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A208133
Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k.
5
1, 2, 2, 8, 2, 4, 2, 12, 9, 4, 2, 16, 2, 4, 4, 31, 2, 18, 2, 16, 4, 4, 2, 24, 11, 4, 14, 16, 2, 8, 2, 42, 4, 4, 4, 72, 2, 4, 4, 24, 2, 8, 2, 16, 18, 4, 2, 62, 13, 22, 4, 16, 2, 28, 4, 24, 4, 4, 2, 32, 2, 4, 18, 90, 4, 8, 2, 16, 4, 8, 2, 108, 2, 4, 22, 16
OFFSET
1,2
COMMENTS
Level function l_tau^2(n) of Bhowmik and Wu.
Records occur at 1, 2, 4, 8, 12, 16, 32, 36, 64, 72, 108, 128, 144, 288, 432, 576, 1152, 1296, 2304, 3600, 5184, 7200, 9216, 10368, 14112, 14400, 20736, 28224, 28800, 32400, 57600, ... and they are: 1, 2, 8, 12, 16, 31, 42, 72, 90, 108, 112, 116, 279, 378, 434, 810, 1044, 1302, 2025, 3069, 3780, 4158, 4644, 4872, 4914, 8910, 9450, 10530, 11484, 14322, 22275, ... - Antti Karttunen, Mar 21 2018
REFERENCES
A. Laurincikas, The universality of Dirichlet series attached to finite Abelian groups, in "Number Theory", Proc. Turku Sympos. on Number Theory, May 31-June 4, 1999, p 179.
LINKS
G. Bhowmik, Jie Wu, Zeta function of subgroups of abelian groups and average orders, J. reine angew. Math. 530 (2001) 1-15.
FORMULA
Dirichlet g.f.: zeta(s)^2*zeta(2s)^2*zeta(2s-1)*Product_{primes p} (1 + 1/p^(2s) - 2/p^(3s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * log(n)^2 * n / 144, where c = A330594 = Product_{primes p} (1 + 1/p^2 - 2/p^3) = 1.10696011195321767665117913000743959294954883365812241904313404497877733241... - Vaclav Kotesovec, Dec 18 2019
More precise asymptotics: Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 2/p^(3*s)), then Sum_{k=1..n} a(k) ~ n*Pi^2 * (Pi^2 * f(1) * log(n)^2 + 2*Pi^2 * log(n) * (f(1) * (-1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) + f'(1)) + Pi^2 * (2*f(1)*(1 + 25*gamma^2 + 576*log(A)^2 + log(A) * (48 - 96*log(2*Pi)) - 8*gamma * (1 + 36*log(A) - 3*log(2*Pi)) - 4*log(2*Pi) + 4*log(2*Pi)^2 - 6*sg1) + 2*(-1 + 8*gamma - 48*log(A) + 4*log(2*Pi))*f'(1) + f''(1)) + 48*f(1)*zeta''(2)) / 144, where f(1) = A330594, f'(1) = A330594 * (-A335705) = f(1) * Sum_{primes p} = -2*(p-3) * log(p) / (p^3 + p - 2) = -0.087825458097278818094375273108270679512035928574..., f''(1) = A330594 * (A335705^2 + A335706) = f'(1)^2/f(1) + f(1) * Sum_{primes p} = 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2) / (p^3 + p - 2)^2 = 0.26722508718782634450711076996710402451611235402675360769..., zeta''(2) = A201994, A is the Glaisher-Kinkelin constant A074962, gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 18 2020
MAPLE
L300828 := [ 1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
] ;
L010052 := [ 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];
L037213 := [ 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ;
Lx := DIRICHLET(L300828, L037213) ;
Lx := DIRICHLET(Lx, L010052) ;
Lx := DIRICHLET(Lx, L010052) ;
Lx := MOBIUSi(Lx) ;
Lx := MOBIUSi(Lx) ;
# Name of initial list L1 changed to L300828 to refer to sequence A300828 by Antti Karttunen, Mar 21 2018
PROG
(PARI)
A037213(n) = if(issquare(n), sqrtint(n), 0);
A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1, d, if(v[i, 2] < 2 || v[i, 2] > 3, return(0)); if (v[i, 2] == 3, val *= -2)); return(val); };
a208133s1(n) = sumdiv(n, d, A037213(n/d)*A300828(d));
a208133s2(n) = sumdiv(n, d, issquare(n/d)*a208133s1(d));
a208133s3(n) = sumdiv(n, d, issquare(n/d)*a208133s2(d));
a208133s4(n) = sumdiv(n, d, a208133s3(d));
A208133(n) = sumdiv(n, d, a208133s4(d)); \\ Antti Karttunen, Mar 21 2018, after R. J. Mathar's Maple code
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X + 2*X^2)/(1 - X)^3/(1 + X)^2/(1 - p*X^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 18 2020
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
R. J. Mathar, Mar 29 2012
STATUS
approved