A306500 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(-A003657(n),q) > 0, or 0 if no such prime exists.

608981813029, 26861, 2, 3, 5, 2, 11, 3, 2, 5, 2, 11, 2, 11, 53, 2, 13, 17, 2, 3, 5, 163, 3, 2, 2, 11, 5, 2, 31, 31, 2, 2, 3, 23, 2, 41, 3, 2, 13, 47, 2, 5, 19, 7, 11, 2, 191, 2, 3, 19, 2, 15073, 3, 2, 29, 5, 2, 41, 109, 2, 11, 2, 31, 59, 3, 2, 19, 2, 11, 53, 2, 1019, 137

Offset: 1

Author

Jianing Song, Feb 19 2019

Keywords

• nonn

Comments

Let D be a fundamental discriminant (only the case where D is fundamental is considered because {Kronecker(D,k)} forms a primitive real Dirichlet character with period |D| if and only if D is fundamental), it seems that Sum_{primes q <= p} Kronecker(D,p) <= 0 occurs much more often than its opposite does. This can be seen as a variation of the well-known "Chebyshev's bias". Sequence gives the least prime that violates the inequality when D runs through all negative discriminants.

For any D, the primes p such that Kronecker(D,p) = 1 has asymptotic density 1/2 in all the primes, so a(n) should be > 0 for all n.

Actually, for most n, a(n) is relatively small compared with A003657(n). There are only 127 n's in [1, 3043] (there are 3043 terms in A003657 below 10000) such that a(n) > A003657(n). The largest terms among the 127 corresponding terms are a(1) = 608981813029 (with A003657(1) = 3), a(1955) = 24996194023 (with A003657(1955) = 6240) and a(847) = 1694759239 (with A003657(847) = 2787).

Examples

			Let D = -A003657(18) = -52, j(k) = Sum_{primes q <= prime(k)} Kronecker(D,q).
For k = 1, Kronecker(-52,2) = 0, so j(1) = 0;
For k = 2, Kronecker(-52,3) = -1, so j(2) = -1;
For k = 3, Kronecker(-52,5) = -1, so j(3) = -2;
For k = 4, Kronecker(-52,7) = +1, so j(4) = -1;
For k = 5, Kronecker(-52,11) = +1, so j(5) = 0;
For k = 6, Kronecker(-52,13) = 0, so j(6) = 0;
For k = 7, Kronecker(-52,17) = +1, so j(7) = 1.
The first time for j > 0 occurs at the prime 17, so a(18) = 17.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..3043
• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A003657, A306499 (the positive discriminants case).

The indices of primes are given in A306503.

Programs

PARI

b(n) = my(i=0); forprime(p=2,oo,i+=kronecker(n,p); if(i>0, return(p)))
print1(608981813029, ", "); for(n=4, 300, if(isfundamental(-n), print1(b(-n), ", ")))