Entropy for some monoids of natural numbers

In abstract number theory and its applications to statistical physics, the concept of entropy plays an important role. Since entropy is equal to the logarithm of the distribution function, studying the entropy behavior of a monoid is equivalent to solving the inverse problem for this monoid.
The paper considers questions about the asymptotics of entropy for some monoids of natural numbers and monoids of natural numbers with a weight function.
First, the problem is solved for two monoids of the geometric progression type.
Secondly, the results obtained with respect to entropy for monoids with an arbitrary exponential sequence of primes of type 𝑞 are based on the solution of the inverse problem for monoids of this type obtained earlier by the authors.
To solve this problem, we consider two homomorphisms of the main monoid 𝑀(P(𝑞)) of type 𝑞 and the distribution problem reduces to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of 𝐶 logarithmic 𝜃-power density is introduced.
It is shown that any monoid 𝑀(P(𝑞)) for a sequence of pseudo-simple numbers P(𝑞) of type 𝑞 has upper and lower bounds for the element distribution function of the main basic monoid 𝑀(P(𝑞)) of type 𝑞.
It is shown that if 𝐶 is a logarithmic 𝜃-power density for the main monoid 𝑀(P(𝑞)) of the type 𝑞 exists, then 𝜃 = 1
2 and for the constant 𝐶 the inequalities are valid 𝜋√︁1/3ln𝑞 <= 𝐶 <= 𝜋√︁2/3ln𝑞 .
The results obtained are similar to those previously obtained by the authors when solving the inverse problem for monoids generated by an arbitrary exponential sequence of primes of type 𝑞.

For basic monoids 𝑀(P(𝑞)) of the type 𝑞, the question remains open about the existence of a 𝐶 logarithmic 1/2 -power density and the value of the constant 𝐶.

1. Bombieria E., Ghoshb A., 2011, “Around the Davenport–Heilbronn function”, Uspekhi Mat.

2. Nauk, 66:2(398) pp. 15–66.

3. Bredikhin, B.M., 1960, “The remainder term in the asymptotic formula for the function 𝜈𝐺(𝑥)”,

4. Izvestiya vuzov Matematika, no. 6, pp. 40–49.

5. Bredikhin, B.M., 1960, “An elementary solution of inverse problems on bases of free semigroups”,

6. matematicheskiy sbornik, 50(92):2, pp. 221–232.

7. Bredikhin, B.M., 1958, “Free numerical semigroups with power densities”, Doklady Akademii

8. nauk SSSR, 118:5, pp. 855–857.

9. Bredikhin, B.M., 1958, “On power densities of some subsets of free semigroups”, Izvestiya vuzov

10. Matematika, no. 3, pp. 24–30.

11. Bredikhin, B.M., 1958, “Free numerical semigroups with power densities”, matematicheskiy

12. sbornik, 46(88):2, pp. 143–158.

13. Bredikhin, B.M., 1956, “An example of a finite homomorphism with a bounded adder function”,

14. UMN, 11:4(70), pp. 119–122.

15. Bredikhin, B.M., 1956, “Some questions of the theory of characters of commutative semigroups”,

16. Trudy 3-go Vsesoyuznogo matematicheskogo s’yezda, vol. 1, Moskva, izdatel’stvo akademii nauk

17. SSSR, no. 3.

18. Bredikhin, B.M., 1954, “On adder functions of characters of numerical semigroups”, DAN 94,

19. pp. 609 – 612.

20. Bredikhin, B.M., 1953, “On the characters of numerical semigroups with a rather rare base”,

21. DAN 90, pp. 707–710.

22. Voronin S. M., Karacuba A. A., 1994, Dzeta-funkcija Rimana, Izd-vo Fiz-matlit, Moskva, 376 p.

23. Gurvic A., Kurant R., 1968, Teorija funkcij, Izd-vo Nauka, Moskva, 618 p.

24. Demidov S. S., Morozova E. A., Chubarikov V. N., Rebrov I. Yu., Balaba I. N., Dobrovol’skii

25. N. N., Dobrovol’skii N. M., Dobrovol’skaya L. P., Rodionov A. V., Pikhtil’kova O. A.,

26. , "Number-theoretic method in approximate analysis" Chebyshevskii Sbornik vol. 18, № 4.

27. pp. 6–85.

28. M. N. Dobrovol’skii, N. N. Dobrovol’skii, N. M. Dobrovol’skii, I. B. Koguhov, I. Yu. Rebrova,

29. , “Monoid of pro ducts of zeta functions of monoids of natural numb ers” , Chebyshevskii

30. sbornik, vol. 23, no. 3, pp. 102–117.

