Open Questions: The Riemann Hypothesis

See also: Number theory

We may -- paraphrasing the famous sentence of George Orwell -- say that 'all mathematics is beautiful, yet some is more beautiful than the other.' But the most beautiful in all mathematics is the Zeta function. There is no doubt about it. Krysztof Maslanka God may not play dice with the universe, but there's something strange going on with the prime numbers! Paul Erdős

Introduction The distribution of prime numbers Properties of the zeta function The prime number theorem The Riemann hypothesis Generalizations of the Riemann hypothesis Bernoulli numbers The Riemann hypothesis and algebraic geometry The Riemann hypothesis and spectral theory Other generalizations of the Riemann hypothesis Reasons for believing the Riemann hypothesis Recommended references: Web sites Recommended references: Magazine/journal articles Recommended references: Books

Introduction

David Hilbert, in his famous speech at the International Congress of Mathematicians at Paris in 1900, included this problem as number 8 in his list of 23 challenging problems for mathematicians in the coming century. Today, over 100 years later, it is one of the few on that list that have not been solved. Many contemporary mathematicians consider it the most important unsolved problem in mathematics. In any case, it has been included among the seven most important problems, for the solution of any of which a prize of $1 million has been offered by the Clay Mathematics Institute.

The definition of the function ζ(s) is not especially difficult. The statement of the Riemann hypothesis is certainly clear enough (if you have any acquaintance with complex numbers). So why on Earth is this hypothesis considered so important? One reason is that it has turned out, in spite of its apparent simplicity, to be so difficult to prove. But a more important reason is that the problem is thoroughly entangled with many questions about prime numbers -- which of course are of intense interest to number theorists, yet at first glance have little to do with functions of a complex variable.

We shall soon see that there is a very close and simple relationship between ζ(s) and prime numbers. But hidden beneath this simple relationship seems to be one much deeper and more profound -- one that connects properties of the zeros of ζ(s) to the way in which prime numbers are distributed among the integers. And not only is there reason to suspect such a relationship, but much more general relationships are suspected to exist between various generalizations of ζ(s) and more general types of algebraic objects.

The function ζ(s) is defined by the infinite sum

ζ(s) = ∑_1≤n<∞ 1/n^s

A great deal of care is required when working with "infinite" sums, because only in certain circumstances do they actually make sense. In this case, it can be shown that the sum actually "converges" only when the real part of s is greater than 1 (i. e. Re(s) > 1). (Recall that any complex number can be written uniquely in the form s = σ + iτ, where σ and τ are both real and i² = -1. We write Re(s) = σ and Im(s) = τ to denote the real and imaginary parts, respectively.)

To say that the infinite sum "converges" for Re(s) > 1 is to say that the complex numbers represented by finite partial sums of the form

S_N(s) = ∑_1≤n<N 1/n^s

Without going into further details, let it suffice to say that there are standard techniques for rigorously talking about infinite sums and proving that they make good sense (i. e. "converge") under appropriate conditions. These rigorous techniques were fully established only in the second half of the 19th century. However, mathematicians had worked with infinte sums long before that -- rearranging terms and performing other kinds of formal operations (e. g. differentiation and integration) -- in order to derive a wide variety of formal identities. Often these formal manipulations could be rigorously justified, at least under suitable conditions. Sometimes, unfortunately, they could not, in which case nonsense results such as 1 = 0 could be derived, just as can happen when improperly allowing division by 0. Indeed, the basic operations of calculus itself (differentiation and integration) themselves involve operations with infinite sums and sequences. It was a most welcome accomplishment of 19th century mathematics to put all such "infinite" operations on a rigorous basis.

With standard rigorous techniques for working with infinite sums (which are also often called "infinite series") in hand, it was clear that the infinte sum used above to define ζ(s) definitely did not converge when Re(s) < 1. Just for example, if s = -1, it would amount to trying to assign a reasonable value to the sum 1 + 2 + 3 + ... . Now, there are roundabout ways for assigning plausible values to such a sum. For instance, the remarakable Indian mathematician Srinivasa Ramanujan (1887-1920) thought that 1 + 2 + 3 + ... should equal -1/12. And soon we shall see a sense in which ζ(-2k), which is formally 1 + 2^2k + 3^2k + ..., has a simple numerical value for integers k &ge 1. But all of this is beyond the standard rigorous concept of convergence.

Instead of representing functions as infinite series, mathematicians developed an entirely different way of defining functions of a complex variable. It is called the method of "analytic continuation". It turns out that complex-valued functions of complex variables have remarkable properties. For one thing, if such a function has even a first derivative (at a given point), it must actually have derivatives of all orders (at that point). Functions with that property are called "analytic" or "holomorphic". Although this property is specific to a given point, for all "reasonable" functions it occurs for all but (at most) an isolated set of points in the complex plane. Another surprising property of analytic functions is that if they are constant on any circular disk in the complex plane (that is, a set of the form {z : |z - z₀| < r}), then they must be constant everywhere in the complex plane that they are analytic. (The notation |w|, for a complex number w, is called the "norm" or "absolute value" of w and is defined as √(Re(w)² + Im(w)²).) Consequently, any two analytic functions that are equal on some disk -- so that their difference is an analytic function equal to the constant 0 on the disk -- must be equal everywhere. Using this fact, the process of analytic continuation consists of extending analytic functions defined on some disk in the complex plane to the whole plane (perhaps with a few isolated singular points) by defining the function on a sequence of overlapping disks.

Now, the infinite series used to define ζ(s) for Re(s) > 1 defines the function not merely on a disk but on an infinite half of the complex plane. It can be shown that anaytic continuation permits ζ(s) to be defined for any complex s except for isolated singularities. (s = 1 is one of these.) The zeta function is therefore well-defined and a generally quite well-behaved function for all but isolated singularities. So it can be meaningfully discussed even when its infinite series doesn't make sense. We shall soon see that there is a "functional equation" which specifies the value of ζ(1-s) in terms of ζ(s) -- in particular, for Re(s) < 0.

Although the function is usually called the "Riemann" zeta function, Riemann, as we noted, was not the first to work with it. Leonhard Euler (1707-83) studied the series 100 years eariler, discovered a number of its properties, and in fact found another extremely important representation of it in terms of an infinite product instead of an infinite series. P. G. L. Dirichlet (1805-59) (who was perhaps Riemann's favorite teacher at the University of Göttingen) studied more general series of the form ∑_n a_n/n^s, for arbitrary complex coefficients a_n. Accordingly, series of this form are called Dirichlet series.

