    MATH  Google Scholar 

  28. Mussardo, G.: The quantum mechanical potential for the prime numbers. arXiv:cond-mat/9712010 (1997)

  29. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)

  30. NIST Digital library of mathematical functions. National Institute of Standards and Technology, dlmf.nist.gov

  31. Odlyzko, A.M.: The 10^22-nd zero of the Riemann zeta function. In: van Frankenhuysen, M., Lapidus, M.L. (eds) dynamical, spectral, and arithmetic zeta functions. Amer. Math. Soc. contemporary math. Series, no. 290, 139-144 (2001)

  32. Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM. 21(2), 120–126 (1978)

    Article  MathSciNet  Google Scholar 

  33. Rosu, H.C.: Quantum hamiltonians and prime numbers. Modern Physics Letters A. 18, 1205–1213 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  34. Saouter, Y., Demichel, P.: A sharp region where π(x) − li(x) is positive. Mathematics of Computation. 79(272), 2395–2405 (2010)

    Article  MathSciNet  Google Scholar 

  35. Schoenfeld, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II Mathematics of Computation. 30(134), 337–360 (1976)

    MathSciNet  MATH  Google Scholar 

  36. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  Google Scholar 

  37. Schumayer, D., Hutchinson, D.A.W.: Physics of the Riemann hypothesis. Rev. Mod. Phys. 83, 307–330 (2011)

    Article  ADS  Google Scholar 

  38. Schumayer, D., van Zyl, B.P., Hutchinson, D.A.W.: Quantum mechanical potentials related to the prime numbers and Riemann zeros. Phys. Rev. E. 78, 056215 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  39. Sierra, G.: A quantum mechanical model of the Riemann zeros. New J. Phys. 10, 033016 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  40. Sierra, G., Rodríguez-Laguna, J.: H=xp model revisited and the Riemann Zeros. Phys. Rev. Lett. 106, 200201 (2011)

    Article  ADS  Google Scholar 

  41. Skewes, S.: On the difference π(x) − li(x). Journal of the London Mathematical Society. 8, 277–283 (1933)

    Article  MathSciNet  Google Scholar 

  42. Skewes, S.: On the difference π(x) − li(x) (II). Proceedings of the London Mathematical Society. 5, 48–70 (1955)

    Article  MathSciNet  Google Scholar 

  43. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://oeis.or, Sequence A006880

  44. Von Koch, H.: Sur la distribution des nombres premiers. Acta Mathematica. 24(1), 159–182 (1901)

    Article  MathSciNet  Google Scholar 

  45. Wintner, A.: On the distribution function of the remainder term of the prime number theorem. Am. J. Math. 63(2), 233–248 (1941)

    Article  MathSciNet  Google Scholar 

  46. Wolf, M.: Application of statistical mechanics in number theory. Physica A. 274, 149–157 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  47. Wolf, M.: Will a Physicist Prove the Riemann Hypothesis? Reports on Progress in Physics. 83(3), 036001 (2020)

    Article  ADS  Google Scholar 

  48. Wu, H., Sprung, D.W.L.: Riemann zeros and a fractal potential. Phys. Rev. E. 48, 2595 (1993)

    Article  ADS  Google Scholar 

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Correspondence to Vito Barbarani.

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Appendices

Appendices

1.1 A - Calculation of the mean value of the position x of the primon in the n-th quantum state.

The probability density function with parameter rn becomes

$$ {\upvarphi}_{\mathrm{n}}\left(\mathrm{x}\right)=\frac{2}{\mathrm{n}}\frac{1}{\left[1-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\right]}{\mathrm{e}}^{-\frac{\mathrm{x}-{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\frac{\mathrm{x}-{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}},\mathrm{x}\ge \mathrm{n} $$

then we have to calculate the following integral

$$ \overline{{\mathrm{q}}_{\mathrm{n}}}=\frac{2}{\mathrm{n}}\frac{1}{\left(1-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\right)}{\mathrm{I}}_{\mathrm{n}} $$
$$ {\mathrm{I}}_{\mathrm{n}}={\int}_{\mathrm{n}}^{\infty }{\mathrm{x}\mathrm{e}}^{-\frac{\mathrm{x}-{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\frac{\mathrm{x}-{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\mathrm{dx}. $$