31. Dobrovolsky N. N., 2017, "The zeta-function is the monoid of natural numbers with unique

32. factorization" , Chebyshevskii Sbornik, vol 18, № 4 pp. 188–208.

33. N. N. Dobrovol’skii, 2018, "On monoids of natural numbers with unique factorization into prime

34. elements" , Chebyshevskii sbornik, vol. 19, no. 1, pp. 79–105.

35. N. N. Dobrovol’skii, 2018, "The zeta function of monoids with a given abscissa of absolute

36. convergence" , Chebyshevskii sbornik, vol. 23, no. 5, pp. 142–150.

37. Добровольский Н. Н. Одна модельная дзета-функция моноида натуральных чисел // Че-

38. бышевcкий сборник. 2019. — Т. 20, вып. 1, С. 148–163.

39. Н. Н. Добровольский, “Об абсциссе абсолютной сходимости одного класса обобщенных

40. произведений Эйлера”, Матем. заметки, 109:3 (2021), 464–469

41. N. N. Dobrovol’skii, 2022, “Distribution of simple elements in some monoids of natural numbers”,

42. Math. Notes (in print).

43. N. N. Dobrovol’skii, M. N. Dobrovol’skii, N. M. Dobrovol’skii, I. N. Balaba, I. Yu. Rebrova, 2018,

44. "About «zagrobelna the series» for the zeta function of monoids with exponential sequence of

45. simple" , Chebyshevskii sbornik, vol. 19, no. 1, pp. 106–123.

46. N. N. Dobrovol’skii, M. N. Dobrovol’skii, N. M. Dobrovol’skii, I. N. Balaba, I. Yu. Rebrova,

47. , "Dirichlet series algebra of a monoid of natural numbers" , Chebyshevskii sbornik, vol. 20,

48. no. 1, pp. 180–196.

49. N. N. Dobrovol’skii, N. M. Dobrovol’skii, I. Yu. Rebrova, A. V. Rodionov, 2019, "Monoids

50. of natural numbers in the numerical-theoretical method in the approximate analysis" ,

51. Chebyshevskii sbornik, vol. 20, no. 1, pp. 164–179.

52. N. N. Dobrovol’skii, A. O. Kalinina, M. N. Dobrovol’skii, N. M. Dobrovol’skii 2018, "On the

53. number of prime elements in certain monoids of natural numbers" , Chebyshevskii sbornik,

54. vol. 19, no. 2, pp. 123–141.

55. N. N. Dobrovol’skii, A. O. Kalinina, M. N. Dobrovol’skii, N. M. Dobrovol’skii 2018, "On the

56. monoid of quadratic residues" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 95–108.

57. Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol’skii N. M., Dobrovolsky N. N., 2014,

58. "On Hyperbolic Zeta Function of Lattices" , In: Continuous and Distributed Systems. Solid

59. Mechanics and Its Applications, Vol. 211. pp. 23–62. DOI:10.1007/978-3-319-03146-0_2.

60. N. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2020, "Inverse problem for a monoid

61. with an exponential sequence of Prime numbers" , Chebyshevskii sbornik, vol. 21, no. 1, pp.

62. –185.

63. N. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2022, "The inverse problem for a basic

64. monoid of type 𝑞" , Chebyshevskii sbornik, vol. 23, no. 4, pp. 59–71.

65. D. S. Minenkov, V. E. Nazaikinskii, 2016, “Remark on the Inverse Abstract Prime Number

66. Theorem”, Math. Notes, 100:4, P. 633–635.

67. Postnikov, A. G., 1971, Introduction to analytical number theory Izd-vo "Nauka" , Moskva,

68. p.

69. Titchmarsh E. K., 1952, Teorija dzeta-funkcii Rimana Izd-vo I-L, Moskva, 407 p.

70. Trost E., 1959, "Prime numbers" , Izd-vo Fiz-matlit, Moskva, 136 p.

71. Chandrasekharan K., 1974, Vvedenie v analiticheskuju teoriju chisel, Izd-vo Mir, Moskva, 188 p.

72. Chudakov N. G., 1947, Introduction to the theory of 𝐿-Dirichlet functions — M.-L.: OGIZ, —

73. p.

74. Davenport H., Heilbronn H., 1936, "On the zeros of certain Dirichlet series" , J. London Math.

75. Soc. Vol. 11. pp. 181–185.