Riemann's contribution, however, was to realize that ζ(s) could be analytically continued to the whole complex plane, to derive many important properties of ζ(s), such as its functional equation, and -- most importantly -- to suspect its deeper relationship to the distribution of prime numbers. And along the way, to state his celebrated hypothesis. So there's plenty of justification for calling ζ(s) the Riemann zeta function.

The product formula

∑_1≤n<∞ 1/n^s = ∏_p (1 - 1/p^s)^-1

If you've read this far, you've probably had enough mathematics to be comfortable with the ∑-notation, but may not be familiar with the product notation on the right side of this equation. It should be intuitively clear, however, that it may be understood in an analogous way as a product of terms of the indicated form. In this case, the product is taken over all positive primes, and only primes. (Note that 1 is not considered a prime, so the product makes sense.) For an infinite product, there are major issues of convergence, just as for infinite sums. Euler worked mostly formally and didn't bother much about such issues. Nevertheless, the issues can be dealt with and a rigorous treatment of infinte products can be given, just as for infinite sums.

To "prove" Euler's equation, in a loose formal sense, we need only recall the so-called "geometric series":

1/(1-x) = ∑_0≤n<∞ xⁿ = 1 + x + x² + ...

1 = (1 - x)(1 + x + x² + ...) = (1 + x + x² + ...) - (x + x² + x³ + ...)

(1 - 1/p^s)^-1 = ∑_0≤k<∞ (1/p^s)^k = ∑_0≤k<∞ p^-ks

∏_p (1 - 1/p^s)^-1 = ∏_p ∑_0≤k<∞ p^-ks = (1 + 2^-s + 2^-2s + ...) × (1 + 3^-s + 3^-2s + ...) × ...

The existence of this product formula of Euler to represent the zeta function is the simplest, most obvious way in which the zeta function is connected with prime numbers. It is certainly not the only way, nor -- by a long shot -- the most profound. To begin to appreciate deeper connections, we have to look into what is known about the distribution of prime numbers.

The distribution of prime numbers

Interestingly enough, Euler's product formula for ζ(s) gives an independent proof that there are infinitely many primes. It is known that the infinite sum ∑_n 1/n diverges, i. e. is not a finite number. By a standard result of calculus, one has the "Taylor series" expansion:

-log(1-x) = ∑_1≤n<∞ xⁿ / n

γ = lim_N→∞ [(∑_1≤n≤N 1/n) - log N]

So already the function ζ(s) tells us something about the distribution of primes. Now, given that there are infinitely many primes, we want to ask ourselves how are they distributed. Simply looking at a table of primes suggests that the primes come farther and farther apart the larger they are. We would like to have a better quantification of this observation.

Intuitively, the reason that primes become sparser as they grow larger is that in some sense it becomes more "difficult" for a large number to be prime the larger it is. This is because there are an ever increasing number of smaller numbers (primes among them) which could divide it. The ancient Greeks knew how to construct tables of primes using what is known as the "sieve of Eratosthenes", after its inventor, who lived in the 3rd century BCE. The method of the sieve is to list all numbers up to whatever limit one wishes to deal with. Then starting with the prime 2, cross out every second number thereafter, every 3rd number after 3, every 5th number after 5, and so on, up to a prime larger than the square root of the largest number in the original list. The numbers which are left are guaranteed to be prime.

One can ask what is the probability that any given number in the list will be crossed out at some point in the process. Of course, for any particular number, it is either prime or it isn't, so the probability is either 0 or 1. But suppose we don't know in advance which numbers are which and we simply pick one at random. What can we say about the probability it is prime, knowing only its size?

Suppose the number is X. The probability it is divisible by 2 is 1/2. The probability it is divisible by 3 is 1/3, and so the probability that it is not divisible by 3 is 2/3. In general, for any prime p ≤ X, the probability X is divisible by p is 1/p, and the probability it is not divisible by p is 1 - 1/p. Therefore, the probability that X is prime is the probability that it is not divisible by any prime p ≤ X, which is the product of all the separate probabilities, which we define as G(X) = ∏_p≤X (1 - 1/p) -- provided the probabilities for divisibility by each prime are independent of each other. A priori we don't know about independence. There's no good reason to think the probabilities are entirely independent. We will see that a slight adjustment is necessary to get the correct probability.

We are interested in the total number of primes ≤ X, and so we give it a name. We will call this number π(X). A very rough approximation is the count of numbers ≤ X times the probability each is prime. We would like to find a simple formula f(X) such that the ratio of π(X) to f(X) for large X is 1. Mathematicians call this notion "asymptotic equivalence" and even have a notation for it. We write π(X) ∼ f(X) provided π(X)/f(X) → 1 as X → ∞. Note that is not to say that the limit of π(X) is f(X) as X → ∞, since that would imply the difference between the two tends to 0 as X → ∞. We are willing to settle for something a little weaker, namely that the ratio tends to 1 in the limit.

Does that product expression for the probability G(X) look familiar? It sure as heck does. It is a finite form of the product in Euler's formula for the zeta function (or rather, its reciprocal, to be precise). We also recall that there is a relationship between the finite sum ∑_1≤n≤X 1/n and log(X) which involves Euler's constant γ. Is there a similar relationship between the finite product and log(X)? Indeed there is. It was discovered by Franz Mertens in 1874, and so it's called Mertens' theorem:

1/G(X) = ∏_p≤X (1 - 1/p)^-1 ∼ e^γ log(X)

π(X) ∼ e^γG(X)X

π(X) ∼ X / log(X)

This formula can be interpreted to say that the probability that a given number less than X is prime is just 1/log(X). Alternatively, it says that the average spacing between primes less than X is about log(X).

The argument here is by no means a rigorous proof. The result is in fact true, but it is a very difficult theorem, now called the prime number theorem. C. F. Gauss (1777-1855) suspected (perhaps as early as 1792 by his own account) that the probability any integer X is prime is about 1/log(X), and hence that π(X) ∼ X / log(X). But a rigorous proof did not come until 1896, when Jacques Hadamard (1865-1963) and C. J. de la Vallée-Poussin (1866-1962) independently came up with proofs. Although various proofs of the prime number theorem have been given since then, none are especially easy.

Interestingly enough, Riemann himself seems not to have worked much on this problem, even though his famous conjecture turns out to be equivalent to a statement about just how precise the approximation of the prime number theorem actually is. To understand somewhat of the nature of this deep connection, we have to talk more about the zeta function.

