Number Theory: An Introduction to Mathematics

6 Alternative Formulations 389

arbitrary closed orbit of the geodesic flow andλ(P)its least period, one shows that the
zeta-function

ζM(s)=

∏

P

( 1 −e−sλ(P))−^1

is nonzero and holomorphic forRs≥h, except for a simple pole ats=h,andthen
applies Ikehara’s theorem. The study of geodesics on a surface of negative curvature
was initiated by Hadamard (1898), but it isunlikely that he realized there was a
connection with the prime number theorem which he had proved two years earlier!

6 AlternativeFormulations

There is an intimate connection between theDirichlet productsconsidered in§3of
Chapter III andDirichlet series. It is easily seen that if the Dirichlet series

f(s)=

∑∞

n= 1

a(n)/ns, g(s)=

∑∞

n= 1

b(n)/ns,

are absolutely convergent forRs>α, then the producth(s)=f(s)g(s)may also be
represented by an absolutely convergent Dirichlet series forRs>α:

h(s)=

∑∞

n= 1

c(n)/ns,

wherec=a∗b,i.e.

c(n)=

∑

d|n

a(d)b(n/d)=

∑

d|n

a(n/d)b(d).

This implies, in particular, that forRs> 1

ζ^2 (s)=

∑∞

n= 1

τ(n)/ns,ζ(s− 1 )ζ(s)=

∑∞

n= 1

σ(n)/ns,

where as in Chapter III (not as in§5),

τ(n)=

∑

d|n

1 ,σ(n)=

∑

d|n

d,

denote respectively the number of positive divisors ofnand the sum of the positive
divisors ofn. The relation for Euler’s phi-function,

σ(n)=

∑

d|n

τ(n/d)φ(d),

which was proved in Chapter III, now yields forRs> 1