9 Selected References 395For a proof thatπ(x)−Li(x)changes sign infinitely often, see Diamond [17].
Estimates for values ofxsuch thatπ(x)>Li(x)are obtained by a technique due to
Lehman [35]; for the most recent estimate, see Bays and Hudson [8].
For the pair correlation conjecture, see Montgomery [40], Goldston [24] and
Odlyzko [45]. Random matrices are thoroughly discussed by Mehta [39]; for a nice
introduction, see Tracy and Widom [58].
For Dedekind zeta functions see Stark [54], besides the books on algebraic number
theory referred to in Chapter III. The prime ideal theorem is proved in Narkiewicz [44],
for example. For consequences of the extended Riemann hypothesis, see Bach [5],
Goldstein [23] and M.R. Murty [41]. Many other generalizations of the zeta function
are discussed in the article on zeta functions in [22].
Function fields are treated in the books of Chevalley [11] and Deuring [16]. The
lengthy review of Chevalley’s book by Weil inBull. Amer. Math. Soc. 57 (1951),
384–398, is useful but over-critical. Even if geometric methods are better adapted for
algebraic varieties of higher dimension, the algebraic methods available for curves
are essentially simpler. Moreover it was the close analogy with number fields that
suggested the possibility ofa Riemann hypothesis for function fields. For a proof of
the latter, see Bombieri [9]. For the Weil conjectures, see Weil [61] and Katz [32].
Stichtenoth [55] gives a good account ofthe theory of function fields with spe-
cial emphasis on its applications to coding theory. For these applications, see also
Goppa [26], Tsfasmanet al.[60], and Tsfasman and Vladut [59]. Curves with a given
genus which have the maximal number ofFq-points are discussed by Cossidente
et al.[14].
For introductions to Ramanujan’s tau-function, see V.K. Murty [42] and Rankin’s
article (pp. 245–268) in Andrewset al.[2]. For analogues of the prime number the-
orem in the theory of dynamical systems, see Katok and Hasselblatt [31] and Parry
and Pollicott [46]. Hadamard’s pioneering study of geodesics on a surface of negative
curvature and his proof of the prime number theorem are both reproduced in [27].
The ‘equivalence’ of Proposition 12 with the prime number theorem is proved in
Ayoub [4]. A proof that the Riemann hypothesis is equivalent toM(x)=O(xα)for
everyα> 1 /2 is contained in the book of Titchmarsh [57]. Good and Churchhouse’s
probabilistic conjectures appeared in [25]. For the central limit theorem and the law of
the iterated logarithm see, for example, Adams [1], Kac [29], Bauer [7] and Lo`eve [36].
Brun’s theorem on twin primes is proved in Narkiewicz [43]. For numerical results,
see Brent [10]. For conjectural asymptotic formulas, see Hardy and Littlewood [28]
and Bateman and Horn [6]. There are several heuristic derivations of the twin prime
formula, the most recent being Rubenstein[51]. It would be useful to try to analyse
these heuristic derivations, so that the conclusion is seen as a consequence of precisely
stated assumptions.
9 SelectedReferences
[1] W.J. Adams,The life and times of the central limit theorem, Kaedmon, New York, 1974.
[2] G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan and R.A. Rankin (ed.),
Ramanujan revisited, Academic Press, London, 1988.
[3] G.E. Andrews, R. Askey and R. Roy,Special functions, Cambridge University Press, 1999.