Number Theory: An Introduction to Mathematics

392 IX The Number of Prime Numbers

Conjecture B

lim

x→∞

M(x)( 2 xlog logx)−^1 /^2 =

√

6 /π

=−lim

x→∞

M(x)( 2 xlog logx)−^1 /^2.

By what has been said, Conjecture B implies not only the Riemann hypothesis,
but also that the zeros ofζ(s)are all simple. These probabilistic conjectures provide
a more interesting reason than symmetry for believing in the validity of the Riemann
hypothesis, but no progress has so far been made towards proving them.

7 SomeFurtherProblems

Aprimepis said to be atwin primeifp+2 is also a prime. For example, 41 is a
twin prime since both 41 and 43 are primes. It is still not known if there are infinitely
many twin primes. However Brun (1919), using the sieve method which he devised
for the purpose, showed that, if infinite, the sum of the reciprocals of all twin primes
converges. Since the sum of the reciprocals of all primes diverges, this means that few
primes are twin primes.
By a formal application of their circle method Hardy and Littlewood (1923) were
led to conjecture that

π 2 (x)∼L 2 (x) forx→∞,

whereπ 2 (x)denotes the number of twin primes≤x,

L 2 (x)= 2 C 2

∫x

2

dt/log^2 t

and

C 2 =

∏

p≥ 3

( 1 − 1 /(p− 1 )^2 )= 0. 66016181 ....

This implies thatπ 2 (x)/π(x)∼ 2 C 2 /logx. Table 2, adapted from Brent (1975),
shows that Hardy and Littlewood’s formula agrees well with the facts. Brent also
calculates
∑

twinp≤ 1010

( 1 /p+ 1 /(p+ 2 ))= 1. 78748 ...

and, using the Hardy–Littlewood formula for the tail, obtains the estimate

∑

all twinp

( 1 /p+ 1 /(p+ 2 ))= 1. 90216 ....

His calculations have been considerably extended by Nicely (1995).