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Dirichlet series for a cubic module of rank 6.
1

%I #7 Aug 28 2022 18:48:13

%S 1,0,0,8,6,0,0,0,10,0,24,0,0,0,0,32,0,0,40,48,0,0,0,0,30,0,0,0,60,0,

%T 64,0,0,0,0,80,0,0,0,0,84,0,0,192,60,0,0,0,51,0,0,0,0,0,144,0,0,0,120,

%U 0,124,0,0,130,0,0,0,0,0,0,144,0,0,0,0,320,0,0,160,192,91,0,0,0,0,0,0,0,180,0,0,0,0,0,240,0,0,0,240,239,204,0,0,0,0,0,0,0,220,0,0,0,0,0,0,480,0,0,0,0,405

%N Dirichlet series for a cubic module of rank 6.

%C Submitted as a substitute for A031365 which appears to display a faulty A031365(16)=24 in the version published 1997.

%H M. Baake, <a href="https://arxiv.org/abs/math/0605222">Solution of the coincidence problem in dimensions d<=4</a>, arxiv:math/0605222 (2006), (5.12)

%p read("transforms") :

%p # expansion of 1/(1-5^(-s)) in (5.10)

%p L1 := [1,seq(0,i=2..200)] :

%p for k from 1 do

%p if 5^k <= nops(L1) then

%p L1 := subsop(5^k=1,L1) ;

%p else

%p break ;

%p end if;

%p end do:

%p # multiplication with 1/(1-p^(-2s)) in (5.10)

%p for i from 1 do

%p p := ithprime(i) ;

%p if modp(p,5) = 2 or modp(p,5)=3 then

%p Laux := [1,seq(0,i=2..200)] :

%p for k from 1 do

%p if p^(2*k) <= nops(Laux) then

%p Laux := subsop(p^(2*k)=1,Laux) ;

%p else

%p break ;

%p end if;

%p end do:

%p L1 := DIRICHLET(L1,Laux) ;

%p end if;

%p if p > nops(L1) then

%p break;

%p end if;

%p end do:

%p # multiplication with 1/(1-p^(-s))^2 in (5.10)

%p for i from 1 do

%p p := ithprime(i) ;

%p if modp(p,5) = 1 or modp(p,5)=4 then

%p Laux := [1,seq(0,i=2..200)] :

%p for k from 1 do

%p if p^k <= nops(Laux) then

%p Laux := subsop(p^k=k+1,Laux) ;

%p else

%p break ;

%p end if;

%p end do:

%p L1 := DIRICHLET(L1,Laux) ;

%p end if;

%p if p > nops(L1) then

%p break;

%p end if;

%p end do:

%p # this is now Zeta_L(s), seems to be A035187

%p # print(L1) ;

%p # generate Zeta_L(s-1)

%p L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;

%p # generate 1/Zeta_L(s)

%p L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :

%p taylor(1/L1x,x=0,nops(L1)) :

%p L1i := gfun[seriestolist](%) ;

%p # generate 1/Zeta_L(2s)

%p L1i2 := [1,seq(0,i=2..nops(L1))] ;

%p for k from 2 to nops(L1i) do

%p if k^2 < nops(L1i2) then

%p L1i2 := subsop(k^2=op(k,L1i),L1i2) ;

%p else

%p break ;

%p end if;

%p end do:

%p # generate Zeta_L(s)*Zeta_L(s-1)

%p DIRICHLET(L1,L1shft) ;

%p # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)

%p L1 := DIRICHLET(%,L1i2) ;

%p # generate 1/(1+4^(-s))

%p Laux := [1,seq(0,i=2..nops(L1))] :

%p for k from 1 do

%p if 4^k <= nops(Laux) then

%p Laux := subsop(4^k=(-1)^k,Laux) ;

%p else

%p break;

%p end if ;

%p end do:

%p # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)/(1+4^(-s))

%p L1 := DIRICHLET(L1,Laux) ;

%p # generate 1+4^(1-s)

%p Laux := [1,seq(0,i=2..3),4,seq(0,i=5..nops(L1))] ;

%p DIRICHLET(L1,Laux) ; # _R. J. Mathar_, Mar 04 2018

%Y Cf. A031365.

%K nonn,less

%O 1,4

%A _R. J. Mathar_, Mar 04 2018