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%I #21 Nov 19 2023 10:23:31
%S 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,
%T 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,
%U 3,3,2,2,1,1,1,1,1,1,2,2,2,2,1,1,1,1,1
%N 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.
%C 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.
%C 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.
%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. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by _Peter Munn_, Nov 19 2023]
%C 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).
%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} Kronecker(12,p) = -Sum_{primes p<=n} A110161(p).
%e 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.
%o (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(12, i))
%Y Cf. A007350, A110161.
%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), A321856 (d=-3), A321857 (d=5), A071838 (d=8), this sequence (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 1,7
%A _Jianing Song_, Nov 20 2018
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