A141454 - OEIS

%I #9 Feb 17 2024 23:44:10

%S 0,2,0,-2,1,2,-2,-1,2,-1,1,-2,1,2,-2,2,-1,1,-2,1,2,-1,2,-1,-2,1,2,-2,

%T -1,2,-2,1,-2,-1,-1,1,-2,2,-2,2,-1,1,1,2,-2,-1,1,2,-2,-1,2,-1,1,1,-2,

%U 2,-1,1,-2,1,2,2,-2,1,2,-2,1,-2,-2,-1,2,-1,-2,2,-1,2,-1,-2,1,-1,-1,1,1,2,-1,2,-1,-2,1,2,-2,-1,-2,1,-1,2,-1,1,2,1

%N A Legendre symbol type assignment of the modulo ten primes to the polynomial: Expand[(x-1)*(x+1)*(x-2)*(x+2)*(x-0)]=4 x - 5 x^3 + x^5; c(n) = If[Mod[n, 10] == 1, 1, If[Mod[n, 10] == 9, -1, If[Mod[n, 10] == 3, 2, If[Mod[n, 10] == 7, -2, 0]]]] such that n is a prime[n].

%C Generalized L functions over polynomials modulo to the primes:

%C Instead of the x^2-1 roots modulo 4 of primes in the Legendre symbol

%C you identify each modulo value of the primes to a root

%C and define the function by that.

%C Modulo 10 of the primes has values {2,5,1,3,7,9};

%C assigning the roots of the polynomial:

%C 1->1

%C 9->-1

%C 3->2

%C 7->-2

%C 2,5->0

%C 5 root polynomial defined over modulo 10 in primes:

%C Expand[(x-1)*(x+1)*(x-2)*(x+2)*x]=4 x - 5 x^3 + x^5;

%C L[s_] = Product[1/(1 - c[Prime[n]]/Prime[n]), {n, 1, Infinity}];

%C Table[N[L[n]], {n, 1, 5}]

%C {1., ComplexInfinity, 1., 0., ComplexInfinity}

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArtinL-Function.html">Artin L-Function</a>.

%F c(n) = If[Mod[n, 10] == 1, 1, If[Mod[n, 10] == 9, -1, If[Mod[n, 10] == 3, 2, If[Mod[n, 10] == 7, -2, 0]]]]; such that n is a prime[n].

%t c[n_] = If[Mod[n, 10] == 1, 1, If[Mod[n, 10] == 9, -1, If[Mod[ n, 10] == 3, 2, If[Mod[n, 10] == 7, -2, 0]]]]; Table[c[Prime[n]], {n, 1, 100}]

%K uned,sign

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 07 2008