A329242 a(n) = Pi(8,3)(n) + Pi(8,5)(n) + Pi(8,7)(n) - 3*Pi(8,1)(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, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10

Offset: 1

Author

Jianing Song, Nov 08 2019

Keywords

• sign

Comments

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]

Define the "Chebyshev's bias sequence mod k" to be sequence q(n), where q(n) = Sum_{b is a quadratic nonresidue mod k, gcd(b, k) = 1} Pi(k,b)(n) - (r-1)*(Sum_{a is a quadratic residue mod k, gcd(a, k) = 1} Pi(k,a)(n)), r is the number of solutions to x^2 == 1 (mod n), then this sequence is the "Chebyshev's bias sequence mod 8". Also the initial terms are nonnegative integers, a(n) is negative for some n ~ 10^28.127. See page 21 of the paper in Journal of Number Theory in the Links section below.

Examples

			Below 2000000, there are 37116 primes congruent to 1 mod 8, 37261 primes congruent to 3 mod 8, 37300 primes congruent to 5 mod 8 and 37255 primes congruent to 7 mod 8, so a(2000000) = 37261 + 37300 + 37255 - 3*37116 = 468.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Bays, Carter & Ford, Kevin & Hudson, Richard & Rubinstein, Michael, Zeros of Dirichlet L-functions near the Real Axis and Chebyshev's Bias. Journal of Number Theory, Vol. 87, Issue 1, Mar 2001, pp. 54-76.
• Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A007350, A329243.

"Chebyshev's bias sequence mod k": A112632 (k=3), A038698 (k=4), A321862 (k=5), A321864 (k=7), A321865 (k=11).

Programs

PARI

a(n) = my(k=0); for(p=1, n, if(isprime(p)&&p>2, if(p%8==1, k-=3, k++))); k