Bertelsen's Number

The number of primes less than x is denoted by pi(x).  In Hardy & 
Wright's book "An Introduction to the Theory of Numbers", 5th 
edition, page 9, it says the vale of pi(10^9) is 50,847,478.  
However, on page 179 of Paulo Ribenboim's book "The Book of Prime 
Number Records" it gives the value 50,847,534, which exceeds H&W's 
count by 56.

Looking for a tie-breaker, I remembered seeing a table of pi(x) in 
"The Mathematical Experience" by Reuben and Hersh.  Actually, the 
book mentions pi(10^9) in two places.  On page 175 it says pi(10^9) 
is known to equal 50,847,478, whereas the table on page 213 lists 
pi(10^9) as 50,847,534.  Curious.  At least two of these three books 
has an error in it.  Perhaps Reuben and Hersh got their first value 
from Hardy and Wright, and then constructed their table from some 
other (more recent?) source, and didn't cross-check the two.  Since 
number theory is a somewhat empirical science, should we consider 
the possibility that the number of primes has increased since 1938? 

Out of curiosity I checked a few more books and found the same 
erroneous value of pi(10^9) in each of the following:

An Introduction To the Theory of Numbers      Hardy/Wright  1938/1988
The Mathematical Experience                   Davis/Hersh   1981/1981
Number Theory and Its History                 Ore           1948/1988
Numbers and Infinity                          Sondheimer    1981/1981
The Nature and Growth of Modern Mathematics   Kramer        1970/1982
Introduction to Algorithms                    Cormen, et al 1993/1993     

The error evidently goes back further then Hardy and Wright.  In 
Oystein Ore's book "Number Theory and Its History" (1st Dover edition,
1988) the table on page 77 gives the old value pi(10^9) = 50,847,478.
Furthermore, it says the values of pi(x) for x up to 10^7 were 
obtained by actual count from tables, whereas the values for x=10^8
and 10^9 "are the values of pi(x) calculated by Meissel and Bertelsen
which we have mentioned previously."

On page 69 Ore discusses a formula for computing pi(N) in terms of 
all the primes less than sqrt(N).  He goes on to say that Meissel 
(1870) improved the method and, through various shortcuts, succeeded 
in finding pi(10^8) = 5,761,455.

  "These computations were continued by the Danish mathematician
   Bertelsen [Niels Peder Bertelsen (1869-1938)], who applied them 
   for the determination of errors in tables.  He announced the 
   following result (1893):

                     pi(10^9) = 50,847,478

   which represents our most extended knowledge of the number
   of primes."

In view of this, it seems clear that there was an error in Bertelsen's
calculation back in 1893 (rather than just a typo), and that it has 
been propogated onwards, appearing in books printed nearly 100 years
later.  S. Sorrento has called my attention to two other references,
first, the 1950 book "Element�r talteori" by Nagell, which gives the
erroneous value, and second, the 1968 book "En bok om primtal" by
Riesel, which gives the correct value, and remarks on the earlier
incorrect value appearing in various texts.  This strikes me as 
interesting, especially since Bertelsen applied his results "for 
the determination of errors in tables".
 
Finally I came across the paper "Calculating pi(x): The Meisell-
Lehmer Method" by Lagarias, Miller, and Odlyzko, Math Comp, 1985,
which reports that the correct value of pi(10^9) is 50,847,534, and
that the erroneous value 50,847,478 originated in Bertelsen's 
application of Meisell's method back in 1893.  Remarkably, the
erroneous value ("Bertelsen's number") was still being propagated 
in text books exactly 100 years later, with at least seven citations
in books published between 1938 and 1993.

Apparently there has been a long hazardous history of computing 
the number of primes up to greater and greater values.  Often the 
greatest "known" value of pi(x) has later turned out to be wrong.
As another illustration of this, the value of pi(10^10) listed in
the "modern" table in Reuben and Hersch is 455052512, which is the
uppermost value they quote, exceeds by 1 the value in Ribenboim's
Book of Prime Number Records (1988).

Anyway, for the record, based on Ribenboim's book, the number of 
primes of the first several orders of magnitude are listed 
(correctly I hope!) below:

                         number of primes
              x          less than x
           -------       ----------------
             10                        4
             10^2                     25
             10^3                    168
             10^4                   1229
             10^5                   9592
             10^6                  78498
             10^7                 664579
             10^8                5761455
             10^9               50847534
             10^10             455052511
             10^11            4118054813
             10^12           37607912018
             10^13          346065536839
             10^14         3204941750802
             10^15        29844570422669
             10^16       279238341033925
           2*10^16       547863431950008
           4*10^16      1075292778753150