An Introduction to Analytic Number Theory

[summary by Cyril Banderier and Ilan Vardi (IHES, Bures-sur-Yvette) of Ilan's talk given at the Algorithms Seminar, December 14, 1998]

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1   Introduction

``Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.'' (Jacques Hadamard)
``The shortest path between two truths in the real domain passes through the complex domain.'' (Jack 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 (several other proofs are given in [7]):

Which of these is the ``best'' proof? One argument would say that it is the one which allows the best generalisation. For example, Euclid's proof easily shows that there are an infinite number of primes of the form 4k+3 (consider 4 p1··· pn -3), but seems to fall flat when trying to prove that the same holds for primes of the form 4k+1 (one has to consider 4 (p1··· pn)2+1). In general, one wants to 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):
 
å
pº a mod q
1
p
=¥ Û there are an infinite number of primes in ak+q.

2   Le théorème de Dirichlet

Soit c un caractère multiplicatif modulo q, c'est-à-dire une fonction à valeurs complexes c(n) vérifiant c(mn)=c(m)c(n), c(1)=1 et c(0)=0 (ce qui implique que si c(n)¹ 0, alors c'est une racine de l'unité et a donc norme 1). Un exemple est le symbole de Legendre (ou de Jacobi si q n'est pas premier)
c(n):= æ
ç
ç
è
n
q
ö
÷
÷
ø		 =
ì
í
î
0 if q | n,
1 if x2 º nmod q for some x,
-1 otherwise.
En fait, pour tout q puissance d'un nombre premier impair, il y a exactement f(q) caractères multiplicatifs modulo q, tous donnés par c(n):=e2 ik p n(n) /f(q) for 0£ k £ f(q)-1 and where n(n) is such that nº gn(n)mod q for any generator of the group of invertible elements of Z/qZ. When q is a power of 2, the definition is little more cumbersome (linked to the ``factorisation'' n º (-1)n1(n) 5n2(n) mod q), and for general q, it is the product of characters of the factors of q. The importance of characters is seen by the following orthogonality relation:
1
f(q)
 
å
c
c(a)
c(n) =
ì
í
î
1 whenever nº a mod q,
0 otherwise,
    (3)
where the sum is over all the characters (the two real ones and the other complex characters). The orthogonality relation allows one to pick out an arithmetic progression. For his proof, Dirichlet introduced what are nowadays called Dirichlet L-functions, defined by
L(s,c):=
 
å
n³ 1
c(n)
ns
=
 
Õ
p
1
1-c(p)p-s
.
Taking logarithm leads to ln L(s,c)=åp -ln(1-c(p)p-s), thus one has
1
f(q)
 
å
c
c(a)
ln L(s,c)=
1
f(q)
 
å
c
c(a)
 
å
p
-ln(1-c(p)p-s)
=
 
å
p
 
å
k³ 1
1
f(q)
 
å
c
c(a)
c(pk )p-sk
k
and a simple application of relation (3) gives
1
f(q)
 
å
c
c(a)
ln L(s,c)=
 
å
p
 
å
k³ 1
pk º a mod q
p-sk
k
=
 
å
p º a mod q
p-s +O(1),    s® 1+.     (4)
Then, by splitting the sum in real and complex characters, one gets
 
å
p º a mod q
p-1=
1
f(q)
æ
ç
ç
ç
ç
ç
è
 
å
c=c0
+
 
å
c= æ
ç
ç
è
n
q
ö
÷
÷
ø
+
 
å
c complex
ö
÷
÷
÷
÷
÷
ø
c(a)
ln L(1,c) + O(1).     (5)
c0 is called the principal character and equals 1 whenever n¬ º0 mod q and 0 otherwise. The first sum (over c0) is +¥, as L(s,c0)=z(s)Õp|q (1-p-s). 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 -¥ at s = 1. The Abel summation criterion shows that L(1, c) is finite. One therefore has to show that L(1, c)¹ 0.

This is definitely true for complex characters since otherwise, by setting a=1 and taking the exponential in (4), one has Õc|L(1, c)|>1 which is incompatible with a zero of order at least 2 (coming from L(1, c)=0 and L(1,c)=0) versus a single pole in s=1. Hence the last sum in relation (5) is bounded.

The real problem is then to bound the middle sum in relation (5), that is to say to show that L(1,(· / q))¹ 0. Dirichlet proved this result by a very ingenious method: He evaluated this number in closed form! This is now known as Dirichlet's class number formula:
0¹ L ( 1,(· / q) ) =
ì
ï
ï
í
ï
ï
î
p h
w (q)1/2
when qº 1mod 4
2hln e
(q)1/2
when q º 3mod 4
where h is the class number of Q(((-1)(q+1)/2q)1/2) and e its fundamental unit and w the number of roots of unity in this field (see the canonical reference [2]). Since each of these quantities counts something, they are positive, the result now follows:
 
å
pº a mod q
p-1=+¥.

