%I #31 Oct 22 2021 12:48:09

%S 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,

%T 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,

%U 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

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

%C 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.

%C Partial sums of A151763. - _Reinhard Zumkeller_, Feb 06 2014

%H T. D. Noe, <a href="/A066520/b066520.txt">Table of n, a(n) for n=1..30000</a> (enough terms to show the first dip into negative territory)

%H Carter Bays and Richard H. Hudson, <a href="https://doi.org/10.1090/S0025-5718-99-01105-9">Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes</a>, Mathematics of Computation, 69 (2000) 861-866.

%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.

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

%F a(2*n+1) = a(2*n+2) = -A156749(n). - _Jonathan Sondow_, May 17 2013

%t a[n_] := Length[Select[Range[3, n, 4], PrimeQ]]-Length[Select[Range[1, n, 4], PrimeQ]]

%t 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 *)

%o (Haskell)

%o a066520 n = a066520_list !! (n-1)

%o a066520_list = scanl1 (+) $ map (negate . a151763) [1..]

%o -- _Reinhard Zumkeller_, Feb 06 2014

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

%Y Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From _Daniel Forgues_, Mar 26 2009]

%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), this sequence (d=-4), A321856 (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,easy,nice,look

%O 1,11

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

%E Edited by _Dean Hickerson_, Mar 05 2002