A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Jianing Song, Nov 20 2018

Keywords

• sign

Comments

a(n) is the number of odd primes <= n that have 3 as a quadratic nonresidue minus the number of primes <= n that have 3 as a quadratic residue.

The first 10000 terms are nonnegative. a(p) = 0 for primes p = 2, 3, 13, 433, 443, 457, 479, 491, 503, 3541, ... The earliest negative term is a(61463) = -1. Conjecturally infinitely many terms should be negative.

In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 19 2023]

Here, although 11 is not a quadratic residue modulo 12, for most n we have Pi(12,7)(n) + Pi(12,11)(n) > Pi(12,1)(n) - Pi(12,5)(n), Pi(12,5)(n) + Pi(12,11)(n) > Pi(12,1)(n) + Pi(12,7)(n) and Pi(12,5)(n) + Pi(12,7)(n) > Pi(12,1)(n) + Pi(12,11)(n).

Examples

			Pi(12,1)(100) = 5, Pi(12,5)(100) = Pi(12,7)(100) = Pi(12,11)(100) = 6, so a(100) = 6 + 6 - 5 - 6 = 1.
		

Links

• Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A007350, A110161.

Let d be a fundamental discriminant.

Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), this sequence (d=12).

Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Programs

PARI

a(n) = -sum(i=1, n, isprime(i)*kronecker(12, i))

Formula

a(n) = -Sum_{primes p<=n} Kronecker(12,p) = -Sum_{primes p<=n} A110161(p).