Properties of the zeta function

To begin with, Riemann showed that ζ(s), as defined by either the Dirichlet series or the Euler product only in case Re(s) > 1, actually had an "analytic continuation" to the entire complex plane except for a single singularity at s=1. (That singularity was to be expected, given that the Dirichlet series reduces to the Taylor expansion of -log(1-s) at s=1.) Riemann did not use an abstract argument based on extending the function by means of defining it piecemeal on overlapping disks. (This procedure originated in the work of Karl Weierstrass (1815-97), which came a little later.) Instead, Riemann came up with an explicit formula for ζ(s) which actually made sense, and defined ζ(s), for all s ≠ 1. That formula can be written as:

ζ(s) = -Γ(1-s)(2πi)^-1 ∫_C (-x)^s-1/(e^x-1) dx

Γ(s) = ∫_0≤x<∞ e^-xx^s-1 dx

Even more importantly, Riemann proved that ζ(s) satisfies a "functional equation" which relates its values at s to its values at 1 - s. The functional equation is this:

ζ(1-s) = (π^1/2-s Γ(s/2) / Γ((1-s)/2)) ζ(s)

Various important facts about ζ(s) follow immediately from the functional equation. Clearly, Γ(s) plays a significant role. One fact about Γ(s) is that it runs off to infinity when s is any integer ≤ 0. (Such a singularity is said to be a "pole" of the function.) Suppose first that s = 1. ζ(1) and Γ(0) both have poles, but because one is in the numerator and the other is in the denominator on the right hand side of the functional equation, they cancel each other out. Γ(1/2) = √π, so as a result ζ(0) is finite and nonzero.

Now suppose s is an odd positive integer ≥ 3. Then Γ(s/2) and ζ(s) are both finite, but there is a pole corresponding to Γ(1-s) in the denominator, so the functional equation shows ζ(-2k) is zero when s = 2k+1 and k is an integer ≥ 1. These values of s = -2k are the only real zeros of ζ(s), as well as the only zeros when Re(s) ≤ 0. They are called the "trivial zeros" of the zeta function. (The Euler product representation of ζ(s) for Re(s) > 1 ensures there are no zeros in that part of the complex plane.)

Finally suppose s = 2k is an even positive integer. Then both Γ(s) and Γ((1-s)/2) are finite and nonzero, as is ζ(s). So ζ(1-2k) is finite and nonzero for integers k ≥ 1. Interestingly enough, those values are rational numbers, which we can write like so:

ζ(1-2k) = -B_2k / 2k

xe^x / (e^x - 1) = ∑_0≤n<∞ B_n xⁿ / n!

(The expression x / (e^x - 1) can also be used as a generating function to define Bernoulli numbers, and the only difference is in the sign of B₁. This fact isn't immediately obvious, and it implies very interesting recurrence relationships among the Bernoulli numbers.)

The functional equation allows us to give an expression for ζ(2k) as well, although Euler had already discovered the formula. It is:

ζ(2k) = (-1)^k+1 B_2k (2π)^2k / (2(2k)!)

However, we haven't yet come to one more important result Riemann stated in his 1859 paper, though he gave no proof of it. (Riemann was the sort of mathematician who "knew" more than he could rigorously prove, as his famous hypothesis shows.) This result is the most important as far as the distribution of prime numbers is concerned. It is an entirely different product formula for ζ(s), which is:

ζ(s) = -e^bs [s(1-s)Γ(s/2)]^-1 ∏_ρ∈Z (1-s/ρ)e^s/ρ

This formula is rather like the expression for a polynomial f(s), which can be written (up to a constant factor) as

f(s) = ∏_ρ (1-s/ρ)

Given that there are in fact two product formulas for ζ(s), that one involves the prime numbers, and that the other involves the zeros of ζ(s), it's hard to avoid the suspicion that there is some relationship between the distribution of the primes and the distribution of the zeros. We'll be more specific about this shortly.

It turns out that many complex analytic functions have a product formula of the above sort, provided they are sufficiently well-behaved. Weierstrass first proved the existence of this kind of representation, and in 1893, just three years before he proved the prime number theorem, Hadamard (motivated by his work on the prime number theorem) proved a more general result, which as a corollary proved the product formula that Riemann conjectured for ζ(s). Such products are now called Hadamard products.

From this representation, together with the functional equation of ζ(s) and some properties of its Dirichlet series, Riemann claimed an asymptotic formula for the number of zeros whose imaginary parts were less than some given value T. More precisely, he claimed

#({ρ∈Z: 0 < Im(ρ) < T}) ∼ (T/2π)log(T/2π) - T/2π

Riemann didn't give a proof of this formula, and it was not rigorously proven until 1905 (by H. von Mangoldt). Nevertheless, he went further and also claimed

#({ρ∈Z: 0 < Im(ρ) < T and Re(ρ) = 1/2}) ∼ (T/2π)log(T/2π) - T/2π

The prime number theorem

That is, based on probabalistic ideas as well as extensive calculations, Gauss conjectured that

π(x) ∼ x/log(x)

There's a similar function which gives a slightly better result, both theoretically and computationally. It is the "logarithmic integral":

Li(x) = ∫_2≤t≤x log(t)^-1 dt

One of Gauss' near contemporaries, A. M. Legendre (1752-1833) reached similar conclusions. His book Theorie des nombres in about 1800 suggested that

π(x) ∼ x/(A log(x) + B)

Riemann's 1859 paper gave an "explicit" formula for π(x) that shows

π(x) = Li(x) + other stuff

Starting with the Euler product for ζ(s), we've already found, for Re(s) > 1,

log ζ(s) = ∑_p ∑_n (1/n)p^-ns

F(x) = ∑_1≤n<∞ π(x^1/n)/n

log ζ(s) = s ∫_0≤x≤∞ F(x)x^-s-1 dx

F(x) = (2πi)^-1 ∫ log ζ(s) (x^-s/s) ds

F(x) = Li(x) - ∑_ρ Li(x^ρ) - log 2 + ∫_x≤t≤∞ (t(t² - 1)log t)^-1 dt

To finish the derivation of a formula for π(x) there is one more "trick", which involves the number-theoretic function μ(n) defined for integers by μ(n) = 0 if n is divisible by the square of a prime, μ(n) = 1 if n is a product of an even number of distinct primes, and μ(n) = -1 if n is a product of an odd number of distinct primes. μ(n) is called the Möbius function. (The same Möbius who was responsible for the Möbius strip.) Using that, we can "invert" the expression for F(x) in terms of π(x) to obtain

π(x) = ∑_1≤n<∞ μ(n)F(x^1/n)/n

π(x) = Li(x) + ∑_2≤n<∞ μ(n)Li(x^1/n)/n + other stuff

Oddly enough, Riemann did not try to use this explicit formula to give an asymptotic formula for π(x). Or perhaps not so oddly, because estimating the size of the "other stuff" is quite difficult. In particular, it entails dealing with with sums like

∑_1≤n<∞ ∑_ρ Li(x^ρ/n)

The bottom line is that it is necessary to describe the location of the zeros of ζ(s) at least crudely even to prove the prime number theorem. And in order to get the "best possible" estimate of π(x) it's necessary to prove Riemann's hypothesis.

