A066520 Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n.

0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 2

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jan 05 2002

Keywords

• sign
• easy
• nice
• look

Comments

Although the initial terms are nonnegative, it has been proved that infinitely many terms are negative. The first two are a(26861)=a(26862)=-1. Next there are 3404 values of n in the range 616841 to 633798 with a(n)<0. Then 27218 values in the range 12306137 to 12382326.

Partial sums of A151763. - Reinhard Zumkeller, Feb 06 2014

Links

• T. D. Noe, Table of n, a(n) for n=1..30000 (enough terms to show the first dip into negative territory)
• Carter Bays and Richard H. Hudson, Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes, Mathematics of Computation, 69 (2000) 861-866.
• A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Crossrefs

Cf. A066339, A066490, A007350, A051024, A051025.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]

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), A320857 (d=-8), A321859 (d=-7), this sequence (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

Haskell

a066520 n = a066520_list !! (n-1)
a066520_list = scanl1 (+) $ map (negate . a151763) [1..]
-- Reinhard Zumkeller, Feb 06 2014

Mathematica

a[n_] := Length[Select[Range[3, n, 4], PrimeQ]]-Length[Select[Range[1, n, 4], PrimeQ]]
f[n_]:=Module[{c=Mod[n,4]},Which[!PrimeQ[n],0,c==3,1,c==1,-1]]; Join[{0,0}, Accumulate[Array[f,110,3]]] (* Harvey P. Dale, Mar 03 2013 *)

Formula

a(n) = A066490(n) - A066339(n).

a(2*n+1) = a(2*n+2) = -A156749(n). - Jonathan Sondow, May 17 2013

Extensions

Edited by Dean Hickerson, Mar 05 2002