Simpler proofs using only complex analysis are also possible. The idea is to use Landau's theorem that a Dirichlet series with positive terms has a pole at its abscissa of convergence and apply it to Õc L(s, c) which has just been shown to have positive coefficients.

3   Prime Number Theorem

The distribution of primes is quite irregular, so it is easier to study their statistical behaviour. In this direction, let p(x) be the number of primes £ x. Gauss conjectured that p(x)~ ò2x dt/ln t =:Li(x). This assertion simply says: ``the probability that n is prime is about 1/ln n.'' This result was finally proved by Hadamard and La Vallée Poussin in 1896. Both of them used fundamental ideas of Riemann who was the first to introduce complex analysis in the study of the distribution of prime numbers.

Using Perron's formula, namely
 
å
pn £ x
ln(p) =
1
2ip
ó
õ
c+i¥


c-i¥
-z'(s)
z(s)
xsds
s
and using residues, Riemann essentially found what is perhaps the most important formula in analytic number theory (the von Mangoldt explicit formula):
 
å
pn £ x
ln(p) = x-
z'(0)
z(0)
-
 
å
z(r)=0
x
r
 
r
= x-ln (2p)-
 
å
Â(r)>0
x
r
 
r
-
1
2
ln(1-x-2),     (6)
where sum on the right is over the zeroes of the Riemann z function. These zeroes can be split up into two types: The trivial zeroes at -2, -4, -6, ..., and the zeroes with 0 £ Â £ 1 (the right hand side of (6) reflects this dichotomy). This formula has many interesting properties and reflects the following principles of analytic number theory:

  1. Primes should always be counted with weight ln(p);

  2. Primes and prime powers should be counted together;

  3. There are much less prime powers than primes;

  4. The zeroes of the z function are the ``fundamental frequencies'' of the primes, and in this sense are dual to the primes.
Following Chebyshev, one defines q(x) = åp £ x ln(p) and y(x) = åpn £ x ln(p) = ån £ x L(n) , where L(n) = ln(p) when n=pm, and zero otherwise. A fairly straightforward partial summation shows that the prime number theorem is equivalent to y(x) ~ x (note that trivially, y(x) = q(x) + O((x)1/2)), and that more generally,
y(x) = x + R(x) ÜÞ p(x) ~ Li(x) + O(R(x)/ln(x)).
One can then see from the explicit formula (6) that the prime number theorem would follow if one can bound  r < 1, since each error term would then be of order < x. The prime number theorem would then be equivalent to showing that z(1 + it) ¹ 0 for t¹ 0. In fact, this is an equivalence (as was later shown by Wiener) and Hadamard and La Vallée Poussin were able to prove that z(1 + it) ¹ 0 using some ingenious trigonometric identities. We will give a proof due to Mertens, in 1898. Set r=1+it, then z(r)=0 Þ Â ln z(s+it)® -¥ when s® 1 (we restrict to Â(s)>1). But, by the Euler identity, one has ln z(s)=åp åm³ 1 m-1 p-ms exp(-itmln(p)) and so
 ln z(s)=
 
å
p
 
å
m³ 1
m-1 p
-ms
 
cos(-tmln(p)).
Mertens' trick consists in noticing that 2(1+cos b)2=3 +4 cos b + cos 2b ³ 0, thus 3 lnz(s)+4Âlnz(s+it)+Âlnz(s+2it)³ 0, hence z3(s) |z4(s+it) z(s+2it)| ³ 1.
But, as s® 1, one has z(s)~ (s-1)-1 and |z(s+it)|~ A (s-1) for a some constant A (by analyticity). So one should have z(s+2it)® ¥, this contradicts the fact that z(1+2it) is bounded (by the Abel summation criterion). In conclusion, the z function has no zero with Â(r)=1, the PNT is proved. Note that by mixing his proof of the PNT and the proof of Dirichlet's theorem, La Vallée Poussin proved also that there is asymptotically p(x)/f(q) primes of the shape a+q n less than x. An elementary (i.e. without complex analysis) proof of the PNT was subsequently found by Erdös and Selberg in 1949 (see [4] and [9]).

4   Chebyshev's Bias

All numerical evidence shows that p(x)<Li(x) and it was long believed that this would be true for all x. Similarly, Chebyshev noted that the number of primes of the form 4k+3 seemed to be more abundant than the primes of the form 4k+1, more precisely, let pq,a(x) = |{p £ x: pº amod q}| then p4, 3(x) ³ p4, 1(x).

In fact, Littlewood proved in 1914 that p(x) - Li(x) changes sign infinitely often and the same is true for p4, 3(x) - p4, 1(x). In 1957 Leech showed that p4, 1 (x) > p4,3(x) is first true for x= 26861. That the similar inequality p3, 1 (x) > p3,2(x) is first true for x=608981813029 was shown by Bays and Hudson in 1978. No example of p(x) > Li(x) is known. Skewes first gave an upper bound eee5 which was later reduced by Sherman-Lehman and then te Riele [10] who gave an upper bound of 10370.

