394 IX The Number of Prime Numbers
Bateman and Horn gave a heuristic derivation of their formula. However, the only
case in which the formula has actually been proved ism=1,n 1 =1. This is the case
of primes in an arithmetic progression which will be considered in the next chapter.
When one considers the vast output of mathematical papers today compared with
previous eras, it is salutary to recall that we still do not know as much about twin
primes as Euclid knew about primes.
8 FurtherRemarks
The historical development of the prime number theorem is traced in Landau [33]. The
original papers of Chebyshev are available in [56]. Pintz [48] has given a simple proof
of Chebyshev’s result thatπ(x)=x/(Alogx−B+o( 1 ))impliesA=B=1.
There is an English translation of Riemann’s memoir in Edwards [20]. Complex
variable proofs of the prime number theorem, with error term, are contained in the
books of Ayoub [4], Ellison and Ellison [21], and Patterson [47]. For a simple complex
variable proof without error term, due to Newman (1980), see Zagier [63].
A proof with error term by the Wiener–Ikehara method is given inCi ̆ ̆zek [12].
Wiener’s general Tauberian theorem is proved in Rudin [52]. For its algebraic
interpretation, see the resum ́e of Fourier analysis in [13]. The development of Selberg’s
method is surveyed in Diamond [18]. An elementary proof of the prime number
theorem which is quite different from that of Selberg and Erd ̋os has been given by
Daboussi [15].
A clear account of Stieltjes integrals is given in Widder [62]. However, we do not
use Stieltjes integrals in any essential way, but only for the formal convenience of
treating integration by parts and summation by parts in the same manner. Widder’s
book also contains the Wiener–Ikehara proof of the prime number theorem.
By a theorem of S. Bernstein (1928), proved in Widder’s book and also in
Mattner [38], the hypotheses of Proposition 7 can be stated without reference to
the functionφ(x). Bernstein’s theorem says that a real-valued functionF(σ)can be
represented in the form
F(σ)=∫∞
0e−σxdφ(x),whereφ(x)is a nondecreasing function forx≥0 and the integral is convergent for
everyσ>1, if and only ifF(σ)has derivatives of all orders and
(− 1 )kF(k)(σ)≥0foreveryσ> 1 (k= 0 , 1 , 2 ,...).
For the Poisson summation formula see, for example, Lasser [34] and Dur ́an
et al. [19]. There is a usefuln-dimensional generalization, discussed more fully in
§7 of Chapter XII, in which a sum over all points of a lattice is related to a sum over
all points of the dual lattice. Further generalizations are mentioned in Chapter X.
More extended treatments of the gamma function are given in Andrewset al.[3]
and Remmert [49].
More information about the Riemann zeta-function is given in the books of
Patterson [47], Titchmarsh [57], and Karatsuba and Voronin [30]. For numerical data,
see Rosser and Schoenfeld [50], van de Luneet al. [37] and Rumely [53].