Number Theory: An Introduction to Mathematics

388 IX The Number of Prime Numbers

The Weil conjectures have a topological significance, since the integerein (ii) is
the Euler characteristic of the original varietyV, regarded as a complex manifold, and
bjin (iv) is itsj-th Betti number.
Conjecture (i) was proved by Dwork (1960). The remaining conjectures were
proved by Deligne (1974), using ideas of Grothendieck. The most difficult part is
the proof that|αjk|=pj/^2 (the Riemann hypothesis for varieties over finite fields).
Deligne’s proof is a major achievement of 20th century mathematics, but unfortunately
of a different order of difficulty than anything which will be proved here.
An analogue for function fields of Artin’s primitive root conjecture was already
proved by Bilharz (1937), assuming the Riemann hypothesis for this case. Function
fields have been used by Goppa (1981) to construct linear codes. Good codes are
obtained when the number of prime divisors is large compared to the genus, and this
can be guaranteed by means of the Riemann ‘hypothesis’.
Carlitz and Uchiyama (1957) used the Riemann hypothesis for function fields
to obtain useful estimates for exponential sums in one variable, and Deligne (1977)
showed that these estimates could be extended to exponential sums in several variables.
LetFpbe the field ofpelements, wherepis a prime, and letf∈Fp[u 1 ,...,un]be
a polynomial innvariables of degreed≥1 with coefficients fromFpwhich is not
of the formgp−g+b,whereb∈Fpandg∈Fp[u 1 ,...,un]. (This condition is
certainly satisfied ifd<p.) Then
∣
∣
∣
∣

∑

x 1 ,...,xn∈Fp

e^2 πif(x^1 ,...,xn)/p

∣

∣

∣

∣≤(d−^1 )p

n− 1 / (^2).
We mention one more application of the Weil conjectures.Ramanujan’s tau-
functionis defined by
q

∏∞

n= 1

( 1 −qn)^24 =

∑∞

n= 1

τ(n)qn.

It was conjectured by Ramanujan (1916), and proved by Mordell (1920), that

∑∞

n= 1

τ(n)/ns=

∏

p

( 1 −τ(p)p−s+p^11 −^2 s)−^1 ,

where the product is over all primesp. Ramanujan additionally conjectured that
|τ(p)|≤ 2 p^11 /^2 for allp, and Deligne (1968/9) showed that this was a consequence
of the (at that time unproven) Weil conjectures.
The prime number theorem also has an interesting analogue in the theory of
dynamical systems. LetMbe a compact Riemannian manifold with negative sectional
curvatures, and letN(T)denote the number of different (oriented) closed geodesics
onMof length≤T. It was first shown by Margulis (1970) that

N(T)∼ehT/hT asT→∞,

where the positive constanthis the topological entropy of the associated geodesic flow.
Although much of the detail is specific to the problem, a proof may be given which
has the same structure as the proof in§3 of the prime number theorem. IfPis an