Prime Numbers

valid certainly for Re(s) > 1. It is interesting that the behavior of the Mertens function runs su ciently deep that the following equivalences are known (in this and subsequent such uses of big-O notation, we mean that the implied constant depends on only):

Theorem 1.4.3. The PNT is equivalent to the statement

M (x) = o(x),

while the Riemann hypothesis is equivalent to the statement

M (x) = O x 12 +

for any fixed > 0.

What a compelling notion, that the Mertens function, which one might envision as something like a random walk, with the M¨obius µ contributing to the summation for M in something like the style of a random coin flip, should be so closely related to the great theorem (PNT) and the great conjecture (RH) in this way. The equivalences in Theorem 1.4.3 can be augmented with various alternative statements. One such is the elegant result that the PNT is equivalent to the statement

∞

µ(n) = 0, n

n=1

as shown by von Mangoldt. Incidentally, it is not hard to show that the sum in relation (1.24) converges absolutely for Re(s) > 1; it is the rigorous sum evaluation at s = 1 that is di cult (see Exercise 1.19). In 1859, Riemann conjectured that for each fixed > 0,

which conjecture is equivalent to the Riemann hypothesis, and perforce to the second statement of Theorem 1.4.3. In fact, the relation (1.25) is equivalent to the assertion that ζ(s) has no zeros in the region Re(s) > 1/2 + . The estimate (1.25) has not been proved for any < 1/2.

In 1901, H. von Koch strengthened (1.25) slightly by showing that the

√

Riemann hypothesis is true if and only if |π(x) − li (x)| = O( x ln x). In fact, for x ≥ 2.01 we can take the big-O constant to be 1 in this assertion; see Exercise 1.37.

Let pn denote the n-th prime. It follows from (1.25) that if the Riemann hypothesis is true, then

pn+1 − pn = O p1n/2+

holds for each fixed > 0. Remarkably, we know rigorously that pn+1 − pn = O p0n.525 , a result of R. Baker, G. Harman, and J. Pintz. But much more is

conjectured. The famous conjecture of H. Cram´er asserts that

lim sup(pn+1 − pn)/ ln2 n = 1.

n→∞

A. Granville has raised some doubt on the value of this limsup, suggesting that it may be as least as large as 2e−γ ≈ 1.123. For primes above 100, the largest known value of (pn+1 − pn)/ ln2 n is ≈ 1.210 when pn = 113. The next highest known values of this quotient are ≈ 1.175 when pn = 1327, and ≈ 1.138 when pn = 1693182318746371, this last being a recent discovery of B. Nyman.

The prime gaps pn+1 − pn, which dramatically showcase the apparent local randomness of the primes, are on average ln n; this follows from the PNT (Theorem 1.1.4). The Cram´er–Granville conjecture, mentioned in the last paragraph, implies that these gaps are infinitely often of magnitude ln2 n, and no larger. However, the best that we can currently prove is that pn+1 − pn is infinitely often at least of magnitude

ln n ln ln n ln ln ln ln n/(ln ln ln n)2,

an old result of P. Erd˝os and R. Rankin. We can also ask about the minimal order of pn+1−pn. The twin-prime conjecture implies that (pn+1−pn)/ ln n has liminf 0, but until very recently the best we knew was the result of H. Maier that the liminf is at most a constant that is slightly less than 1/4. As we go to press for this 2nd book edition, a spectacular new result has been announced by D. Goldston, J. Pintz, and C. Yildirim: Yes, the liminf of (pn+1 − pn)/ ln n is indeed 0.

1.4.2Computational successes

The Riemann hypothesis (RH) remains open to this day. However, it became known after decades of technical development and a great deal of computer time that the first 1.5 billion zeros in the critical strip (ordered by increasing positive imaginary part) all lie precisely on the critical line Re(s) = 1/2 [van de Lune et al. 1986]. It is highly intriguing—and such is possible due to a certain symmetry inherent in the zeta function—that one can numerically derive rigorous placement of the zeros with arithmetic of finite (yet perhaps high) precision. This is accomplished via rigorous counts of the number of zeros to various heights T (that is, the number of zeros σ + it with imaginary part t (0, T ]), and then an investigation of sign changes of a certain real function that is zero if and only if zeta is zero on the critical line. If the sign changes match the count, all of the zeros to that height T are accounted for in rigorous fashion [Brent 1979].