This behaviour can easily be explained using explicit formulas. In the case of p(x), the point is the following: The explicit formula (6) expresses y(x) as a sum of powers xr. Assuming the Riemann Hypothesis, one can write this as
y(x) = x - x1/2 æ
ç
ç
è
 
å
z(1/2+ig) = 0
x
ig
 
1/2+ig
ö
÷
÷
ø		 +o(x1/2).
One can now see the reason for the bias: The function y(x) does not count primes but prime powers so what one really wants is the behaviour of q(x) which is given by
q(x) = y(x) - q((x)1/2) +O(x1/3),
so that
q(x) = x - x1/2 æ
ç
ç
è		 1+
 
å
z(1/2+ig) = 0
x
ig
 
1/2+ig
ö
÷
÷
ø		 +o(x1/2).
The function
 
å
z(1/2+ig) = 0
e
igln(x)
 
1/2+ig
,
is a very slowly oscillating trigonometric series which should be zero on average, so the extra term biases q(x) to be smaller than x on average. A simple description is that Li(x) counts the number of prime powers £ x, so the number of primes should be slightly less since the number of prime squares is of the same order as the error term.

There is a similar explanation for the bias in arithmetic progressions. There is an explicit formula
 
å
pn£ x
c(n) ln(p) = - x1/2 æ
ç
ç
ç
è
 
å
L(1/2+ig
 
c
, c) = 0
x
ig
 
c
 
1/2+ig
 
c
ö
÷
÷
÷
ø		 + o(x1/2),
where the Generalised Riemann Hypothesis has been assumed (there is no x term since L(1, c) is no longer a pole if c ¹ c0). As before one has
yq, a(x) =
 
å
pn º amod q
pn £ x
ln(p) =
x
f(q)
-
x1/2
f(q)
 
å
c
c(a)
 
å
L(1/2+ig
 
c
) = 0
x
ig
 
c
 
1/2+ig
 
c
but one really wants to look at
qq,a (x) =
 
å
p º a mod q
p £ x
ln (p) = yq, a (x) -
 
å
p2 º amod q
p2 £ x
ln(p) + O(x1/3) = yq, a(x) - cq, a x1/2 + O(x1/3),
where cq, a is the number of solutions of y2 º amod q. In particular, the same argument shows that there will always be fewer primes in the progression qn + a when a is a residue than when a is a nonresidue. Simply put, the ``balanced'' count is the set of prime powers º amod q so there are fewer primes º amod q when a is quadratic residue since the number of prime squares congruent to a is of the same order as the error term in the analytic formulas.

In 1994, Rubinstein and Sarnak [8] were able to make Chebyshev's bias precise. Assuming GRH (if this is false, then there is no bias) and also the Grand Simplicity Hypothesis (GSH: All the ordinates of zeroes of L-function are linearly independent over Q), then
1
ln(x)
 
å
p(n) > Li n
n £ x
® .00000026,     
1
ln(x)
 
å
p4,3(n) > p4,1(n)
n £ x
® .9959.

References

[1]
Daboussi (Hédi). -- Sur le théorème des nombres premiers. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, vol. 298, n°8, 1984, pp. 161--164.

[2]
Davenport (Harold). -- Multiplicative Number Theory. -- Springer-Verlag, New York, 1980, second edition, xiii+177p. Revised by Hugh L. Montgomery.

[3]
Dirichlet (L.). -- Beweis des Satzes, das jede unbegrenzte arithmetische Progression... Abh. König. Preuss. Akad., vol. 34, 1837, pp. 45--81.

[4]
Erdös (P.). -- On a New Method in Elementary Number Theory which leads to an Elementary Proof of the Prime Number Theorem. Proceedings of the National Academy of Sciences. U.S.A., vol. 35, 1949, pp. 374--384.

[5]
Friedlander (John) and Iwaniec (Henryk). -- Using a Parity-Sensitive Sieve to Count Prime Values of a Polynomial. Proceedings of the National Academy of Sciences. U.S.A., vol. 94, n°4, 1997, pp. 1054--1058.

[6]
Niven (Ivan). -- A Simple Proof that p is Irrational. Bulletin of the American Mathematical Society, vol. 53, 1947, p. 509.

[7]
Ribenboim (Paulo). -- The New Book of Prime Number Records. -- Springer-Verlag, New York, 1996, xxiv+541p.

[8]
Rubinstein (Michael) and Sarnak (Peter). -- Chebyshev's Bias. Experimental Mathematics, vol. 3, n°3, 1994, pp. 173--197.

[9]
Selberg (Atle). -- An Elementary Proof of the Prime-Number Theorem. Annals of Mathematics (2), vol. 50, 1949, pp. 305--313.

[10]
te Riele (Herman J. J.). -- On the Sign of the Difference p(x)-Li(x). Mathematics of Computation, vol. 48, n°177, 1987, pp. 323--328.

[11]
Tenenbaum (Gérald) and Mendès France (Michel). -- Les nombres premiers. -- Presses Universitaires de France, Paris, 1997, 128p.