Let’s make the change of variable \( \mathrm{s}=\frac{\mathrm{x}-{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}} \) and set

$$ {\mathrm{J}}_{\mathrm{n}}\left(\mathrm{u}\right)=\mathrm{n}{\mathrm{r}}_{\mathrm{n}}{\int}_{1-\left(\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}\right)}^{\mathrm{u}}{\mathrm{e}}^{-\mathrm{s}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}+{\mathrm{n}}^2{\int}_{1-\left(\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}\right)}^{\mathrm{u}}{\mathrm{se}}^{-\mathrm{s}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}. $$

For the first indefinite integral we get

$$ \int {\mathrm{e}}^{-\mathrm{s}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}=\frac{1}{2}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}+\mathrm{const}. $$

while integrating by parts the second one

$$ \int {\mathrm{s}\mathrm{e}}^{-\mathrm{s}}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}=\frac{\mathrm{s}}{2}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}-\frac{1}{2}\int {\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}+\mathrm{const}. $$

It is

$$ \int {\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{s}}}\mathrm{ds}={\mathrm{E}}_1\left(2{\mathrm{e}}^{-\mathrm{s}}\right)+\mathrm{const}. $$

where E1 is the function defined in Section 3 (upper incomplete gamma function).

It is

$$ {\mathrm{J}}_{\mathrm{n}}\left(\mathrm{u}\right)=\frac{\mathrm{n}{\mathrm{r}}_{\mathrm{n}}}{2}\left({\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{u}}}-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\right)+\frac{{\mathrm{n}}^2}{2}\left[\mathrm{u}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{u}}}-\left(1-\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}\right){\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}-{\mathrm{E}}_1\left(2{\mathrm{e}}^{-\mathrm{u}}\right)++{\mathrm{E}}_1\left(\frac{2}{\mathrm{e}}{\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}\right)\right] $$

and obviously \( {\mathrm{I}}_{\mathrm{n}}=\underset{\mathrm{u}\to \infty }{\lim }{\mathrm{J}}_{\mathrm{n}}\left(\mathrm{u}\right) \). In order to calculate this limit it is useful to use the series expansion (28) which gives

\( {\mathrm{J}}_{\mathrm{n}}\left(\mathrm{u}\right)=\frac{\mathrm{n}{\mathrm{r}}_{\mathrm{n}}}{2} \)(\( {\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{u}}}-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\Big)+ \)

$$ +\frac{{\mathrm{n}}^2}{2}\left[\mathrm{u}{\mathrm{e}}^{-2{\mathrm{e}}^{-\mathrm{u}}}-\left(1-\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}\right){\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}-\mathrm{u}+{\sum}_{\mathrm{k}=1}^{\infty}\frac{{\left(-1\right)}^{\mathrm{k}}{\left(2{\mathrm{e}}^{-\mathrm{u}}\right)}^{\mathrm{k}}}{\mathrm{k}\mathrm{k}!}+1-\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}--{\sum}_{\mathrm{k}=1}^{\infty}\frac{{\left(-1\right)}^{\mathrm{k}}}{\mathrm{k}\mathrm{k}!}{\left(\frac{2}{\mathrm{e}}{\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}\right)}^{\mathrm{k}}\right]. $$

Finally taking the limit as u → ∞

$$ {\mathrm{I}}_{\mathrm{n}}=\frac{{\mathrm{n}}^2}{2}\left[1-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}-{\sum}_{\mathrm{k}=1}^{\infty}\frac{{\left(-1\right)}^{\mathrm{k}}}{\mathrm{k}\mathrm{k}!}{\left(\frac{2}{\mathrm{e}}{\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}\right)}^{\mathrm{k}}\right] $$

and

$$ \overline{{\mathrm{q}}_{\mathrm{n}}}=\mathrm{n}-\frac{\mathrm{n}}{\left(1-{\mathrm{e}}^{-\left(2/\mathrm{e}\right){\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}}\right)}{\sum}_{\mathrm{k}=1}^{\infty}\frac{{\left(-1\right)}^{\mathrm{k}}}{\mathrm{k}\mathrm{k}!}{\left(\frac{2}{\mathrm{e}}{\mathrm{e}}^{\frac{{\mathrm{r}}_{\mathrm{n}}}{\mathrm{n}}}\right)}^{\mathrm{k}} $$

which can be more conveniently written as in (27).

