A160499 Number of quartic primitive Dirichlet characters modulo n.

1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 3, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 3, 6, 3, 0, 1, 1, 0, 0, 1, 4, 0, 0, 3, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 3, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 12, 0

Offset: 1

Author

Steven Finch, May 15 2009

Keywords

• mult
• nonn

Comments

Also called biquadratic primitive Dirichlet characters.

Primitive Dirichlet characters of both order 2 & order 4 are included.

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fourth-power root of unity (1, i, -1 and -i). - Jianing Song, Feb 27 2019

Mobius transform of A073103. - Jianing Song, Mar 02 2019

Examples

			From _Jianing Song_, Mar 02 2019: (Start)
For n = 5, the 3 quartic primitive Dirichlet characters modulo n are [0, 1, -1, -1, 1], [0, 1, i, -i, -1] and [0, 1, -i, i, -1], so a(5) = 3.
For n = 16, the 4 quartic primitive Dirichlet characters modulo n are [0, 1, 0, i, 0, i, 0, 1, 0, -1, 0, -i, 0, -i, 0, -1], [0, 1, 0, -i, 0, -i, 0, 1, 0, -1, 0, i, 0, i, 0, -1], [0, 1, 0, i, 0, -i, 0, -1, 0, -1, 0, -i, 0, i, 0, 1] and [0, 1, 0, -i, 0, i, 0, -1, 0, -1, 0, i, 0, -i, 0, 1], so a(16) = 4. (End)
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Cubic and quartic characters.
• Steven R. Finch, Cubic and quartic characters.
• Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2016.

Crossrefs

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160498 (number of cubic primitive Dirichlet characters modulo n).

Cf. A073103 (number of solutions to x^4 == 1 (mod n)).

Cf. A064533.

Programs

Mathematica

f[n_] := Sum[If[Mod[k^4 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*f[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-François Alcover, Jun 19 2013 *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e == 4, 4, e >= 5, 0]; f[p_, 1] := If[Mod[p, 4] == 1, 3, 1]; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

PARI

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^4-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

Formula

Multiplicative with a(4) = 1, a(8) = 2, a(16) = 4, a(2^e) = 0 for e = 1 or e >= 5; for odd primes p, a(p) = 3 if p == 1 (mod 4) and 1 if p == 3 (mod 4), a(p^e) = 0 for e >= 2. - Jianing Song, Mar 02 2019

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (7/(16*Pi*K^2)) * Product_{primes p == 1 (mod 4)} (1 - (5*p-3)/(p^2*(p+1))) = 0.1908767211685284480112237..., and K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Sep 16 2020

Extensions

a(1) = 1 prepended by Jianing Song, Feb 27 2019