You are currently browsing the tag archive for the ‘Mertens’ theorems’ tag.

In analytic number theory, an arithmetic function is simply a function {f: {\bf N} \rightarrow {\bf C}} from the natural numbers {{\bf N} = \{1,2,3,\dots\}} to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than {{\bf R}} or {{\bf C}}, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions {f: {\bf N} \rightarrow {\bf C}} with the additional property that

\displaystyle  f(nm) = f(n) f(m) \ \ \ \ \ (1)

whenever {n,m \in{\bf N}} are coprime. (One also considers arithmetic functions, such as the logarithm function {L(n) := \log n} or the von Mangoldt function, that are not genuinely multiplicative, but interact closely with multiplicative functions, and can be viewed as “derived” versions of multiplicative functions; see this previous post.) A typical example of a multiplicative function is the divisor function

\displaystyle  \tau(n) := \sum_{d|n} 1 \ \ \ \ \ (2)

that counts the number of divisors of a natural number {n}. (The divisor function {n \mapsto \tau(n)} is also denoted {n \mapsto d(n)} in the literature.) The study of asymptotic behaviour of multiplicative functions (and their relatives) is known as multiplicative number theory, and is a basic cornerstone of modern analytic number theory.

There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions {f}. In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function {f: {\bf N} \rightarrow {\bf C}}:

  1. The summatory functions

    \displaystyle  \sum_{n \leq x} f(n)

    of an arithmetic function {f}, as well as the associated natural density

    \displaystyle  \lim_{x \rightarrow \infty} \frac{1}{x} \sum_{n \leq x} f(n)

    (if it exists).

  2. The logarithmic sums

    \displaystyle  \sum_{n\leq x} \frac{f(n)}{n}

    of an arithmetic function {f}, as well as the associated logarithmic density

    \displaystyle  \lim_{x \rightarrow \infty} \frac{1}{\log x} \sum_{n \leq x} \frac{f(n)}{n}

    (if it exists).

Here, we are normalising the arithmetic function {f} being studied to be of roughly unit size up to logarithms, obeying bounds such as {f(n)=O(1)}, {f(n) = O(\log^{O(1)} n)}, or at worst

\displaystyle  f(n) = O(n^{o(1)}). \ \ \ \ \ (3)

A classical case of interest is when {f} is an indicator function {f=1_A} of some set {A} of natural numbers, in which case we also refer to the natural or logarithmic density of {f} as the natural or logarithmic density of {A} respectively. However, in analytic number theory it is usually more convenient to replace such indicator functions with other related functions that have better multiplicative properties. For instance, the indicator function {1_{\mathcal P}} of the primes is often replaced with the von Mangoldt function {\Lambda}.

Typically, the logarithmic sums are relatively easy to control, but the summatory functions require more effort in order to obtain satisfactory estimates; see Exercise 7 below.

If an arithmetic function {f} is multiplicative (or closely related to a multiplicative function), then there is an important further statistic on an arithmetic function {f} beyond the summatory function and the logarithmic sum, namely the Dirichlet series

\displaystyle  {\mathcal D}f(s) := \sum_{n=1}^\infty \frac{f(n)}{n^s} \ \ \ \ \ (4)

for various real or complex numbers {s}. Under the hypothesis (3), this series is absolutely convergent for real numbers {s>1}, or more generally for complex numbers {s} with {\hbox{Re}(s)>1}. As we will see below the fold, when {f} is multiplicative then the Dirichlet series enjoys an important Euler product factorisation which has many consequences for analytic number theory.

In the elementary approach to multiplicative number theory presented in this set of notes, we consider Dirichlet series only for real numbers {s>1} (and focusing particularly on the asymptotic behaviour as {s \rightarrow 1^+}); in later notes we will focus instead on the important complex-analytic approach to multiplicative number theory, in which the Dirichlet series (4) play a central role, and are defined not only for complex numbers with large real part, but are often extended analytically or meromorphically to the rest of the complex plane as well.

Remark 1 The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such as

\displaystyle  \sum_n \frac{f(n)}{n} G( t - \log n ) \ \ \ \ \ (5)

as {t \rightarrow \infty} for various cutoff functions {G: {\bf R} \rightarrow {\bf C}}, such as smooth, compactly supported functions. See this previous blog post for an instance of such an approach. Another related approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville-Soundararajan and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.

To reverse the process and derive control on summatory functions or logarithmic sums starting from control of Dirichlet series is trickier, and usually requires one to allow {s} to be complex-valued rather than real-valued if one wants to obtain really accurate estimates; we will return to this point in subsequent notes. However, there is a cheap way to get upper bounds on such sums, known as Rankin’s trick, which we will discuss later in these notes.