1.2 B - The Hellmann-Feynman Theorem applied to the quantum system.

Considering φn(x) given by (25) and \( \left(\frac{\partial {\mathrm{H}}_{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{\partial n}}\right) \) given by (31), the integral on the left side of (30) becomes

$$ {\int}_{\mathrm{n}}^{\infty }{\upvarphi}_{\mathrm{n}}\left(\mathrm{x}\right)\left(\frac{\partial {\mathrm{H}}_{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{\partial n}}\right)\mathrm{dx}= $$
$$ =\frac{\uplambda_{\mathrm{n}}}{\mathrm{n}}\frac{1}{\left[1-{\mathrm{e}}^{-\left({\uplambda}_{\mathrm{n}}/\mathrm{e}\right)}\right]}\times \left(\frac{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}{\uplambda}_{\mathrm{n}}^{\prime }-\mathrm{n}{\uplambda}_{\mathrm{n}}^2}{2{\mathrm{n}}^4}{\mathrm{I}}_3+\frac{\uplambda_{\mathrm{n}}^2}{2{\mathrm{n}}^4}{\mathrm{J}}_3-\frac{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}^{\prime }-2\mathrm{n}{\uplambda}_{\mathrm{n}}}{{\mathrm{n}}^4}{\mathrm{I}}_2-\frac{\uplambda_{\mathrm{n}}}{{\mathrm{n}}^4}{\mathrm{J}}_2\right) $$
(B1)

with

$$ {\mathrm{I}}_3={\int}_{\mathrm{n}}^{\infty }{\mathrm{e}}^{-\frac{3\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\mathrm{dx}= $$
$$ =\frac{1}{\uplambda_{\mathrm{n}}^3}{\left[\mathrm{n}{\uplambda}_{\mathrm{n}}^2{\mathrm{e}}^{-\frac{2\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+2\mathrm{n}{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+2\mathrm{n}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty }= $$
$$ =\frac{2\mathrm{n}}{\uplambda_{\mathrm{n}}^3}-\frac{\mathrm{n}}{\uplambda_{\mathrm{n}}^3}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}/\mathrm{e}}\left[{\left(\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}\right)}^2+2\left(\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}\right)+2\right] $$
$$ {\mathrm{I}}_2={\int}_{\mathrm{n}}^{\infty }{\mathrm{e}}^{-\frac{2\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\mathrm{dx}= $$
$$ =\frac{1}{\uplambda_{\mathrm{n}}^2}{\left[\mathrm{n}{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+\mathrm{n}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty }= $$
$$ =\frac{\mathrm{n}}{\uplambda_{\mathrm{n}}^2}-\frac{\mathrm{n}}{\uplambda_{\mathrm{n}}^2}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}/\mathrm{e}}\left[\left(\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}\right)+1\right]. $$

The total term within (B1) containing I3 and I2 becomes

$$ \frac{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}{\uplambda}_{\mathrm{n}}^{\prime }-\mathrm{n}{\uplambda}_{\mathrm{n}}^2}{2{\mathrm{n}}^4}{\mathrm{I}}_3-\frac{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}^{\prime }-2\mathrm{n}{\uplambda}_{\mathrm{n}}}{{\mathrm{n}}^4}{\mathrm{I}}_2= $$
$$ =\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}-\frac{1}{2}\frac{\uplambda_{\mathrm{n}}^{\prime }}{\mathrm{n}{\mathrm{e}}^2}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}+\frac{1}{2}\frac{\uplambda_{\mathrm{n}}}{{\mathrm{n}}^2{\mathrm{e}}^2}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{{\mathrm{n}}^2\mathrm{e}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}} $$
(B2)

