primes mod 3

[Joseph S. Myers]

On Sat, 20 May 2006, Joshua Zucker wrote:

> Up to 10^8, at least according to my program, once 2 gets the lead in
> the camp of "primes = 2 mod 3", there are always more primes = 2 mod 3
> between 1 and n than there are primes = 1 mod 3.
> 
> Does anyone have a reference about some future crossing point, where
> the primes = 1 mod 3 catch up?  My data say that 233 is the last time
> it gets really close, 1889 is the last time it's within 2, and 198593
> is the last time that it's within 3 (again, searching up to 10^8).

See:

   MR0476616 (57 #16175)
   Bays, Carter; Hudson, Richard H.
   Details of the first region of integers $x$ with $\pi \sb{3,2}(x)<\pi
   \sb{3,1}(x)$. 
   Math. Comp. 32 (1978), no. 142, 571--576.

   Authors' summary: "Since the time of \v Ceby\v sev, there has been
   interest in the magnitude of the smallest integer $x$ with
   $\pi_{3,2}(x)<\pi_{3,1}(x)$, where $\pi_{b,c}(x)$ denotes the number 
   of positive $\text{primes}\leqq\,x$ and $\equiv c\,(\text{mod}\,b)$.
   The authors have recently reached this threshold with the discovery
   that $\pi_{3,2}(608981813029)-\pi_{3,1}(608981813029)=-1$. This paper
   includes a detailed numerical and graphical description of values of
   $\pi_{3,2}(x)-\pi_{3,1}(x)$ in the vicinity of this long sought
   number."

-- 
Joseph S. Myers
jsm at polyomino.org.uk