About «zagrobelna the series» for the zeta function of monoids with exponential sequence of simple

The work continues the study of a new class of Dirichlet series — Zeta function of monoids of natural numbers. First of all, we study in detail the Zeta function ζ(M(q)|α) of geometric progress M(q) with the first term equal to 1 and an arbitrary natural denominator q > 1, which is the simplest monoid of natural numbers with a unique decomposition into simple elements of the monoid. For a meromorphic function ζ(M(q)|α) = qα /qα−1 with many poles S(M(q)) ={︂2πik lnq⃒ ⃒ ⃒ ⃒k ∈Z}︂

representations are received:

ζ(M(q)|α) =qα 2 αlnq∞ ∏︁ n=1(︂1 + α2 ln2 q 4π2n2 )︂−1 = 1 2+1 αlnq

+∞ ∑︁ n=12αlnq α2 ln2 q + 4n2π2==qα 2 αlnq 4π2

Γ(︂αilnq 2π )︂Γ(︂−αilnq 2π )︂.

For the Zeta function ζ(M(p~)|α) of the monoid M(p~) with a finite number of primesp~ = (p1,...,pn) the decomposition into an infinite product is obtained

ζ(M(p~)|α) =P(p~)α 2 αnQ(p~)n ∏︁ ν=1∞ ∏︁ m=1(︂1 + α2 ln2 pν 4π2m2 )︂−1 ,

where P(p~) = p1 ...pn, Q(p~) = lnp1 ...lnpn, and a functional equation is found ζ(M(p~)|−α) = (−1)n ζ(M(p~)|α) P(p~)α .

For the monoid of positive integers M*(p~) = N·M−1(p~) with a unique Prime factorization consisting of positive integers n mutually Prime with P(p~) = p1 ...pn, and for the Euler product P(M*(p~)/ alpha), consisting of factors for all primes other than p1,...,pn, a functional equation is found

ζ(M*(p~)|α) = M(p~,α)ζ(M*(p~)|1−α),

where

M(p~,α) = M(α)· M1(p~,α) M1(p~,1−α) , M1(p~,α) = n ∏︁ ν=1(︂1− 1 pα ν)︂.

It is proved that for any infinite set of Prime P1 there is no analytic function equal to

lim /n→∞ζ(M(p~n)|α)

on the whole complex plane.

The protective series conjecture is formulated for any exponential set of PE primes.

In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

1. Bombieria E., Ghoshb A., 2011, “Around the Davenport–Heilbronn function”, Uspekhi Mat. Nauk, 66:2(398) pp. 15–66.

2. Voronin S. M., 2006, Izbrannye trudy: Matematika. Pod red. A. A. Karacuby, Izd-vo MGTU im. N. Je. Baumana, Moskva, 480 p.

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

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

5. Demidov S. S., Morozova E. A., Chubarikov V. N., Rebrov I. Yu., Balaba I. N., Dobrovol’skii N. N., Dobrovol’skii N. M., Dobrovol’skaya L. P., Rodionov A. V., Pikhtil’kova O. A., 2017, "Number-theoretic method in approximate analysis" Chebyshevskii Sbornik vol. 18, № 4. pp. 6–85.

6. Dobrovol’skaja L. P., Dobrovol’skij M. N., Dobrovol’skij N. M., Dobrovol’skij N. N., 2012, "Giperbolicheskie dzeta-funkcii setok i reshjotok i vychislenie optimal’nyh kojefficientov" Chebyshevskii Sbornik vol 13, №4(44) pp. 4–107.

7. Dobrovol’skij M. N., 2007, "Funkcional’noe uravnenie dlja giperbolicheskoj dzeta-funkcii celochislennyh reshetok", Doklady akademii nauk, vol 412, № 3, pp. 302–304.

8. Dobrovolsky N. M., Dobrovolsky N. N., Soboleva V. N., Sobolev D. K., Dobrovol’skaya L. P., Bocharova O. E., 2016, "On hyperbolic Hurwitz zeta function", Chebyshevskii Sbornik, vol. 17, № 3 pp. 72–105.

9. Dobrovolsky N. N., 2017, The zeta-function is the monoid of natural numbers with unique factorization Chebyshevskii Sbornik, vol. 18, № 4. P. 187–207.

10. Dobrovolsky N. N., 2018, "On monoids of natural numbers with unique factorization into prime elements", Chebyshevskii Sbornik, vol. 19, № 1. P.

11. Davenport H., 1971, Mul’tiplikativnaja teorija chisel, Izd-vo Nauka, Moskva, 200 p.

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

13. Whittaker E. T., Watson D. N., 1963, A Course in modern analysis. Part two. Transcendental function. — Moscow: Fizmatgiz, 516 p.

14. Shabat B. V., 1969, Introduction to complex analysis — M.: Science, — 576 p.

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

16. Davenport H., Heilbronn H., 1936, "On the zeros of certain Dirichlet series", J. London Math. Soc. Vol. 11. pp. 181–185.

17. Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol’skii N. M., Dobrovolsky N. N., 2014, "On Hyperbolic Zeta Function of Lattices", In: Continuous and Distributed Systems. Solid Mechanics and Its Applications, Vol. 211. pp. 23–62. DOI:10.1007/978-3-319-03146-0_2.