Let’s calculate J3 and J2

$$ {\mathrm{J}}_3={\int}_{\mathrm{n}}^{\infty}\mathrm{x}{\mathrm{e}}^{-\frac{3\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\mathrm{dx}= $$
$$ =\frac{1}{\uplambda_{\mathrm{n}}^3}{\left[2{\mathrm{n}}^2{\mathrm{E}}_{\mathrm{i}}\left(-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}\right)-3{\mathrm{n}}^2{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+2\mathrm{nx}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty }+\frac{1}{\uplambda_{\mathrm{n}}^2}{\left[-{\mathrm{n}}^2{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}++2\mathrm{nx}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty }+\frac{1}{\uplambda_{\mathrm{n}}}{\left[\mathrm{nx}{\mathrm{e}}^{-\frac{2\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty } $$
$$ {\mathrm{J}}_2={\int}_{\mathrm{n}}^{\infty}\mathrm{x}{\mathrm{e}}^{-\frac{2\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\mathrm{dx}= $$
$$ =\frac{1}{\uplambda_{\mathrm{n}}^2}{\left[{\mathrm{n}}^2{\mathrm{E}}_{\mathrm{i}}\left(-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}\right)-{\mathrm{n}}^2{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+\mathrm{nx}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty }+\frac{1}{\uplambda_{\mathrm{n}}}{\left[\mathrm{nx}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty } $$

The total term within (B1) containing J3 and J2 becomes

\( \frac{\uplambda_{\mathrm{n}}^2}{2{\mathrm{n}}^4}{\mathrm{J}}_3-\frac{\uplambda_{\mathrm{n}}}{{\mathrm{n}}^4}{\mathrm{J}}_2 \)=

$$ ={\left[-\frac{3}{2{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}-\frac{1}{2{\mathrm{n}}^2}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+\frac{\uplambda_{\mathrm{n}}}{2{\mathrm{n}}^3}\mathrm{x}{\mathrm{e}}^{-\frac{2\mathrm{x}}{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}+\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-{\uplambda}_{\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathrm{n}}}}\right]}_{\mathrm{n}}^{\infty } $$
$$ \frac{\uplambda_{\mathrm{n}}^2}{2{\mathrm{n}}^4}{\mathrm{J}}_3-\frac{\uplambda_{\mathrm{n}}}{{\mathrm{n}}^4}{\mathrm{J}}_2= $$
$$ =-\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}+\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}+\frac{1}{2}\frac{1}{{\mathrm{n}}^2\mathrm{e}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{2}\frac{\uplambda_{\mathrm{n}}}{{\mathrm{n}}^2{\mathrm{e}}^2}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}} $$
(B3)

Putting (B2) and (B3) all together in (B1) one gets

$$ {\int}_{\mathrm{N}}^{\infty }{\upvarphi}_{\mathrm{n}}\left(\mathrm{x}\right)\left(\frac{\partial {\mathrm{H}}_{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{\partial n}}\right)\mathrm{dx}= $$
$$ =\frac{\uplambda_{\mathrm{n}}}{\mathrm{n}}\frac{1}{\left[1-{\mathrm{e}}^{-\left({\uplambda}_{\mathrm{n}}/\mathrm{e}\right)}\right]}\left[\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}-\frac{1}{2}\frac{\uplambda_{\mathrm{n}}^{\prime }}{\mathrm{n}{\mathrm{e}}^2}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{2}\frac{1}{{\mathrm{n}}^2\mathrm{e}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}\right] $$

and Eq. (30) can be rewritten as

$$ \left[1-{\mathrm{e}}^{-\left({\uplambda}_{\mathrm{n}}/\mathrm{e}\right)}\right]\frac{1}{2{\mathrm{n}}^3}=\frac{\uplambda_{\mathrm{n}}}{\mathrm{n}}\left[\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}-\frac{1}{2}\frac{\uplambda_{\mathrm{n}}^{\prime }}{\mathrm{n}{\mathrm{e}}^2}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{2}\frac{1}{{\mathrm{n}}^2\mathrm{e}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}-\frac{1}{2}\frac{1}{{\mathrm{n}}^2{\uplambda}_{\mathrm{n}}}{\mathrm{e}}^{-\frac{\uplambda_{\mathrm{n}}}{\mathrm{e}}}\right] $$

from which finally Eq. (32) follows.

1.3 C - Proof of Theorem 1 of Section 4.

The following lemma must be stated before Theorem 1.

