The inverse problem for a basic monoid of type 𝑞
https://doi.org/10.22405/2226-8383-2022-23-4-64-76
Abstract
In the paper for an arbitrary basic monoid 𝑀(P(𝑞)) of type 𝑞 the inverse problem is solved, that is, finding the asymptotics for the distribution function of the elements of the monoid 𝑀(P(𝑞)), based on the asymptotics of the distribution of pseudo-prime numbers P(𝑞) of type 𝑞.
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 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.
Keywords
Riemann zeta function,
Dirichlet series,
zeta function of the monoid of natural numbers,
Euler product,
exponential sequence of primes,
the basic monoid 𝑀(P(𝑞)) of type 𝑞,
𝐶 logarithmic 𝜃-power density.
Supporting agencies: Acknowledgments: This work was prepared under a grant from the RSF № 22-21-00544.
About the Authors
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University; Tula State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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For citations:
Dobrovol’skii N.N., Rebrova I.Yu., Dobrovol’skii N.M. The inverse problem for a basic monoid of type 𝑞. Chebyshevskii Sbornik. 2022;23(4):64-76. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-4-64-76
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