396
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
and other Greek geniuses, and even if you don't, the right hand side is thus,
apart from the term n = 1, arbitrarily small for n large so long as the series
(( s) converges, i.e. for Re( s) > 1. We conclude that
(17.2)
lim (1 - l/pD ... (1 - pk) ((s) = 1.
We have thus almost established the following result:
Theorem 14. 28 We have ((s) f:. 0 for Re(s) > 1 and
(17.3)
1/((s) = II (1 - l/pS),
p prime
the infinite product being absolutely convergent.
The absolute convergence of the infinite product reduces to that of the
series L: II/pSI; obvious, since it is a subseries of the series L: II/n s l.
The preceding theorem explains the role that the function has played,
since Riemann's famous memoir, in the problem of the distribution of prime
numbers. It has given rise to an immense literature. "Analytic number theory" consists of using the methods of analysis to establish results in arithmetic.
The preceding formula, for example, implies the existence of infinitely
many prime numbers. If such were not the case, the function ( would be the
restriction to Re( s) > 1 of a function analytic on C except at the points
s = 2ki7r / log P which annul one of the factors, finite in number, of the product. In particular, it would tend to a finite limit when s E IR tends to 1, the
limit of convergence of the series.
Now the function (( s) is decreasing for s > 1, like the functions 1/ n S •
When s > 1 tends to 1, the number ((s) thus tends, increasing, to a limit
M :::; +00. If the latter were finite, all the partial sums of the series with
positive terms ((s) would be :::; M for all s > 1. But if the relation
1 + 1/2 s + ... + l/n s :::; M
is valid for all s > 1, it remains valid at the limit for s = 1. The hypothesis
M < +00 would thus imply the convergence of the harmonic series. [In fact,
we shall prove later that the product (s - 1)((s) tends to 1 with s.]
This result, known for more than twenty centuries, is not very impressive.
But by examining combinations of partial series of the formL: 1/(an + b)S,
where a and b are given relatively prime integers, Dirichlet was able to prove
by the same method, with the help of arithmetic calculations that the XX th
century has widely generalised, that there are infinitely many prime numbers
in the "arithmetic progression" an + b, as had been long conjectured. Experimental verification can surely be programmed on a computer and surely
28 Euler, Introductio ... , I, Chap. XV.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
and other Greek geniuses, and even if you don't, the right hand side is thus,
apart from the term n = 1, arbitrarily small for n large so long as the series
(( s) converges, i.e. for Re( s) > 1. We conclude that
(17.2)
lim (1 - l/pD ... (1 - pk) ((s) = 1.
We have thus almost established the following result:
Theorem 14. 28 We have ((s) f:. 0 for Re(s) > 1 and
(17.3)
1/((s) = II (1 - l/pS),
p prime
the infinite product being absolutely convergent.
The absolute convergence of the infinite product reduces to that of the
series L: II/pSI; obvious, since it is a subseries of the series L: II/n s l.
The preceding theorem explains the role that the function has played,
since Riemann's famous memoir, in the problem of the distribution of prime
numbers. It has given rise to an immense literature. "Analytic number theory" consists of using the methods of analysis to establish results in arithmetic.
The preceding formula, for example, implies the existence of infinitely
many prime numbers. If such were not the case, the function ( would be the
restriction to Re( s) > 1 of a function analytic on C except at the points
s = 2ki7r / log P which annul one of the factors, finite in number, of the product. In particular, it would tend to a finite limit when s E IR tends to 1, the
limit of convergence of the series.
Now the function (( s) is decreasing for s > 1, like the functions 1/ n S •
When s > 1 tends to 1, the number ((s) thus tends, increasing, to a limit
M :::; +00. If the latter were finite, all the partial sums of the series with
positive terms ((s) would be :::; M for all s > 1. But if the relation
1 + 1/2 s + ... + l/n s :::; M
is valid for all s > 1, it remains valid at the limit for s = 1. The hypothesis
M < +00 would thus imply the convergence of the harmonic series. [In fact,
we shall prove later that the product (s - 1)((s) tends to 1 with s.]
This result, known for more than twenty centuries, is not very impressive.
But by examining combinations of partial series of the formL: 1/(an + b)S,
where a and b are given relatively prime integers, Dirichlet was able to prove
by the same method, with the help of arithmetic calculations that the XX th
century has widely generalised, that there are infinitely many prime numbers
in the "arithmetic progression" an + b, as had been long conjectured. Experimental verification can surely be programmed on a computer and surely
28 Euler, Introductio ... , I, Chap. XV.