Lemma 1

Given L > 0 constant, rL < r0 such that

$$ \mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left({\mathrm{r}}_{\mathrm{L}}\right)=\mathrm{L}, $$

it is

li(r0) − li(r) ≤ L, for rL ≤ r ≤ r0

$$ \underset{{\mathrm{r}}_{0\to \infty }}{\lim }{\mathrm{r}}_{\mathrm{L}}=\infty $$
$$ \underset{{\mathrm{r}}_{0\to \infty }}{\lim}\Delta \left(\mathrm{L},{\mathrm{r}}_0\right)=\infty $$

where (L, r0) = (r0 − rL) is the interval length.

Proof of Lemma1.

The first inequality is obvious since li(r) is an increasing function. As far as the value of rL, from Lagrange’s theorem we get

$$ \frac{\mathrm{L}}{{\mathrm{r}}_0-{\mathrm{r}}_{\mathrm{L}}}=\frac{1}{\ln \overline{\mathrm{r}}}, $$
$$ \left(\mathrm{with}\right)\ {\mathrm{r}}_{\mathrm{L}}\le \overline{\mathrm{r}}\le {\mathrm{r}}_0 $$

hence \( \mathrm{L}\cdotp \ln \overline{\mathrm{r}}={\mathrm{r}}_0-{\mathrm{r}}_{\mathrm{L}} \), from which it is easy to derive

$$ \mathrm{L}\cdotp \ln {\mathrm{r}}_{\mathrm{L}}\le {\mathrm{r}}_0-{\mathrm{r}}_{\mathrm{L}}\le \mathrm{L}\cdotp \ln {\mathrm{r}}_0 $$
(C1)

and

$$ {\mathrm{r}}_0-\mathrm{L}\cdotp \ln {\mathrm{r}}_0\le {\mathrm{r}}_{\mathrm{L}}\le {\mathrm{r}}_0-\mathrm{L}\cdotp \ln {\mathrm{r}}_{\mathrm{L}} $$
(C2)

Since (r0 − L · ln r0) → ∞ as r0 → ∞, from (C2) it follows \( \underset{{\mathrm{r}}_{0\to \infty }}{\lim }{\mathrm{r}}_{\mathrm{L}}=\infty \), and \( \underset{{\mathrm{r}}_{0\to \infty }}{\lim}\Delta \left(\mathrm{L},{\mathrm{r}}_0\right)=\infty \) from (C1).

As far as the increasing rate of the interval length, it can be proved it is (L, r0) ≅ L · ln r0.

Proof of Theorem 1.

The HFT ode Eq. (36) can be written as

$$ \frac{\mathrm{dn}}{\mathrm{dr}}=\frac{1}{\ln \mathrm{r}\left(1-\updelta \left(\mathrm{r}\right)-\frac{\mathrm{e}}{2}\ \frac{{\mathrm{e}}^{-\mathrm{r}/\mathrm{n}}}{\ln \mathrm{r}}\right)} $$

where δ(r) is the function defined by (37). Since n(r) is an increasing function of r, it can be stated δ(r) increases with r. Indeed

$$ {\partial}_{\mathrm{r}}\updelta \left(\mathrm{r}\right)=-\left[\frac{1}{\ln \mathrm{r}}{\partial}_{\mathrm{n}}\left(\frac{\mathrm{r}}{\mathrm{n}}\right){\partial}_{\mathrm{r}}\mathrm{n}\left(\mathrm{r}\right)-\left(\frac{\mathrm{r}}{\mathrm{n}}\right)\frac{1}{\mathrm{r}{\ln}^2\mathrm{r}}\right], $$

from (10) and (32)

$$ {\partial}_{\mathrm{n}}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)=-\frac{\mathrm{e}}{2}\ \frac{{\mathrm{e}}^{-\mathrm{r}/\mathrm{n}}}{\mathrm{n}} $$

hence

$$ {\partial}_{\mathrm{r}}\updelta \left(\mathrm{r}\right)=\frac{\mathrm{e}}{2}\ \frac{{\mathrm{e}}^{-\mathrm{r}/\mathrm{n}}}{\mathrm{n}\mathrm{lnr}}{\partial}_{\mathrm{r}}\mathrm{n}\left(\mathrm{r}\right)+\left(\frac{\mathrm{r}}{\mathrm{n}}\right)\frac{1}{\mathrm{r}{\ln}^2\mathrm{r}}>0. $$

Furthermore from the definition (37) it also follows δ(r) ≥ 0 if n(r) ≥ r/ ln r.

