7 Some Further Problems 393Table 2.
x π 2 (x) L 2 (x) π 2 (x)/L 2 (x)
103 35 46 0.76
104 205 214 0.96
105 1224 1249 0.980
106 8169 8248 0.9904
107 58980 58754 1.0038
108 440312 440368 0.99987
109 3424506 3425308 0.99977
1010 27412679 27411417 1.000046Besides the twin prime formula many other asymptotic formulae were conjectured
by Hardy and Littlewood. Most of them are contained in a general conjecture, which
will now be described.
Let f(t)be a polynomial intof positive degree with integer coefficients. Iff(n)
is prime for infinitely many positive integersn,thenf has positive leading coeffi-
cient,f is irreducible over the fieldQof rational numbers and, for each primep,
there is a positive integernfor whichf(n)is not divisible byp. It was conjectured by
Bouniakowsky (1857) that conversely, if these three conditions are satisfied, thenf(n)
is prime for infinitely many positive integersn. Schinzel (1958) extended the conjec-
ture to several polynomials and Bateman and Horn (1962) gave Schinzel’s conjecture
the following quantitative form.
Letfj(t)be a polynomial intof degreedj≥1, with integer coefficients and posi-
tive leading coefficient, which is irreducible over the fieldQof rational numbers(j=
1 ,...,m). Suppose also that the polynomialsf 1 (t),...,fm(t)are distinct and that,
for each primep, there is a positive integernfor which the productf 1 (n)···fm(n)is
not divisible byp. Bateman and Horn’s conjecture states that, ifN(x)is the number
of positive integersn≤xfor which f 1 (n),...,fm(n)are all primes, then
N(x)∼(d 1 ···dm)−^1 C(f 1 ,...,fm)∫x2dt/logmt,where
C(f 1 ,...,fm)=∏
p{( 1 − 1 /p)−m( 1 −ω(p)/p)},the product being taken over all primespandω(p)denoting the number ofu∈Fp(the
field ofpelements) such thatf 1 (u)···fm(u)=0. (The convergence of the infinite
product when the primes are taken in their natural order follows from the prime ideal
theorem.)
The twin prime formula is obtained by takingm=2andf 1 (t)=t,f 2 (t)=t+2.
By taking insteadf 1 (t)=t,f 2 (t)= 2 t+1, the Bateman–Horn conjecture gives the
same asymptotic formulaπG(x)∼L 2 (x)for the numberπG(x)of primesp≤xfor
which 2p+1 is also a prime (‘Sophie Germain’ primes). By takingm=1andf 1 (t)=
t^2 +1 one obtains an asymptotic formula for the number of primes of the formn^2 +1.