# Rigorous C = sup_{0<y<=1423} (y - theta(y))/sqrt(y), and check theta(y) < y on (0,1423].
# On each gap [p_i, p_{i+1}) theta is constant, (y-th)/sqrt(y) is increasing in y (derivative (y+th)/(2y^{3/2})>0),
# so the sup over the gap is the left limit at p_{i+1}; on (0,2) theta=0 and value sqrt(y)<sqrt2.
from flint import arb, ctx
ctx.prec = 200
N=1423
s=[1]*(N+2); s[0]=s[1]=0
for i in range(2,40):
    if s[i]:
        for j in range(i*i,N+2,i): s[j]=0
P=[p for p in range(2,N+1) if s[p]]
th=arb(0); best=arb(2).sqrt(); arg=2; mn=None
for i,p in enumerate(P):
    v=(arb(p)-th)/arb(p).sqrt()          # left limit at p
    if v.upper()>best.upper(): best=v; arg=p
    th+=arb(p).log()
    m=(arb(p)-th)/arb(p).sqrt()          # value at p (min on gap)
    mn = m if mn is None or m.lower()<mn.lower() else mn
# right end y=1423 (prime, included above); 
print("sup", best, "at left limit of", arg)
print("min (y-theta)/sqrt y at primes", mn)