The current height to which Riemann-critical-zero computations have been pressed is that in [Gourdon and Sebah 2004], namely the RH is intact up to the 1013-th zero. Gourdon has also calculated 2 billion zeros near t = 1024. This advanced work uses a variant of the parallel-zeta method of [Odlyzko and Sch¨onhage 1988] discussed in Section 3.7.2. Another important pioneer

Definition 1.4.4. Suppose D is a positive integer and χ is a function from the integers to the complex numbers such that

(1)For all integers m, n, χ(mn) = χ(m)χ(n).

(2)χ is periodic modulo D.

(3)χ(n) = 0 if and only if gcd(n, D) > 1.

Then χ is said to be a Dirichlet character to the modulus D.

For example, if D > 1 is an odd integer, then the Jacobi symbol n is a

D

Dirichlet character to the modulus D (see Definition 2.3.3).

It is a simple consequence of the definition that if χ is a Dirichlet character (mod D) and if gcd(n, D) = 1, then χ(n)ϕ(D) = 1; that is, χ(n) is a root of unity. Indeed, χ(n)ϕ(D) = χ nϕ(D) = χ(1), where the last equality follows from the Euler theorem (see (2.2)) that for gcd(n, D) = 1 we have nϕ(D) ≡ 1 (mod D). But χ(1) = 1, since χ(1) = χ(1)2 and χ(1) = 0.

If χ1 is a Dirichlet character to the modulus D1 and χ2 is one to the modulus D2, then χ1χ2 is a Dirichlet character to the modulus lcm [D1, D2], where by (χ1χ2)(n) we simply mean χ1(n)χ2(n). Thus, the Dirichlet characters to the modulus D are closed under multiplication. In fact, they form a multiplicative group, where the identity is χ0, the “principal character” to the modulus D. We have χ0(n) = 1 when gcd(n, D) = 1, and 0 otherwise. The multiplicative inverse of a character χ to the modulus D is its complex conjugate, χ.

As with integers, characters can be uniquely factored. If D has the prime factorization pa11 · · · pakk , then a character χ (mod D) can be uniquely factored as χ1 · · · χk, where χj is a character (mod paj j ).

In addition, characters modulo prime powers are easy to construct and understand. Let q = pa be an odd prime power or 2 or 4. There are primitive roots (mod q), say one of them is g. (A primitive root for a modulus D is a cyclic generator of the multiplicative group ZD of residues modulo D that are coprime to D. This group is cyclic if and only if D is not properly divisible by 4 and not divisible by two di erent odd primes.) Then the powers of g (mod q) run over all the residue classes (mod q) coprime to q. So, if we pick a ϕ(q)-th root of 1, call it η, then we have picked the unique character χ (mod q) with χ(g) = η. We see there are ϕ(q) di erent characters χ (mod q).

It is a touch more di cult in the case that q = 2a with a > 2, since then there is no primitive root. However, the order of 3 (mod 2a) for a > 2 is always 2a−2, and 2a−1 + 1, which has order 2, is not in the cyclic subgroup generated by 3. Thus these two residues, 3 and 2a−1 + 1, freely generate the multiplicative group of odd residues (mod 2a). We can then construct the characters (mod 2a) by choosing a 2a−2-th root of 1, say η, and choosing ε {1, −1}, and then we have picked the unique character χ (mod 2a) with χ(3) = η, χ(2a−1 + 1) = ε. Again there are ϕ(q) characters χ (mod q).

Thus, there are exactly ϕ(D) characters (mod D), and the above proof not only lets us construct them, but it shows that the group of characters

(mod D) is isomorphic to the multiplicative group ZD of residues (mod D) coprime to D. To conclude our brief tour of Dirichlet characters we record the following two (dual) identities, which express a kind of orthogonality:

Now we can turn to the main topic of this section, Dirichlet L-functions. If χ is a Dirichlet character modulo D, let

∞

χ(n) L(s, χ) = n=1 ns .

The sum converges in the region Re(s) > 1, and if χ is nonprincipal, then (1.28) implies that the domain of convergence is Re(s) > 0. In analogy to (1.18) we have

It is easy to see from this formula that if χ = χ0 is the principal character (mod D), then L(s, χ0) = ζ(s) p|D(1 − p−s), that is, L(s, χ0) is almost the

same as ζ(s).