As intriguing as it is to have an actual "explicit" formula for π(x), making use of the formula is another matter. The first term of the formula is Li(x), which is precisely what we want to have as the asymptotic value of π(x). However, the rest of the formula makes it too cumbersome to use for deriving the prime number theorem -- since, as it turns out, Chebyshev had in 1850 already found an equivalent form of the theorem which was much easier to work with.

The basis of the alternative formulation is the function ψ(x) defined by the finite sum

ψ(x) = ∑_pⁿ≤x log p

Where did this strange definition of ψ(x) come from? It isn't quite as weird as it may seem. For one thing, ψ(x) may be thought of as a function that counts the number of primes, only in a slightly different way than π(x) does. The latter function increases by 1 every time x is a prime. ψ(x), on the other hand, increases by log(x) every time x is a power of a prime. In exchange for this odd way of counting, the conjectured asymptotic relationship π(x) ∼ Li(x) is equivalent to the relationship ψ(x) ∼ x, which is a much simpler function of x -- almost as simple as possible, in fact. One might therefore expect ψ(x) to be easier to work with.

In any case, we can write ψ(x) = ∑_1<n≤x Λ(n) where

Λ(n) = log p if n = p^k for some k ≥ 1, otherwise Λ(n) = 0

-ζ′(s)/ζ(s) = (∑_p log (1-p^-s))′ = ∑_p p^-s(log p) / (1 - p^-s) = ∑_p ∑_k≥1 (log p)p^-ks = ∑_n≥2 Λ(n)n^-s

The expression that relates ζ(s) to ψ(x) is

-ζ′(s)/ζ(s) = s ∫_0≤x<∞ ψ(x)x^-s-1 dx

-ζ′(s)/ζ(s) = ∑_n≥2 Λ(n)n^-s = ∑_n≥2 s&Lambda(n) ∫_n≤x<∞ x^-s-1 dx = s∫_0≤x<∞ (∑_1<n≤x Λ(n)) x^-s-1 dx = s ∫_0≤x<∞ ψ(x)x^-s-1 dx)

By manipulations similar to those Riemann used to obtain an explicit formula for π(x), von Mangoldt found from this, in 1894, that

ψ(x) = x - ∑_ρ x^ρ/ρ + ∑_1≤n<∞ x^-2n/2n - ζ′(0)/ζ(0)

ψ(x) = x - ∑_ρ x^ρ/ρ + log(x²/(x²-1))/2 - log(2π)

It is required to calculate the limit of ψ(x)/x as x → ∞. The ratio of the last two terms to x in the limit is 0, so in order to have ψ(x) ∼ x as desired, it suffices to show that ∑_ρ x^ρ-1/ρ → 0 also as x → ∞. Suppose the limit of that sum can be evaluated termwise, so it suffices to show that x^ρ-1 → 0 for each ρ. Then it only remains to show Re(ρ) < 1 for all roots of ζ(s), because then Re(ρ-1) < 0 and so the absolute value

|x^ρ-1| = |x|^Re(ρ-1) → 0

It was these last two steps that both Hadamard and de la Vallée Poussin independently accomplished in 1896, completing the proof of the prime number theorem. In other words, considerably less than the Riemann hypothesis needs to be known about the zeros of ζ(s). All that's required is that none of the nontrivial zeros lie on the boundary of the critical strip, Re(s) = 1 (or equivalently, by the functional equation, Re(s) = 0). The fact that there are no zeros with Re(s) = 1 turns out to be pretty easy to deduce in a variety of ways from the Euler product.

In the time since 1896, many other proofs of the prime number theorem have been found, some of them fairly short. There are even proofs which are "elementary" in the sense that they don't use complex analysis or Fourier transforms. The first such "elementary" proof was devised by Paul Erdös and Atle Selberg in the late 1940s. There are a number of variations of this proof, but none are especially straightforward or "natural", nor do they provide much insight into the theorem. Various other short proofs use more sophisticated techniques involving Fourier transfoms and "Tauberian theorems". Proofs of this sort were pioneered by Norbert Wiener and Shikao Ikehara around 1930.

Yet another kind of short proof of the prime number theorem is based on a Tauberian theorem (that is, a theorem about conditions for convergence of a series) due to A. E. Ingham. The reason that Tauberian theorems play a prominent role in this theory is that they relate the convergence of a series (such as a Dirichlet series) to the size of its coefficients. In particular, the prime number theorem is equivalent to ψ(x) ∼ x, which means that ψ(x)/x → 1 as x → ∞. But ψ(N)/N = (1/N)∑_1<n≤N Λ(n) is simply the average value (up to the Nth term) of the coeffiecients of the Dirichlet series for -ζ(s)′/ζ(s). Then the average value of the coefficients of that series being 1 is equivalent to the average value of the coefficients of -ζ(s)′/ζ(s) - ζ(s) being 0. So one can write out what those coefficients are exactly, make other estimates on them, and eventually establish the desired result.

However, there is a shortcoming of most of these shorter proofs, in that they are generally unable to say much about the absolute error in the approximation. That is, π(x) ∼ Li(x) means

π(x) = Li(x) + O(f(x))

So we want to know as much as possible about the location of the zeros of ζ(s), to know the best estimates that can be made for bounds on the approximation of π(x) by Li(x) -- with the best bounds being achieved if the Riemann hypothesis is true. So let's take a closer look at that.

The Riemann hypothesis

Let's review a bit. The first clue that the Riemann zeta function is related to prime numbers came from the Euler product formula, which exhibits ζ(s) explicitly in terms of an infinite product (that converges for Re(s) > 1) which has factors involving the prime numbers in a simple way. From this formula it is possible to derive fairly straightforwardly many identites that relate ζ(s) to a variety of arithmetic functions. In fact, it can be shown that

log ζ(s) = s ∫_2≤x<∞ π(x)/(x(x^s-1)) dx

When the prime number theorem was eventually proved by Hadamard and de la Vallée Poussin the plot thickened considerably. Not only was ζ(s) related to the prime numbers in a formal way, but its properties turned out to play a governing role in the asymptotic behavior of π(x). More specifically, it was critical for the proof of the prime number theorem that ζ(s) should not have any zeros with Re(s) = 1.