Without limiting the generality of our reasoning we can assume the solution n(r) for

r1 ≤ r ≤ r0, with r1 defined by n(r1) = r1/ ln r1, where it is

$$ \mathrm{r}/\ln \mathrm{r}\le \mathrm{n}\left(\mathrm{r}\right)\le \mathrm{li}\left(\mathrm{r}\right), $$
$$ 0\le \updelta \left(\mathrm{r}\right)<1 $$

and r0 sufficiently large so that δ(r0) ≪ 1. Then we can write

$$ \frac{\mathrm{dn}}{\mathrm{dr}}\cong \frac{1}{\ln \mathrm{r}}\left(1+\updelta \left(\mathrm{r}\right)+\frac{\mathrm{e}}{2}\ \frac{{\mathrm{e}}^{-\frac{\mathrm{r}}{\mathrm{n}}}}{\ln \mathrm{r}}\right) $$

where I have omitted the higher terms in \( \mathcal{o}\left(\frac{\updelta \left(\mathrm{r}\right)}{\ln \mathrm{r}}\right) \) and \( \mathcal{o}\left(\frac{{\mathrm{e}}^{-\frac{\mathrm{r}}{\mathrm{n}}}}{\ln^2\mathrm{r}}\right) \).

Hence

$$ \mathrm{n}\left({\mathrm{r}}_0\right)-\mathrm{n}\left(\mathrm{r}\right)\cong {\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{1}{\ln \mathrm{s}}\mathrm{ds}+{\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{\updelta \left(\mathrm{s}\right)}{\ln \mathrm{s}}\mathrm{ds}+\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right) $$

with

$$ \mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)=\frac{\mathrm{e}}{2}{\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{{\mathrm{e}}^{-\frac{\mathrm{s}}{\mathrm{n}}}}{\ln^2\mathrm{s}}\mathrm{ds}. $$

Remembering δ(r) and li(r) are non-negative increasing functions we can deduce the following bounds

$$ \mathrm{n}\left({\mathrm{r}}_0\right)-\mathrm{n}\left(\mathrm{r}\right)\le \mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)+\updelta \left({\mathrm{r}}_0\right)\left(\mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)\right)+\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right) $$

and

$$ \mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)+\updelta \left(\mathrm{r}\right)\left(\mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)\right)+\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)\le \mathrm{n}\left({\mathrm{r}}_0\right)-\mathrm{n}\left(\mathrm{r}\right) $$

from which after some manipulations we get

$$ -\updelta \left({\mathrm{r}}_0\right)\left(\mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)\right)-\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)\le \mathrm{n}\left(\mathrm{r}\right)-\mathrm{li}\left(\mathrm{r}\right)\le -\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right). $$
(C1)

About the integral I(r, r0) we can write:

$$ \frac{2}{\mathrm{e}}\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)={\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{1}{{\mathrm{s}}^{1-\updelta \left(\mathrm{s}\right)}{\ln}^2\mathrm{s}}\mathrm{ds}. $$

By assumptions we are working within the interval of values of r such that

0 ≤ δ(r) ≤ δ(r0) < 1, hence in r ≤ s ≤ r0 it is

$$ {\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{1}{{\mathrm{s}}^{1-\updelta \left(\mathrm{r}\right)}{\ln}^2\mathrm{s}}\mathrm{ds}\le \frac{2}{\mathrm{e}}\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)\le {\int}_{\mathrm{r}}^{{\mathrm{r}}_0}\frac{1}{{\mathrm{s}}^{1-\updelta \left({\mathrm{r}}_0\right)}{\ln}^2\mathrm{s}}\mathrm{ds}. $$

