OFFSET
1,31
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).
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.
FORMULA
a(n) = -Sum_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).
EXAMPLE
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.
MATHEMATICA
Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)
PROG
(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-2, i))
CROSSREFS
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).
KEYWORD
sign
AUTHOR
Jianing Song, Nov 24 2018
STATUS
approved