## D(-1) quadruples, Theorem 3
N=1.641*10^55
M=18*((12/float(pi^2))*N*log(N)+4.332*N)
print N
print log(N)
print M

#1.64100000000000e55
#127.137485926780
#4.69396175860710e58

##----------------------------------------------------
## Improvement of Theorem 2, for N\geq 1000
for n in range(1000,1701):
       if sum([(moebius(m))^2 for m in range(1,n+1)])>0.62*n:
           print n
##-----------------------------------------------------
## the function K(x) < (8/float(pi^2))*x*log(x)+0.648*x  for x\geq 2
#c=0.647
c=0.648
x=2
(c-0.496)*x-0.608/x
plot(0.648*x-(0.496*x+0.608/x),(x,0.5,3))

##----------------------------------------------------
## EXAMPLES COMPUTATIONS
## \lambda(\delta) - the coefficient in McKee's formula
def l(d):
    print 12*pari(4*d).qfbhclassno()/(3.1416*sqrt(4*d))

#Constants (C_1,C_2,C_3) from Theorem 1

def Const(b,c,d):
    if mod(d,4)!=2:
        if mod(d,4)!=3 :
            print "Not good discriminant"

    q=4*abs(d)
    kappa=(4/float(pi^2))*float(sqrt(q))*float(log(q))+0.648*float(sqrt(q))
    ksi=float(sqrt(1+2*abs(b)+abs(c)))
    A=ceil(max(abs(b),float(sqrt(abs(c)))))

    c1=(12/float(pi^2))*(log(kappa)+1)
    c2=2*(kappa+(log(kappa)+1)*((6/float(pi^2))*float(log(ksi))+1.166))
    c3=2*kappa*A
    print (c1,c2,c3)
    
#EXAMPLES FROM THE TABLE
##n^2+1
print 'n^2+1'
l(1)
Const(0,1,-1)
## n^2+10n+27
print 'n^2+10n+27'
l(2)
Const(5,27,-2)
## n^2+4n+10
print 'n^2+4n+10'
l(6)
Const(2,10,-6)
## n^2+52n+706
print 'n^2+52n+706'
l(30)
Const(26,706,-30)
## n^2+10n-26 #discriminant is positive, no McKee's formula
print 'n^2+10n-26'
Const(5,-26,51)

#n^2+1
#0.954927425515661
#(2.29022985955444, 10.0257540720166, 4.83937576872585)
#n^2+10n+27
#1.35047131624627
#(2.96548152735700, 19.5144159446370, 50.5982729489532)
#n^2+4n+10
#1.55938995593531
#(3.95111220213397, 31.8971233487115, 75.8760488968092)
#n^2+52n+706
#1.39476077796070
#(5.28257784085363, 84.3557338925090, 1531.08233404361)
#n^2+10n-26
#(5.70220562480180, 101.311664340512, 480.478649736826)