But the connection between the zeros of ζ(s) and the asymptotic behavior of π(x) turned out to be far stronger than that. Using the alternative product formula for ζ(s) that Riemann conjectured and Hadamard proved, it could be shown that the precise distribution of the zeros of ζ(s) affected the goodness of the approximation of π(x) by Li(x). And in the best possible case, if all zeros of ζ(s) lie on the line Re(s) = 1/2 as Riemann conjectured, then the error term, i. e. the difference π(x) - Li(x), is as small as possible. Not only that, but the converse is also true: if the error term has the conjectured smallest possible form, then the Riemann hypothesis is true.

So the main conjecture which is equivalent to the Riemann hypothesis is formulated in terms of the absolute error in the approximation of π(x) by Li(x). Specifically it is the conjecture that we can write

π(x) = Li(x) + E(x), where |E(x)| ≤ Cx^1/2 log(x)

OK, if we can't yet say for sure that Re(s) = 1/2 for all s such that ζ(s) = 0, what can we say? Progress towards establishing the Riemann hypothesis should be viewed in terms of giving tighter limits on Re(s). As before, we shall let Z be the set of zeros of ζ(s) in the critical strip {s ∈ C: 0 ≤ Re(s) ≤ 1}. ρ will stand for any member of Z. It can be fairly easily shown that there is a real constant c > 0 such that if ρ ∈ Z then

Re(ρ) < 1 - c Min(1, (log |Im(ρ)|)^-1)

In spite of that, most zeros must be located close to the line for large values of Im(s). It has been shown that more than 99% of zeros satisfy

|Re(ρ) - 1/2| ≤ 8 / log|Im(ρ)|

Even from this relatively weak information about ρ it is possible to make an estimate for the error term E(x) in the relation π(x) = Li(x) + E(x). Namely it can be shown that

|E(x)| ≤ C x e^{-c √(log x)}

E(x) = O(x e^{-c √(log x)})

It is possible to make tests of the Riemann hypothesis by counting the number of zeros whose imaginary part is less than any given T > 0. That is, we look at the number N(T) which Riemann considered (as described above) that is defined by

N(T) = #({ρ∈Z: 0 < Im(ρ) < T})

N(T) = (T/2π)log(T/2π) - T/2π + O(log T)

In a slightly different direction, it is possible to make estimates of how many zeros lie off the line Re(ρ) = 1/2. One defines, for T > 0 and σ > 1/2

N(σ,T) = #({ρ ∈ Z: Re(ρ) > σ and 0 < Im(ρ) < T})

N(σ,T)/N(T) → 0 as T → ∞

There have been a few additional results regarding how many zeros are on the line. G. H. Hardy and J. E. Littlewood showed that infinitely many are. Later Hardy and Atle Selberg showed that a nonzero proportion are on the line. The best currently proven estimate is that this proportion is more than 40%.

Equivalents of the Riemann hypothesis

|π(x) - Li(x)| = O(x^1/2 log(x))

Given this importance of the hypothesis, it is often helpful to have other equivalent formulations available. So we'll just mention some of them now. It's fairly easy to guess at some equivalent statements. For example, think of functions that can be defined in terms of ζ(s) which would have singularities if ζ(s) = 0 for some s with Re(s) > 1/2. The logarithmic derivative (log ζ(s))′ = ζ′(s)/ζ(s) is one such function. We can easily eliminate the pole of ζ(s) at s = 1, and therefore state that

ζ′(s)/ζ(s) + (s-1)^-1

We worked a bit with ζ′(s)/ζ(s) in connection with the function ψ(x). When all the gory details are processed, it turns out that two things are true:

If for some α < 1 one has

ψ(x) = x + O(x^α) as x → ∞

then all zeros ρ of ζ(s) in the critical strip have Re(ρ) ≤ α

If all zeros of ζ(s) are contained in the strip {s ∈ C: 1-θ ≤ Re(s) ≤ θ} for some θ < 1, then

ψ(x) = x + O(x^θ (log(x))²) as x → ∞

On the other hand, if the Riemann hypothesis is correct, we can apply the second statement with θ = 1/2 to conclude that in fact ψ(x) = x + O(x^1/2 (log(x))²) and hence π(x) = Li(x) + O(x^1/2 log(x)). Conversely, by the first statement, this estimate implies the Riemann hypothesis.

In other words, the Riemann hypothesis is equivalent to the estimate π(x) = Li(x) + O(x^1/2 log(x)), and no better estimate of this sort is possible.

Unfortunately, we must point out that no results have actually been proven of the form that all zeroes of ζ(s) must have Re(ρ) ≤ α for some α < 1. Even now, the best result we have for a zero-free region is Hadamard's, which illustrates how little actual progress on the Riemann hypothesis has been made in almost 100 years.

The Lindelöf hypothesis

One of these was proposed by E. Lindelöf in 1908 and is accordingly known as the Lindelöf hypothesis. Interestingly enough, it is concerned with how large |ζ(s)| can be on the line s = 1/2. There are several equivalent ways to state this hypothesis. Considering ζ(1/2+iτ) as a function of a real variable τ, one formulation is the statement that

|ζ(1/2 + iτ)| = O(τ^ε)

|ζ(σ + iτ)| = O(τ^ε)

The relation of the hypotheses of Riemann and Lindelöf is made clearer given the provable fact that the Lindelöf hypothesis is equivalent to the requirement that the number of zeros of ζ(s) in the region 1/2 < Re(s) < 1 and T < Im(s) < T+1, grows more slowly than log(T), or more precisely,

(N(σ,T+1) - N(σ,T))/log(T) → 0 as T → ∞

Nevertheless, even the Lindelöf hypothesis remains an open problem, demonstrating just how difficult the Riemann hypothesis must be. The best result currently known along these lines, due to Hermann Weyl, is that

|ζ(1/2 + iτ)| = O(τ^1/6+ε)

There are a number of other results which could be proven if the Riemann hypothesis is assumed. Apart from better estimates on the distribution of prime numbers, most such results, like the Lindelöf hypothesis, concern either the distribution of the imaginary parts of zeros of ζ(s), or the absolute values of ζ(s) or log(ζ(s)) as Im(s) becomes large. Most such results, likewise, remain unproven without assuming the Riemann hypothesis. Of course, if any were ever disproven, the Riemann hypothesis would be also -- and that hasn't happened.

Generalizations of the Riemann hypothesis

Algebraic number theory is a huge and deep subject, with a plethora of difficult open questions of its own. We go into more detail on this subject elsewhere, but there are straightforward analogues of ζ(s) which play a large role in the theory and which are relevant to mention here.