Calculating the indefinite integral

$$ \int \frac{1}{{\mathrm{s}}^{1-\mathrm{a}}{\ln}^2\mathrm{s}}\mathrm{ds}=\mathrm{a}\bullet {\mathrm{E}}_{\mathrm{i}}\left(\mathrm{a}\bullet \mathrm{lns}\right)-\frac{{\mathrm{s}}^{\mathrm{a}}}{\ln \mathrm{s}}+\mathrm{const}.,\mathrm{a}\ne 0 $$

and neglecting the terms with positive sign on the left side, those with negative sign on the right side, from the above inequality it follows

$$ -\frac{\mathrm{e}}{2}\updelta \left({\mathrm{r}}_0\right){\mathrm{E}}_{\mathrm{i}}\left(\updelta \left({\mathrm{r}}_0\right)\ln {\mathrm{r}}_0\right)-\frac{\mathrm{e}}{2}\frac{{\mathrm{r}}^{\updelta \left(\mathrm{r}0\right)}}{\ln \mathrm{r}}\le -\mathrm{I}\left(\mathrm{r},{\mathrm{r}}_0\right)\le 0. $$

Inserting this in (C1) after observing that r ≤ r0 implies \( -{\mathrm{r}}_0^{\updelta \left({\mathrm{r}}_0\right)}\le -{\mathrm{r}}^{\updelta \left({\mathrm{r}}_0\right)} \), we finally get

$$ -\updelta \left({\mathrm{r}}_0\right)\left(\mathrm{li}\left({\mathrm{r}}_0\right)-\mathrm{li}\left(\mathrm{r}\right)\right)-\frac{\mathrm{e}}{2}\updelta \left({\mathrm{r}}_0\right){\mathrm{E}}_{\mathrm{i}}\left(\updelta \left({\mathrm{r}}_0\right)\ln {\mathrm{r}}_0\right)-\frac{\mathrm{e}}{2}\frac{{\mathrm{r}}_0^{\updelta \left({\mathrm{r}}_0\right)}}{\ln \mathrm{r}}\le $$
$$ \le \mathrm{n}\left(\mathrm{r}\right)-\mathrm{li}\left(\mathrm{r}\right)\le 0. $$
(C2)

Let’s choose a constant L > 0 such that li(r0) − li(rL) = L with r1 ≤ rL < r0, then inequality (C2) becomes for rL ≤ r ≤ r0

$$ -\updelta \left({\mathrm{r}}_0\right)\mathrm{L}-\frac{\mathrm{e}}{2}\updelta \left({\mathrm{r}}_0\right){\mathrm{E}}_{\mathrm{i}}\left(\updelta \left({\mathrm{r}}_0\right)\ln {\mathrm{r}}_0\right)-\frac{\mathrm{e}}{2}\frac{{\mathrm{r}}_0^{\updelta \left({\mathrm{r}}_0\right)}}{\ln {\mathrm{r}}_{\mathrm{L}}}\le \mathrm{n}\left(\mathrm{r}\right)-\mathrm{li}\left(\mathrm{r}\right)\le 0. $$
(C3)

The thesis of Theorem 1 follows directly from (C3) and Lemma 1 after observing for r0 sufficiently large

$$ \updelta \left({\mathrm{r}}_0\right)=1-\frac{{\mathrm{r}}_0}{\mathrm{li}\left({\mathrm{r}}_0\right)\ln {\mathrm{r}}_0}\cong \frac{1}{\ln {\mathrm{r}}_0}, $$
$$ {\mathrm{r}}_0^{\updelta \left({\mathrm{r}}_0\right)}\cong \mathrm{e} $$
$$ {\mathrm{E}}_{\mathrm{i}}\left(\updelta \left({\mathrm{r}}_0\right)\ln {\mathrm{r}}_0\right)\cong {\mathrm{E}}_{\mathrm{i}}(1),\left(\mathrm{finite}\right). $$

1.4 D - Proof of Theorem 2 of Section 5.

Given n1(x) and n2(x) solution of (47)–(48), the following proposition holds true

$$ {\lim}_{\mathrm{x}\to \infty}\frac{\mathrm{x}}{{\mathrm{n}}_2\left(\mathrm{x}\right)\ln \left(\frac{{\mathrm{n}}_2\left(\mathrm{x}\right)}{\mathrm{A}}\right)}=1 $$
(D1)
$$ {\lim}_{\mathrm{x}\to \infty}\frac{\mathrm{x}}{{\mathrm{n}}_1\left(\mathrm{x}\right)\ln \left(\mathrm{B}{\mathrm{n}}_1\left(\mathrm{x}\right)\right)}=1 $$
(D2)

with

$$ \mathrm{A}=\left(\frac{2+\sqrt{3}}{2}\right),\mathrm{B}=\left(\frac{2}{2-\sqrt{3}}\right). $$
(D3)

