## Legendre's Prime Number Conjecture

```Most historical accounts of the Prime Number Theorem mention Legendre's
experimental conjecture (made in 1798 and again in 1808) that

x
pi(x)  =   ---------------
log(x) - A(x)

where pi(x) is the number of primes less than x, and the limit of
A(x) as x goes to infinity is 1.08366....  In 1850, Tschebycheff
proved that Legendre's conjecture cannot be true unless 1.08366...
is replaced by 1.

Aside from the comment that Legendre's conjecture was based on
"experimental evidence", I've never seen an explanation of how he
actually arrived at the number 1.08366...  Here's a table giving
the actual values of A(x) for several values of x:

x        A(x)
-----     ------
10^2      0.6052
10^3      0.9553
10^4      1.0736
10^5      1.0876
10^6      1.0763
10^7      1.0710
10^8      1.0639
10^9      1.0566
10^10     1.0504

I've never been able to see how anyone could infer a limit of 1.08366
from this table, or even from any truncated version of this table
(allowing for the possibility that Legendre may not have had the
values of pi(x) for very large values of x).  Notice that he gave
the "constant" to five significant digits, which seems remarkable
working from this kind of data.

This raises some questions:
(1) How did Legendre arrive at the constant 1.08366...?
(2) For what precise value of x does A(x) achieve it's maximum value?

Regarding (1), I wonder if there is any connection with the limit of

1         /     1 \
---  PROD ( 1 + --- )
ln(x)       \     p /

where the product is evaluated over all primes p < x.  I believe that
the infinite product is known to equal

6 e^(gamma)
-----------  =~  1.082762...
pi^2

which is fairly close to Legendre's constant 1.08366...  Is it possible
that Legendre was aware of this infinite product (or some estimate of
it) when he made his Prime Number conjecture?

I don't have an explicit reference for the above evaluation of the
infinite product, but it follows closely from a combination of Theorem
302 in Hardy & Wright's "Introduction to the Theory of Numbers"

zeta(s)
-------  =    PROD (1 + p^-s)           (s>1)
zeta(2s)

and "a formula of Mertens" given on page 162 of Ribenboim's "Book of
Prime Number Records"
1                1
e^gamma   =   lim    -------   PROD  ----------
n->inf   log(n)   i < n   1 - 1/p_i

along with the fact that zeta(2)=pi^2/6.

The only other information I've found is in Tchebyshev's paper where
he says Legendre

"..begins by comparing his formula with the result of
counting the primes in the most extended tables,
namely those from 10,000 up to 1,000,000, after
which he applies his formula to the solution of many
problems".

This doesn't clear up the mystery for me, because by 1,000,000 the
value A(x) has already passed its maximum and is down to 1.076...
So I still don't see how  Legendre arrived at the precise value
1.08366...

Regarding my question (2), which asked for the maximum value of A(x),
I've computed A(p) for p < 10^6 and found that the maximum value is
1.1119625..., occurring at p = 24137, which is the 2688th prime.
```