Many of the important questions of algebraic number theory revolve around the properties of suitable analogues of prime numbers. It is not hard to define the meaning of "integers" within the class of all algebraic numbers. However, it turns out, in general algebraic number theory one does not usually have a set of numbers (even algebraic numbers) in terms of which any arbitrary algebraic integer can be expressed uniquely as a product of "prime" numbers of some sort. One must instead work with certain sets of algebraic numbers called "ideals". As you might expect, this makes the theory much more complicated. It is therefore, nevertheless, striking that a fairly simple analogue of the Riemann zeta function exists. The analogue is called a Dedekind zeta function, after Richard Dedekind (1831-1916) who did much to develop algebraic number theory, and was a close friend of Riemann and his first biographer as well.

To talk about Dedekind zeta functions we need a little terminology. First, a "field" is a collection of mathematical objects (such as numbers or functions) for which there are concepts analogous to the addition and multiplication of ordinary numbers. There are certain axiomatic rules which specify how addition and multiplication should behave. For instance, there is the "distributive" rule of multiplication with respect to addition: a(b + c) = ab + ac for any a, b, and c belonging to the field. The rational numbers Q form a field, as do the real numbers R and the complex numbers C.

Fields can contain, and be contained in, other fields. For present purposes we are concerned with fields K that contain the rationals Q. Symbolically, Q ⊆ K. In this case, one says that Q is a subfield of K, and K is an extension of Q. An "algebraic number" is simply any root of any polynomial f(x) whose coefficients are rational numbers in Q. Polynomials, of course, are expressions of the form

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ where a_i ∈ Q for 0 ≤ i ≤ n

To say that the algebraic number &alpha is a "root" of f(x) means just that f(α) = 0. If n is the smallest integer such that a_n ≠ 0, while a_m = 0 if m > n, n is said to be the "degree" of f(x). An algebraic number α is said to be an "algebraic integer" if there is a polynomial f(x) of degree n where a_n = 1, and all of the other coefficients are ordinary integers (elements of Z). It isn't hard to show that the subset of an extension K ⊇ Q consisting only of algebraic integers is what is called a "ring", which is a very much like a field, except that its members do not necessarily have multiplicative inverses (i. e. reciprocals) that are also algebraic integers (although the inverses are still algebraic numbers in K). This ring is called the ring of algebraic integers of K/Q. (Which is read "K over Q".) It is this ring which generalizes the ring Z of ordinary integers.

In general, an algebraic number α can be a root of infinitely many polynomials, but there is always a polynomial of smallest degree n whose leading coefficient a_n = 1. This is called the "minimal" polynomial of α. There is a function called the "norm" that maps all algebraic numbers of an extension field K ⊇ Q to Q. If K is the smallest extension field of Q that contains α and all other complex roots of the minimal polynomial f(x) of α (known as the "splitting field" of f(x)), then the norm of α is simply the constant term of f(x), i. e. f(0). It is denoted by N_K/Q(α) and it is a member of Q by definition. If α is an algebraic integer, its norm is an ordinary integer.

Now, as we mentioned, the ring of integers R of some K ⊇ Q doesn't necessarily have the property that all its members are products of "primes" in a unique way. This fact makes arithmetic in R much messier than arithmetic in Z. Fortunately, algebraists in the 19th century discovered that it is possible to define certain subsets of R called "ideals" that have a property analogous to unique factorization. An ideal I ⊆ R is itself a ring with the further property that all products of elements of I by elements of R are members of I. (Symbolically, IR ⊆ I.) An important type of ideal is a "principal" ideal, which consists of all products of an algebraic integer and other elements of the ring, and is usually written as αR or (α). Although ideals are not numbers, they play an analogous role in the theory. For instance, it is possible to define the norm of an ideal, and the value of the norm is an ordinary integer.

It turns out that unique factorization can be recovered for rings of integers as far as ideals are concerned. That is, there are ideals which are called prime ideals such that any ideal in the ring can be expressed as a "product" of prime ideals in a unique way. The way that an algebraic integer α factors in the ring can be expressed in terms of the prime ideals that are factors of the principal ideal (α). Most of algebraic number theory is devoted to describing the prime ideals of a given ring of algebraic integers in greater detail.

One of the key tools used in algebraic number theory is the Dedekind zeta function, which we now have enough terminology to define. The zeta function of the algebraic number field K ⊇ Q is given by

ζ_K(s) = ∑_A (N_K/QA)^-s

Many of the properties of ζ(s) carry over to ζ_K(s). In particular, precisely because one has unique factorization of ideals, there is a product formula:

ζ_K(s) = ∏_P (1 - 1/(N_K/QP)^s)^-1

ζ_K(s) can also be represented as an ordinary Dirichlet series:

ζ_K(s) = ∑_1≤n<∞ f(n)/n^s

The similarities to ζ(s) don't stop there. There are many others:

1. The series and product expressions for ζ_K(s) converge in the half plane Re(s) > 1, and the function is holomorphic there.
2. ζ_K(s) has a pole at s = 1.
3. ζ_K(s) has a simple functional equation involving the Γ function.
4. ζ_K(s) has zeros at s = -2k and sometimes (depending on the field K) at s = -2k-1 for integers k ≥ 1.
5. All other zeros of ζ_K(s) lie in the critical strip 0 ≤ Re(s) ≤ 1.
6. ζ_K(s) has no zeros with Re(s) = 0, and in fact ζ_K(s) ≠ 0 if Re(s) ≥ 1 - A/(n log(|T|)), for sufficiently large T = Im(s), a constant A, and n being the "degree" of the extension K over Q.

Naturally, the precise characteristics of ζ_K(s) depend on the specific field K. Indeed, important quantities related to K can be expressed in terms of ζ_K(s). For instance, the very important "ideal class number" of K, which roughly expresses how badly the ring of integers of K fails to have all of its ideals be principal ideals, enters into the formula for the "residue" of ζ_K(s) at its pole at s = 1.

