Geodesic
The inverse problem for a basic monoid of type $q$
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 64-76.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper for an arbitrary basic monoid ${M(\mathbb{P}(q))}$ of type $q$ the inverse problem is solved, that is, finding the asymptotics for the distribution function of the elements of the monoid ${M(\mathbb{P}(q))}$, based on the asymptotics of the distribution of pseudo-prime numbers $\mathbb{P}(q)$ of type $q$. To solve this problem, we consider two homomorphisms of the main monoid ${M(\mathbb{P}(q))}$ 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 $\theta$-power density is introduced. It is shown that any monoid ${M(\mathbb{P}(q))}$ for a sequence of pseudo-simple numbers $\mathbb{P}(q)$ of type $q$ has upper and lower bounds for the element distribution function of the main monoid ${M(\mathbb{P}(q))}$ of type $q$. It is shown that if $C$ is a logarithmic $\theta$-power density for the main monoid ${M(\mathbb{P}(q))}$ of the type $q$ exists, then $\theta=\frac{1}{2}$ and for the constant $C$ the inequalities are valid $ \pi\sqrt{\frac{1}{3\ln q}}\le C\le \pi\sqrt{\frac{2}{3\ln q}}. $ 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 ${M(\mathbb{P}(q))}$ of the type $q$, the question remains open about the existence of a $C$ logarithmic $\frac{1}{2}$-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 ${M(\mathbb{P}(q))}$ of type $q$, $C$ logarithmic $\theta$-power density. 
@article{CHEB_2022_23_4_a5,
     author = {N. N. Dobrovol'skii and I. Yu. Rebrova and N. M. Dobrovol'skii},
     title = {The inverse problem for a basic monoid of type $q$},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {64--76},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a5/}
}
TY  - JOUR
AU  - N. N. Dobrovol'skii
AU  - I. Yu. Rebrova
AU  - N. M. Dobrovol'skii
TI  - The inverse problem for a basic monoid of type $q$
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 64
EP  - 76
VL  - 23
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a5/
LA  - ru
ID  - CHEB_2022_23_4_a5
ER  - 
%0 Journal Article
%A N. N. Dobrovol'skii
%A I. Yu. Rebrova
%A N. M. Dobrovol'skii
%T The inverse problem for a basic monoid of type $q$
%J Čebyševskij sbornik
%D 2022
%P 64-76
%V 23
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a5/
%G ru
%F CHEB_2022_23_4_a5
N. N. Dobrovol'skii; I. Yu. Rebrova; N. M. Dobrovol'skii. The inverse problem for a basic monoid of type $q$. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 64-76. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a5/

[1] E. Bomberi, A. Gosh, “Vokrug funktsii Devenporta-Kheilbronna”, UMN, 66:2(398) (2011), 15–66 | DOI | MR

[2] B. M. Bredikhin, “Ostatochnyi chlen v asimptoticheskoi formule dlya funktsii $\nu_G(x)$”, Izv. vuzov. Matem., 1960, no. 6, 40–49 | Zbl

[3] B. M. Bredikhin, “Elementarnoe reshenie obratnykh zadach o bazisakh svobodnykh polugrupp”, Matem. sb., 50(92):2 (1960), 221–232 | Zbl

[4] B. M. Bredikhin, “Svobodnye chislovye polugruppy so stepennymi plotnostyami”, Dokl. AN SSSR, 118:5 (1958), 855–857

[5] B. M. Bredikhin, “O stepennykh plotnostyakh nekotorykh podmnozhestv svobodnykh polugrupp”, Izv. vuzov. Matem., 1958, no. 3, 24–30 | Zbl

[6] B. M. Bredikhin, “Svobodnye chislovye polugruppy so stepennymi plotnostyami”, Matem. sb., 46(88):2 (1958), 143–158 | Zbl

[7] B. M. Bredikhin, “Primer konechnogo gomomorfizma s ogranichennoi summatornoi funktsiei”, UMN, 11:4(70) (1956), 119–122 | MR

[8] B. M. Bredikhin, “Nekotorye voprosy teorii kharakterov kommutativnykh polugrupp”, Trudy 3-go Vsesoyuzn. matem. s'ezda, v. I, Izd. AN SSSR, M., 1956, 3

[9] B. M. Bredikhin, “O summatornykh funktsiyakh kharakterov chislovykh polugrupp”, DAN, 94 (1954), 609–612

[10] B. M. Bredikhin, “O kharakterakh chislovykh polugrupp s dostatochno redkoi bazoi”, DAN, 90 (1953), 707–710

[11] Voronin S. M., Karatsuba A. A., Dzeta-funktsiya Rimana, Fizmatlit, M., 1994, 376 pp. | MR

[12] Gurvits A., Kurant R., Teoriya funktsii, Nauka, M., 1968, 618 pp.

