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An Introduction to Analytic Number Theory

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An Introduction to Analytic Number Theory
Ilan Vardi IHES, Bures-sur-Yvette December 14, 1998 Summary by Cyril Banderier and Ilan Vardi

``Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.'' ``The shortest path between two truths in the real domain passes through the complexdomain.'' J. Hadamard. The above quote captures the depth analysis can bring when one is confronted by number theoretic questions. The oldest and most fundamental of such questions is the study of prime numbers. The first question to be answered is: Are there an infinite number of primes? This can be answered by a number of simple proofs: Euclid: Assume there are a finite number of primes


is not divisible by any of the pi's, so any of its

prime divisors yields a new prime number (Euclid only considered the case n=3). Pólya: The Fermat numbers Fn=22 +1 are pairwise relatively prime, so the set of their prime divisors must be infinite. Erdos: Fix x and consider the primes can write every integer as .Since every integer is the product of a perfect square and a squarefreenumber, one ,where and . There 2n choices for the ei and

choices for Q, so it follows that Euler: One has the formal identity

which in fact holds for

. As

, the left hand side of (1) tends to

since the harmonic series diverges, so there must be

an infinite number of factors on the right. This proof can be modified by noting that then (1) would imply that , where . If there were onlya finite number of primes,

is rational, proved false by Legendre in 1797, see also [6]. Several other proofs are given in [7].

A stronger version of this is due to Mertens: The finite version of (1) gives

and taking logs will give


and so there are an infinite number of primes. Which of these is the ``best'' proof? One argument would say that it is the one which allows the bestgeneralisation. For example, Euclid's proof easily


An Introduction to Analytic Number Theory
shows that there are an infinite number of primes of the form 4k+3 (consider same holds for primes of the form 4k+1 (one has to consider

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), but seems to fall flat when trying to prove that the ). In general, one wantsto demonstrate Dirichlet's assertion (that

he proved in 1837, in [3]) ``there are an infinite number of primes of the form ak+b, where a and b are relatively prime.'' It turns out that the proof of this deep fact uses a generalisation of Euler's method, i.e., equation (2):

Dirichlet's theorem
Let be a multiplicative character modulo q, that it is to say a complex valued function (thisimplies that if prime) symbol satisfying and , then it is a root of unity and so has norm one). An example is the Legendre (or Jacobi if q is not a

In fact, there are exactly that

multiplicative characters modulo q, all given by


is a primitive root and

is such

.The importance of characters is seen by the following orthogonality relation:


which allows one to pick out anarithmetic progression. For his proof, Dirichlet introduced what are nowadays called Dirichlet L-functions, defined by

Taking logarithm leads to

, thus one has

and a simple application of relation (3) gives

Then, by splitting the sum in real and complex characters, one gets


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is called the principal character and equals 1 whenever

and 0 otherwise. The first sum (over

) is

, as

.This infinite term should imply that there are an infinite number of primes in the arithmetic progression. The only problem is that one of the other terms could cancel this one by being zero at s = 1 (partial summation shows that therefore has to show...
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