This class number is number of elements of the "ideal class group" of the field. This group consists of classes of ideals within the set of all ideals of the ring of integers of K. We won't attempt to explain that idea more here -- it's part of the very deep theory called "class field theory -- except to note that Dedekind zeta functions can also be defined for each of these ideal classes, by letting the sum which defines the function run only over those ideals which are members of a given class:

ζ(s; H) = ∑_A∈H (N_K/QA)^-s

ζ_K(s) = ∑_1≤i≤h ζ_K(s; H_i)

Dirichlet L-functions

Instead, Dirichlet was interested in certain simple arithmetic functions called "characters". A character (specifically, a "Dirichlet" character) is a complex-valued function defined for integers and having the multiplicative property: if χ is a character, then χ(ab) = χ(a)χ(b). For any character, there is an integer m such that χ is "periodic modulo m", i. e. χ(a) = χ(a + km) for any integer k. In modern terminology that we won't attempt to explain in detail here, a Dirichlet character corresponds to "homomorphism" from the multiplicative group of integers modulo m, denoted by (Z/mZ)*, to a group of roots of unity -- complex numbers some power of which is 1. Given such a homomorphism, χ(n) can easily be defined for all integers n that are "relatively prime" to m (i. e., have no prime factors in common with m, which one expresses symbolically in terms of the greatest common divisor: (m,n) = 1), while for integers n that aren't relatively prime to m, one sets χ(n) = 0. (The group (Z/mZ)* is a group of φ(m) elements, where φ(m) is Euler's φ function: the number of positive integers less than m and relatively prime to m. Therefore, the value of χ(n) are either φ(m)-th roots of unity or zero.) Dirichlet characters are useful tools for studying multiplicative properties of the integers.

Given any Dirichlet character, the obvious thing to do is to define a series, called an L-series or L-function:

L(s, χ) = ∑_1≤n<∞ χ(n)/n^s

If we take χ(n) to be the "trivial" character, corresponding to m=1 such that χ(n) = 1 for all n, we get the Riemann zeta function. Not too surprisingly, then, L-functions also have many properties in common with ζ(s).

First of all, because of the multiplicative property of χ(n), there is an Euler product formula:

L(s, χ) = ∏_p (1 - χ(p)p^-s)^-1

Another way that L(s, χ) is analogous to ζ(s) is the existence of a simple formula for values of the function at negative integers. Specifically, one has

L(1-k, χ) = - B_k,χ / k for k ≥ 1

∑_1≤a≤m χ(a) x e^ax / (e^mx - 1) = ∑_0≤n<∞ B_n,χ xⁿ / n!

Why pay so much attention to these seemingly arcane L-functions? The answer is several fold. First, historically, they played a key role in Dirichlet's noteworthy theorem that there are actually an infinite number of primes in every arithmetic progression of the form {a + km: k ∈ Z} where a and m are relatively prime. This is obviously a nice improvement on Euclid's old result on the infinity of primes.

But far more significantly than that, L-functions have proven to have a profound importance for expressing very deep facts about algebraic number fields through a series of generalizations. This comes about (in part) because the Dedekind zeta functions can be expressed as a finite product of suitable L-functions. The model for such results is the formula

ζ_K = Z(s) ∏_χ L(s, χ)

Now, K is a relatively simple sort of extension of Q, called a "cyclotomic" field. It is very important in algebraic number theory, but it is also rather a special case. Most extensions of Q have a more complicated structure, which is reflected in the so-called Galois group G(K/Q). Galois groups, named after Évariste Galois (1811-32), play a fundamental role in the theory of polynomial equations. Characters can be defined for any group, and indeed they are a fundamental tool in the theory of "group representations", which is a way of studying abstract groups in terms of groups of matrices.

It is possible to generalize the Dirichlet L-functions in various ways, which involve using characters of certain groups that can be defined for different types of field extensions. One such kind of generalization was discovered by Erich Hecke, and hence the corresponding characters are called Hecke characters, and the L-functions are called Hecke L-functions. Another type of generalization was discovered by Emil Artin, yielding Artin L-functions.

Many of the mysteries of advanced algebraic number theory are bound up in the properties of Artin's L-functions. They are defined in terms of certain products, they are meromorphic functions, and they satisfy a functional equation. Their zeros are easy to describe, and Dedekind L-functions can be expressed as a product of ζ(s) and Artin L-functions. For certain types of field extensions (abelian extensions, which have abelian Galois groups), the Artin L-functions can be related to Hecke L-functions. Artin L-functions, however, do not in general have an infinite series expansions.

But there remain important open questions about the Artin L-functions. In particular, there is Artin's conjecture: that the functions are in fact holomorphic everywhere. This is known to be true for abelian field extensions, because this part of the theory (known as "abelian class field theory") is fairly well understood. However, the Artin L-functions encode facts about nonabelian field extensions as well, and nonabelian class field theory is an active research area.

Bernoulli numbers

The Riemann hypothesis and algebraic geometry

The Riemann hypothesis and spectral theory

Other generalizations of the Riemann hypothesis

Reasons for believing the Riemann hypothesis

Recommended references: Web sites

Site indexes

Galaxy: Riemann Hypothesis

Categorized site directory. Entries usually include descriptive annotations.

Sites with general resources

The Riemann Hypothesis

An outline (literally) of some of the conjectures and open problems concerning The Riemann hypothesis. Explanations are sketchy, but the outline is very helpful if you have a general understanding of the theory. Includes material on related hypotheses and generalizations of the Riemann hypothesis. There is also a similar, related ouline of L-functions and Random Matrix Theory.

Andrew Odlyzko: Home Page

Odlyzko has done fundamental computational and theoretical work on RH, as well as many other areas of number theory, combinatorics, probability, algorithms, cryptography, and network theory. Of particular relevance here are papers on the zeros of ζ(s), tables of some of the zeros, and some of Odlyzko's correspondence about the Hilbert-Polya conjecture.

The Riemann Hypothesis

Brief page by Daniel Bump with links to other sites. Bump also has some lecture notes on the Riemann Zeta Function.

Riemann hypothesis

Article from Wikipedia.

Noncommutative Geometry, Trace Formulas and the Zeros of the Riemann Zeta Function

Description of a course at Ohio State University, by Alain Connes. Has links to related downloadable papers by Connes.

Inexplicable Secrets of Creation

Very interesting site by Matthew Watkins deailing with prime numbers and especially the Riemann zeta function, the Riemann hypothesis, and connections between physics and the distribution of primes.

Louis de Branges de Bourcia

de Branges established his credentials with a proof of the celebrated Bieberbach conjecture. Unfortunately, his efforts to prove RH have not been considered successful by the mathematical community. Even so, many of the ideas contained in (highly technical) papers available from his Web site are interesting. Two relevant papers (PDF format) are: Apology for the Proof of the Riemann Hypothesis, and Riemann Zeta Functions.

Thoughts on the Riemann Hypothesis

An essay on the hypothesis by Gregory Chaitin, who applies his iconoclastic views on mathmatics to the question of whether RH should be taken as a new axiom. Has some good external links.

Surveys, overviews, tutorials

The Riemann Hypothesis

Brief description of the problem at the Clay Mathematics Institute site by Enrico Bombieri. (A more complete description is available as a PDF file.)