Let’s prove the result for n2(x). In agreement with (40) and (41) from Theorem 1, for n2 sufficiently large we can assume r(n2) = ali(n2), hence by applying (42), M ≥ 0 constant,

$$ -\mathrm{M}{\mathrm{n}}_2\ln \ln {\mathrm{n}}_2\le {\mathrm{n}}_2\ln {\mathrm{n}}_2-{\mathrm{r}}_{\mathrm{n}2}\le \mathrm{M}{\mathrm{n}}_2\ln \ln {\mathrm{n}}_2 $$

and (48)

$$ -\mathrm{M}\frac{\ln \ln {\mathrm{n}}_2}{\ln \left({\mathrm{n}}_2/\mathrm{A}\right)}+1\le \frac{\mathrm{x}}{{\mathrm{n}}_2\ln \left({\mathrm{n}}_2/\mathrm{A}\right)}\le 1+\mathrm{M}\frac{\ln \ln {\mathrm{n}}_2}{\ln \left({\mathrm{n}}_2/\mathrm{A}\right)} $$

from which (D1) follows. In the same way the analogous result (D2) can be proved for n1(x).

Since

$$ \frac{\mathrm{x}}{{\mathrm{n}}_2\ln {\mathrm{n}}_2}=\frac{\mathrm{x}}{{\mathrm{n}}_2\ln \left(\frac{{\mathrm{n}}_2}{\mathrm{A}}\right)}\left(1-\frac{\ln \mathrm{A}}{\ln {\mathrm{n}}_2}\right) $$

the limit (D1) implies

$$ {\lim}_{\mathrm{x}\to \infty}\frac{\mathrm{x}}{{\mathrm{n}}_2\left(\mathrm{x}\right)\ln {\mathrm{n}}_2\left(\mathrm{x}\right)}=1 $$

and through a similar observation an analogous result follows from (D2) for n1(x)

$$ {\lim}_{\mathrm{x}\to \infty}\frac{\mathrm{x}}{{\mathrm{n}}_1\left(\mathrm{x}\right)\ln {\mathrm{n}}_1\left(\mathrm{x}\right)}=1. $$

From (44) we can write

$$ \frac{\mathrm{x}}{{\mathrm{n}}_2\ln {\mathrm{n}}_2}\le \frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}\le \frac{\mathrm{x}}{{\mathrm{n}}_1\ln {\mathrm{n}}_1} $$

which implies, due to the above limits

$$ {\lim}_{\mathrm{x}\to \infty}\frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}=1. $$
(D4)

Let’s define now the function δ(x) as

$$ \frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}=1+\updelta \left(\mathrm{x}\right) $$

from which it is easy to get

$$ \frac{\ln \mathrm{x}}{\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}=1+\frac{\ln \ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}{\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}+\frac{\ln \left(1+\updelta \left(\mathrm{x}\right)\right)}{\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}. $$

Because of (D4) it is δ(x) → 0 and \( \frac{\ln \mathrm{x}}{\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}\to 1 \) as x → ∞.

By observing that

$$ \frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}=\frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln \mathrm{x}}\left(\frac{\ln \mathrm{x}}{\ln {\uppi}_{\mathrm{Q}}\left(\mathrm{x}\right)}\right) $$

we can conclude (D4) implies also

$$ \underset{\mathrm{x}\to \infty }{\lim}\frac{\mathrm{x}}{\uppi_{\mathrm{Q}}\left(\mathrm{x}\right)\ln \mathrm{x}}=1 $$

which is the usual form of the PNT with πQ(x) instead of π(x).

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Barbarani, V. A Quantum Model of the Distribution of Prime Numbers and the Riemann Hypothesis. Int J Theor Phys 59, 2425–2470 (2020). https://doi.org/10.1007/s10773-020-04512-2

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