A141454 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].

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, -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, 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

Offset

1,2

Comments

Generalized L functions over polynomials modulo to the primes:

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

you identify each modulo value of the primes to a root

and define the function by that.

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

assigning the roots of the polynomial:

1->1

9->-1

3->2

7->-2

2,5->0

5 root polynomial defined over modulo 10 in primes:

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

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

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

{1., ComplexInfinity, 1., 0., ComplexInfinity}

Links

Table of n, a(n) for n=1..100.

Eric Weisstein's World of Mathematics, Artin L-Function.

Formula

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].

Mathematica

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}]

Crossrefs

Sequence in context: A305576 A129679 A319514 * A301564 A334361 A360568

Adjacent sequences: A141451 A141452 A141453 * A141455 A141456 A141457

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Aug 07 2008

Status

approved