The Riemann Hypothesis

Description of the hypothesis from The Prime Pages.

The Riemann's Zeta function z(s)

Brief introductory page by Xavier Gourdon

This Week's Finds in Mathematical Physics (Week 216)

In spite of the opaque title, this is an excellent condensed summary of zeta functions and the Riemann hypothesis, from John Baez' This Week's Finds in Mathematical Physics.

A Prime Case of Chaos

An article by Barry Cipra (downloadable in PDF format). It describes conjectural links between the Riemann zeta function and chaotic quantum-mechanical systems.

Recommended references: Magazine/journal articles

Prime Numbers Get Hitched
Marchs Du Satoy
Seed Magazine, February-March 2006

The Riemann Hypothesis
J. Brian Conrey
Notices of the AMS, March 2003, pp. 341-353

The Riemann Hypothesis originated in Riemann's investigations into explicit formulas for π(x). Extensive computations have found no counterexample of a nontrivial zero off the critical line, and there is other supporting evidence as well. The idea that the zeros may be the eigenvalues of a Hermitian operator has received some support from studies of random matrices. However, a proof of the hypothesis may ultimately depend on fundamentally new ideas encompassing many types of zeta- and L-functions.
[Article in PDF format]

Zeroes of Zeta Functions and Symmetry
Nicholas M. Katz; Peter Sarnak
Bulletin of the AMS, January 1999, pp. 1-26

Hilbert and Polya suggested that there might be a natural relationship between spectral theory and the zeros of the Riemann zeta function. This is known to be true in the analogous case of zeta and L-Functions of function fields, where results have been easier to obtain than in the number field case.
[Abstract, references, downloadable text]

Selberg's Eigenvalue Conjecture
Peter Sarnak
Notices of the AMS, November 1995, pp. 1272-1277

Selberg's conjecture concerns Fourier coefficients of modular forms and eigenvalues of the Laplacian operator on certain surfaces, some of which are significant in number theory. There are connections to the Riemann hypothesis, given the expectation that L-functions arise as eigenfunctions.
[Article in PDF format]

Prime Territory
Enrico Bombieri
The Sciences, September/October 1992, pp. 30-36

The asymptotic behavior of the prime-counting function π(x) can more easily be studied in terms of the asymptotic behavior of a related function ψ(x). This in turn can be understood using "harmonic analysis" involving infinite sums of certain oscillating function whose behavior depends on the location of the imaginary zeros of ζ(s). (In spite of the technical terminology of this summary, the article is actually very accessible to a general audience.)

The First 50 Million Prime Numbers
Don Zagier
Mathematical Intelligencer, August 1977, pp. 7-19

Studies of the distribution of prime numbers have led to deep theoretical results, such as the prime number theorem. Now that it is possible to compute millions of primes, it is especially interesting to compare the actual distribution with theoretical approximations.

Recommended references: Books

Dan Rockmore – Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers
Vintage Books, 2005

Although it follows in the wake of three other books on the subject for a general audience, and although (like them) it shies away from the most important mathematical details, Rockmore's book does provide another useful perspective. In addition to going over the history of the problem, it sketches the efforts of contemporary mathematicans who are working on the hypothesis, and it talks about the interesting, and perhaps significant, connections with quantum chaos theory.

Marcus du Sautoy -- The Music of the Primes: Searching to Solve the Greatest Unsolved Mystery in Mathematics
HarperCollins, 2003

Among authors of the three books which appeared in 2003 on the Riemann hypothesis, du Sautoy has the strongest credentials, as a professor of mathematics at Oxford, so his account is authoritative. But unfortunately, he does not attempt to seriously explain the underlying mathematics. Nevertheless, he presents a solid historical account of the study of prime numbers from the time of Euler and Gauss up through current attempts to prove the Riemann hypothesis. Vignettes of contemporary mathematicians working on the problem put a human face on the story. Some of the ideas they are pursuing are introduced... but leave the reader wanting more in the way of details.
[Book review]

John Derbyshire -- Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Joseph Henry Press, 2003

Derbyshire's book is unusual among mathematics books for a general audience in that it actually discusses non-elementary, but essential, concepts like infinite series, calculus, and functions of a complex variable. Some of the actual formulas Riemann discovered are shown and, to a degree, explained. Readers may find that such things are not as forbidding as they might fear. And on top of that, some of the current methods of attack on the problem are explained, the historical background of the problem is presented, including sketches of the lives of the most important mathematicians who have worked on the problem -- especially Riemann himself.
[Book review]
[Book home page]

Karl Sabbagh -- The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Farrar, Straus and Giroux, 2003

This is a surprisingly good book on the subject, even though it is much more about the contemporary mathematicians who are working on the problem rather than the problem itself. It does a good job of describing how (and to some extent) why mathematicians work as they do. But the repeated insinuation that mathematical thinking is completely unlike "ordinary" thinking is irritating. So too is the relegation of rather elementary mathematical details to appendices, with its implication that "you wouldn't understand the real stuff anyhow."
[Book review]

Serge Lang -- Math Talks for Undergraduates
Springer-Verlag, 1999

This informal book deals with a variety of sophisticated mathematical topics using only prerequisites from undergraduate mathematics. The prime number theorem, the Riemann hypothesis, the functional equation of the zeta function, and the abc conjecture are the topics of interest with respect to number theory.

S. J. Patterson -- An Intorduction to the Theory of the Riemann Zeta-Function
Cambridge University Press, 1988

The author has produced a concise, modern treatment of the key facts about the zeta function, beginning with a good historical sketch of Riemann's contribution. Other main topics are the Poisson summation formula and the functional equation, the Hadamard product formula, the explicit formula for π(x), zeros of ζ(s) and the prime number theorem, the Riemann hypothesis, the Lindelöf hypothesis, and the approximate functional equation. All of this occupies little more than 100 pages, with additional space devoted to appendices on background topics such as Fourier theory and the gamma function.

E. C. Titchmarsh; D. R. Heath-Brown -- The Theory of the Riemann Zeta-function
Oxford University Press, 1986

In this second edition of a classic, encyclopedic reference on the subject, the theory is brought up to date with numerous new results since the first edition in 1951. The mathematics is technically demanding. However, the subject remains accessible if one takes many of the details on faith, since it consists of mostly classical real and complex analysis without highly abstract modern concepts.

H. M. Edwards -- Riemann's Zeta Function
Academic Press, 1974

Edwards presents the theory of the zeta function specifically in relation to its applications in number theory. The book contains an English translation of Riemann's original paper, together with a thorough, rigorous proof of its results. The inexpensive Dover edition of 2001 provides a very convenient means of becoming introduced to the subject.

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