A320857 a(n) = Pi(8,5)(n) + Pi(8,7)(n) - Pi(8,1)(n) - Pi(8,3)(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, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2

Offset: 1

Author

Jianing Song, Nov 24 2018

Keywords

• sign

Comments

a(n) is the number of odd primes <= n that have -2 as a quadratic nonresidue minus the number of primes <= n that have -2 as a quadratic residue.

It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 212 ones are 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 18 2023]

Here, although 3 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n).

Examples

			Pi(8,1)(200) = 8, Pi(8,5)(200) = 13, Pi(8,3)(200) = Pi(8,7)(200) = 12, so a(200) = 13 + 12 - 8 - 12 = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
• Wikipedia, Chebyshev's bias.

Crossrefs

Cf. A007350, A188510.

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), this sequence (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (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

Mathematica

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)

PARI

a(n) = -sum(i=1, n, isprime(i)*kronecker(-2, i))

Python

from sympy import isprime; from numpy import sign
def A320857(n): return sum(isprime(i)*(i%2)*sign(i%8-4) for i in range(1,n+1)) # Ya-Ping Lu, Jan 25 2025

Formula

a(n) = -Sum_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).