Dirichlet used his L-functions to prove Theorem 1.1.5 on primes in a residue class. The idea is to take the logarithm of (1.29) just as in (1.19), getting

χ(p)

ln(L(s, χ)) = ps + O(1), (1.30)

p

uniformly for Re(s) > 1 and all Dirichlet characters χ. Then, if a is an integer coprime to D, we have

(mod D)

where the second equality follows from (1.27) and from the fact that χ(a)χ(p) = χ(bp), where b is such that ba ≡ 1 (mod D). Equation (1.31) thus contains the magic that is necessary to isolate the primes p in the residue class a (mod D). If we can show the left side of (1.31) tends to infinity as s → 1+, then it will follow that there are infinitely many primes p ≡ a (mod D), and in fact, they have an infinite reciprocal sum. We already know that the term

on the left corresponding to the principal character χ0 tends to infinity, but the other terms could cancel this. Thus, and this is the heart of the proof of Theorem 1.1.5, it remains to show that if χ is not a principal character (mod D), then L(1, χ) = 0. See [Davenport 1980] for a proof.

Just as the zeros of ζ(s) say much about the distribution of all of the primes, the zeros of the Dirichlet L-functions L(s, χ) say much about the distribution of primes in a residue class. In fact, the Riemann hypothesis has the following extension:

Conjecture 1.4.2 (The extended Riemann hypothesis (ERH)). Let χ be an arbitrary Dirichlet character. Then the zeros of L(s, χ) in the region Re(s) > 0 lie on the vertical line Re(s) = 12 .

We note that an even more general hypothesis, the generalized Riemann hypothesis (GRH) is relevant for more general algebraic domains, but we limit the scope of our discussion to the ERH above. (Note that one qualitative way to think of the ERH/GRH dichotomy is: The GRH says essentially that every general zeta-like function that should reasonably be expected not to have zeros in an interesting specifiable region indeed does not have any [Bach and Shallit 1996].) Conjecture 1.4.2 is of fundamental importance also in computational number theory. For example, one has the following conditional theorem.

Theorem 1.4.5. Assume the ERH holds. For each positive integer D and each nonprincipal character χ (mod D), there is a positive integer n < 2 ln2 D with χ(n) = 1 and a positive integer m < 3 ln2 D with χ(m) = 1 and

χ(m) = 0.

the ERH, was originally due to N. Ankeny in 1952. Theorem

This result is in [Bach 1990]. That both estimates are O ln2 D , assuming 1.4.5 is what

is behind ERH-conditional “polynomial time” primality tests, and it is also useful in other contexts.

The ERH has been checked computationally, but not as far as the Riemann hypothesis has. We know that it is true up to height 10000 for all characters χ with moduli up to 13, and up to height 2500 for all characters χ with moduli up to 72, and for various other moduli [Rumely 1993]. Using these calculations, [Ramar´e and Rumely 1996] obtain explicit estimates for the distribution of primes in certain residue classes. (In recent unpublished calculations, Rumely has verified the ERH up to height 100000 for all characters with moduli up through 9.) Incidentally, the ERH implies an explicit estimate of the error in (1.5), the prime number theorem for residue classes; namely, for x ≥ 2, d ≥ 2,

We note the important fact that there is here not only a tight error bound, but an explicit bounding constant (as opposed to the appearance of just an implied, nonspecific constant on the right-hand side). It is this sort of hard

Vinogradov’s proof was an extension of the earlier work of Hardy and Littlewood (see the monumental collection [Hardy 1966]), whose “circle method” was a tour de force of analytic number theory, essentially connecting exponential sums with general problems of additive number theory such as, but not limited to, the Goldbach problem.

Let us take a moment to give an overview of Vinogradov’s method for estimating the integral (1.36). The guiding observation is that there is a strong correspondence between the distribution of primes and the spectral information embodied in En(t). Assume that we have a general estimate on primes not exceeding n and belonging to an arithmetic progression {a, a + d, a + 2d, . . .} with gcd(a, d) = 1, in the form

which estimate, we assume, will be “good” in the sense that the error term will be suitably small for the problem at hand. (We have given a possible estimate in the form of the ERH relation (1.32) and the weaker, but unconditional Theorem 1.4.6.) Then for rational t = a/q we develop an estimate for the sum (1.35) as

=π(n; q, f )e2πif a/q + O(q),

gcd(f,q)=1

where it is understood that the sums involving gcd run over the elements f [1, q − 1] that are coprime with q. It turns out that such estimates are of greatest value when the denominator q is relatively small. In such cases one may use the chosen estimate on primes in a residue class to arrive at

En(a/q) = cq (a) π(n) + O(q + | |ϕ(q)),

ϕ(q)

where | | denotes the maximum of | (n; q, f )| taken over all residues f coprime to q, and cq is the well-studied Ramanujan sum

cq (a) = e2πif a/q . (1.37)

gcd(f,q)=1

We shall encounter this Ramanujan sum later, during our tour of discrete convolution methods, as in equation (9.26). For the moment, we observe that [Hardy and Wright 1979]