OFFSET
1,4
COMMENTS
LINKS
M. Baake, Solution of the coincidence problem in dimensions d<=4, arxiv:math/0605222 (2006), (5.12)
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
CROSSREFS
KEYWORD
nonn,less
AUTHOR
R. J. Mathar, Mar 04 2018
STATUS
approved