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A160499 Number of quartic primitive Dirichlet characters modulo n. 8

%I #40 Sep 16 2020 05:05:35

%S 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,

%T 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,

%U 1,0,1,0,3,0,0,1,1,0,1,12,0

%N Number of quartic primitive Dirichlet characters modulo n.

%C Also called biquadratic primitive Dirichlet characters.

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

%C 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

%C Mobius transform of A073103. - _Jianing Song_, Mar 02 2019

%H Jianing Song, <a href="/A160499/b160499.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="https://web.archive.org/web/20160511040501/http://www.people.fas.harvard.edu/~sfinch/csolve/char.pdf">Cubic and quartic characters</a>.

%H Steven R. Finch, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.638.5547">Cubic and quartic characters</a>.

%H Steven R. Finch, <a href="https://arxiv.org/abs/0907.4894">Quartic and Octic Characters Modulo n</a>, arXiv:0907.4894 [math.NT], 2016.

%F 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

%F 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

%e From _Jianing Song_, Mar 02 2019: (Start)

%e 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.

%e 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)

%t 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 *)

%t 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 *)

%o (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

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

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

%Y Cf. A064533.

%K mult,nonn

%O 1,5

%A _Steven Finch_, May 15 2009

%E a(1) = 1 prepended by _Jianing Song_, Feb 27 2019

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)