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A300382 Dirichlet series for a cubic module of rank 6.

%I A300382 #7 Aug 28 2022 18:48:13
%S A300382 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 A300382 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 A300382 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 A300382 Dirichlet series for a cubic module of rank 6.
%C A300382 Submitted as a substitute for A031365 which appears to display a faulty A031365(16)=24 in the version published 1997.
%H A300382 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 A300382 read("transforms") :
%p A300382 # expansion of 1/(1-5^(-s)) in (5.10)
%p A300382 L1 := [1,seq(0,i=2..200)] :
%p A300382 for k from 1 do
%p A300382     if 5^k <= nops(L1) then
%p A300382         L1 := subsop(5^k=1,L1) ;
%p A300382     else
%p A300382         break ;
%p A300382     end if;
%p A300382 end do:
%p A300382 # multiplication with 1/(1-p^(-2s)) in (5.10)
%p A300382 for i from 1 do
%p A300382     p := ithprime(i) ;
%p A300382     if modp(p,5) = 2 or modp(p,5)=3 then
%p A300382         Laux := [1,seq(0,i=2..200)] :
%p A300382         for k from 1 do
%p A300382             if p^(2*k) <= nops(Laux) then
%p A300382                 Laux := subsop(p^(2*k)=1,Laux) ;
%p A300382             else
%p A300382                 break ;
%p A300382             end if;
%p A300382         end do:
%p A300382         L1 := DIRICHLET(L1,Laux) ;
%p A300382     end if;
%p A300382     if p > nops(L1) then
%p A300382         break;
%p A300382     end if;
%p A300382 end do:
%p A300382 # multiplication with 1/(1-p^(-s))^2 in (5.10)
%p A300382 for i from 1 do
%p A300382     p := ithprime(i) ;
%p A300382     if modp(p,5) = 1 or modp(p,5)=4 then
%p A300382         Laux := [1,seq(0,i=2..200)] :
%p A300382         for k from 1 do
%p A300382             if p^k <= nops(Laux) then
%p A300382                 Laux := subsop(p^k=k+1,Laux) ;
%p A300382             else
%p A300382                 break ;
%p A300382             end if;
%p A300382         end do:
%p A300382         L1 := DIRICHLET(L1,Laux) ;
%p A300382     end if;
%p A300382     if p > nops(L1) then
%p A300382         break;
%p A300382     end if;
%p A300382 end do:
%p A300382 # this is now Zeta_L(s), seems to be A035187
%p A300382 # print(L1) ;
%p A300382 # generate Zeta_L(s-1)
%p A300382 L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
%p A300382 # generate 1/Zeta_L(s)
%p A300382 L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
%p A300382 taylor(1/L1x,x=0,nops(L1)) :
%p A300382 L1i := gfun[seriestolist](%) ;
%p A300382 # generate 1/Zeta_L(2s)
%p A300382 L1i2 := [1,seq(0,i=2..nops(L1))] ;
%p A300382 for k from 2 to nops(L1i) do
%p A300382     if k^2 < nops(L1i2) then
%p A300382         L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
%p A300382     else
%p A300382         break ;
%p A300382     end if;
%p A300382 end do:
%p A300382 # generate Zeta_L(s)*Zeta_L(s-1)
%p A300382 DIRICHLET(L1,L1shft) ;
%p A300382 # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)
%p A300382 L1 := DIRICHLET(%,L1i2) ;
%p A300382 # generate 1/(1+4^(-s))
%p A300382 Laux := [1,seq(0,i=2..nops(L1))] :
%p A300382 for k from 1 do
%p A300382     if 4^k <= nops(Laux) then
%p A300382         Laux := subsop(4^k=(-1)^k,Laux) ;
%p A300382     else
%p A300382         break;
%p A300382     end if ;
%p A300382 end do:
%p A300382 # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s)/(1+4^(-s))
%p A300382 L1 := DIRICHLET(L1,Laux) ;
%p A300382 # generate 1+4^(1-s)
%p A300382 Laux := [1,seq(0,i=2..3),4,seq(0,i=5..nops(L1))] ;
%p A300382 DIRICHLET(L1,Laux) ; # _R. J. Mathar_, Mar 04 2018
%Y A300382 Cf. A031365.
%K A300382 nonn,less
%O A300382 1,4
%A A300382 _R. J. Mathar_, Mar 04 2018