[13] Demidov S. S., Morozova E. A., Chubarikov V. N., Rebrova I. Yu., Balaba I. N., Dobrovolskii N. N., Dobrovolskii N. M., Dobrovolskaya L. P., Rodionov A. V., Pikhtilkova O. A., “Teoretiko-chislovoi metod v priblizhennom analize”, Chebyshevskii sb., 18:4 (2017), 6–85 | DOI | MR | Zbl

[14] M. N. Dobrovolskii, N. N. Dobrovolskii, N. M. Dobrovolskii, I. B. Kozhukhov, I. Yu. Rebrova, “Monoid proizvedenii dzeta-funktsii monoidov naturalnykh chisel”, Chebyshevckii sbornik, 23:3 (2022), 102–117 | DOI | MR

[15] N. N. Dobrovolskii, “Dzeta-funktsiya monoidov naturalnykh chisel s odnoznachnym razlozheniem na prostye mnozhiteli”, Chebyshevskii sb., 18:4 (2017), 187–207 | DOI | MR

[16] Dobrovolskii N. N., “O monoidakh naturalnykh chisel s odnoznachnym razlozheniem na prostye elementy”, Chebyshevskii sb., 19:1 (2018), 79–105 | DOI | MR | Zbl

[17] Dobrovolskii N. N., “Dzeta-funktsiya monoidov s zadannoi abstsissoi absolyutnoi skhodimosti”, Chebyshevskii sb., 19:2 (2018), 142–150 | DOI | MR | Zbl

[18] Dobrovolskii N. N., “Odna modelnaya dzeta-funktsiya monoida naturalnykh chisel”, Chebyshevckii sbornik, 20:1 (2019), 148–163 | DOI | Zbl

[19] N. N. Dobrovolskii, “Ob abstsisse absolyutnoi skhodimosti odnogo klassa obobschennykh proizvedenii Eilera”, Matem. zametki, 109:3 (2021), 464–469 | DOI | MR | Zbl

[20] N. N. Dobrovolskii, “Raspredelenie prostykh elementov v nekotorykh monoidakh naturalnykh chisel”, Matem. zametki (to appear)

[21] Dobrovolskii N. N., Dobrovolskii M. N., Dobrovolskii N. M., Balaba I. N., Rebrova I. Yu., “Gipoteza o "zagraditelnom ryade" dlya dzeta-funktsii monoidov s eksponentsialnoi posledovatelnostyu prostykh”, Chebyshevskii sb., 19:1 (2018), 106–123 | DOI | MR | Zbl

[22] N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, I. N. Balaba, I. Yu. Rebrova, “Algebra ryadov Dirikhle monoida naturalnykh chisel”, Chebyshevckii sbornik, 20:1 (2019), 180–196 | MR | Zbl

[23] N. N. Dobrovolskii, N. M. Dobrovolskii, I. Yu. Rebrova, A. V. Rodionov, “Monoidy naturalnykh chisel v teoretiko-chislovom metode v priblizhennom analize”, Chebyshevckii sbornik, 20:1 (2019), 164–179 | DOI

[24] Dobrovolskii N. N., Kalinina A. O., Dobrovolskii M. N., Dobrovolskii N. M., “O kolichestve prostykh elementov v nekotorykh monoidakh naturalnykh chisel”, Chebyshevckii sbornik, 19:2 (2018), 123–141 | DOI | Zbl

[25] Dobrovolskii N. N., Kalinina A. O., Dobrovolskii M. N., Dobrovolskii N. M., “O monoide kvadratichnykh vychetov”, Chebyshevckii sbornik, 19:3 (2018), 95–108 | DOI | Zbl

[26] N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “Obratnaya zadacha dlya monoida s eksponentsialnoi posledovatelnostyu prostykh”, Chebyshevckii sbornik, 21:1 (2020), 165–185 | DOI | MR | Zbl

[27] A. G. Postnikov, Vvedenie v analiticheskuyu teoriyu chisel, Nauka, M., 1971, 416 pp.

[28] E. K. Titchmarsh, Teoriya dzeta-funktsii Rimana, IL, M., 1952, 407 pp.

[29] E. Trost, Prostye chisla, Fizmatlit, M., 1959, 136 pp.

[30] Chandrasekkharan K., Vvedenie v analiticheskuyu teoriyu chisel, Mir, M., 1974, 188 pp.

[31] Chudakov N. G., Vvedenie v teoriyu $L$-funktsii Dirikhle, OGIZ, M.–L., 1947, 204 pp. | MR

[32] H. Davenport, H. Heilbronn, “On the zeros of certain Dirichlet series”, J. London Math. Soc., 11 (1936), 181–185 | DOI | MR | Zbl