A321856 - OEIS

%I #26 Nov 18 2023 08:28:48

%S 0,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,3,3,2,2,2,2,

%T 2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,3,3,3,3,3,3,2,2,

%U 2,2,3,3,2,2,2,2,2,2,1,1,1,1,2,2,2,2,2

%N Number of primes of the form 3*m + 2 <= n minus number of primes of the form 3*m + 1 <= n.

%C a(n) is the number of primes <= n that are quadratic nonresidues modulo 3 minus the number of primes <= n that are quadratic residues modulo 3.

%C Conjecturally infinitely many terms are negative. The earliest negative term is a(608981813029) = -1, see A112632.

%C 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. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by _Peter Munn_, Nov 05 2023]

%H Andrew Granville and Greg Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

%F a(n) = -Sum_{primes p<=n} Legendre(p,3) = -Sum_{primes p<=n} Kronecker(-3,p) = -Sum_{primes p<=n} A102283(p).

%F a(n) = A340764(n) - A340763(n). - _Jianing Song_, May 06 2021

%e Below 100, there are 11 primes congruent to 1 modulo 3 and 13 primes congruent to 2 modulo 3, so a(100) = 13 - 11 = 2.

%o (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-3, i))

%Y Cf. A007350, A102283, A340763, A340764.

%Y Let d be a fundamental discriminant.

%Y 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), this sequence (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).

%Y 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).

%K sign

%O 1,5

%A _Jianing Song_, Nov 20 2018