A300382 Dirichlet series for a cubic module of rank 6.

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, 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, 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

Offset: 1

Author

R. J. Mathar, Mar 04 2018

Keywords

• nonn
• less

Comments

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

Links

• M. Baake, Solution of the coincidence problem in dimensions d<=4, arxiv:math/0605222 (2006), (5.12)

Crossrefs

Cf. A031365.

Programs

Maple

read("transforms") :
# expansion of 1/(1-5^(-s)) in (5.10)
L1 := [1,seq(0,i=2..200)] :
for k from 1 do
    if 5^k <= nops(L1) then
        L1 := subsop(5^k=1,L1) ;
    else
        break ;
    end if;
end do:
# multiplication with 1/(1-p^(-2s)) in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 2 or modp(p,5)=3 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^(2*k) <= nops(Laux) then
                Laux := subsop(p^(2*k)=1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# multiplication with 1/(1-p^(-s))^2 in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 1 or modp(p,5)=4 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^k <= nops(Laux) then
                Laux := subsop(p^k=k+1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# this is now Zeta_L(s), seems to be A035187
# print(L1) ;
# generate Zeta_L(s-1)
L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
# generate 1/Zeta_L(s)
L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
taylor(1/L1x,x=0,nops(L1)) :
L1i := gfun[seriestolist](%) ;
# generate 1/Zeta_L(2s)
L1i2 := [1,seq(0,i=2..nops(L1))] ;
for k from 2 to nops(L1i) do
    if k^2 < nops(L1i2) then
        L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
    else
        break ;
    end if;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)
DIRICHLET(L1,L1shft) ;
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)
L1 := DIRICHLET(%,L1i2) ;
# generate 1/(1+4^(-s))
Laux := [1,seq(0,i=2..nops(L1))] :
for k from 1 do
    if 4^k <= nops(Laux) then
        Laux := subsop(4^k=(-1)^k,Laux) ;
    else
        break;
    end if ;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)/(1+4^(-s))
L1 := DIRICHLET(L1,Laux) ;
# generate 1+4^(1-s)
Laux := [1,seq(0,i=2..3),4,seq(0,i=5..nops(L1))] ;
DIRICHLET(L1,Laux) ; # R. J. Mathar, Mar 04 2018