The basic strategy of elementary multiplicative theory is to first gather useful estimates on the statistics of “smooth” or “non-oscillatory” functions, such as the constant function {n \mapsto 1}, the harmonic function {n \mapsto \frac{1}{n}}, or the logarithm function {n \mapsto \log n}; one also considers the statistics of periodic functions such as Dirichlet characters. These functions can be understood without any multiplicative number theory, using basic tools from real analysis such as the (quantitative version of the) integral test or summation by parts. Once one understands the statistics of these basic functions, one can then move on to statistics of more arithmetically interesting functions, such as the divisor function (2) or the von Mangoldt function {\Lambda} that we will discuss below. A key tool to relate these functions to each other is that of Dirichlet convolution, which is an operation that interacts well with summatory functions, logarithmic sums, and particularly well with Dirichlet series.

This is only an introduction to elementary multiplicative number theory techniques. More in-depth treatments may be found in this text of Montgomery-Vaughan, or this text of Bateman-Diamond.

Read the rest of this entry »

Mertens’ theorems are a set of classical estimates concerning the asymptotic distribution of the prime numbers:

Theorem 1 (Mertens’ theorems) In the asymptotic limit {x \rightarrow \infty}, we have

\displaystyle  \sum_{p\leq x} \frac{\log p}{p} = \log x + O(1), \ \ \ \ \ (1)

\displaystyle  \sum_{p\leq x} \frac{1}{p} = \log \log x + O(1), \ \ \ \ \ (2)

and

\displaystyle  \sum_{p\leq x} \log(1-\frac{1}{p}) = -\log \log x - \gamma + o(1) \ \ \ \ \ (3)

where {\gamma} is the Euler-Mascheroni constant, defined by requiring that

\displaystyle  1 + \frac{1}{2} + \ldots + \frac{1}{n} = \log n + \gamma + o(1) \ \ \ \ \ (4)

in the limit {n \rightarrow \infty}.

The third theorem (3) is usually stated in exponentiated form

\displaystyle  \prod_{p \leq x} (1-\frac{1}{p}) = \frac{e^{-\gamma}+o(1)}{\log x},

but in the logarithmic form (3) we see that it is strictly stronger than (2), in view of the asymptotic {\log(1-\frac{1}{p}) = -\frac{1}{p} + O(\frac{1}{p^2})}.

Remarkably, these theorems can be proven without the assistance of the prime number theorem

\displaystyle  \sum_{p \leq x} 1 = \frac{x}{\log x} + o( \frac{x}{\log x} ),

which was proven about two decades after Mertens’ work. (But one can certainly use versions of the prime number theorem with good error term, together with summation by parts, to obtain good estimates on the various errors in Mertens’ theorems.) Roughly speaking, the reason for this is that Mertens’ theorems only require control on the Riemann zeta function {\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}} in the neighbourhood of the pole at {s=1}, whereas (as discussed in this previous post) the prime number theorem requires control on the zeta function on (a neighbourhood of) the line {\{ 1+it: t \in {\bf R} \}}. Specifically, Mertens’ theorem is ultimately deduced from the Euler product formula

\displaystyle  \zeta(s) = \prod_p (1-\frac{1}{p^s})^{-1}, \ \ \ \ \ (5)

valid in the region {\hbox{Re}(s) > 1} (which is ultimately a Fourier-Dirichlet transform of the fundamental theorem of arithmetic), and following crude asymptotics:

Proposition 2 (Simple pole) For {s} sufficiently close to {1} with {\hbox{Re}(s) > 1}, we have

\displaystyle  \zeta(s) = \frac{1}{s-1} + O(1) \ \ \ \ \ (6)

and

\displaystyle  \zeta'(s) = \frac{-1}{(s-1)^2} + O(1).

Proof: For {s} as in the proposition, we have {\frac{1}{n^s} = \frac{1}{t^s} + O(\frac{1}{n^2})} for any natural number {n} and {n \leq t \leq n+1}, and hence

\displaystyle  \frac{1}{n^s} = \int_n^{n+1} \frac{1}{t^s}\ dt + O( \frac{1}{n^2} ).

Summing in {n} and using the identity {\int_1^\infty \frac{1}{t^s}\ dt = \frac{1}{s-1}}, we obtain the first claim. Similarly, we have

\displaystyle  \frac{-\log n}{n^s} = \int_n^{n+1} \frac{-\log t}{t^s}\ dt + O( \frac{\log n}{n^2} ),

and by summing in {n} and using the identity {\int_1^\infty \frac{-\log t}{t^s}\ dt = \frac{-1}{(s-1)^2}} (the derivative of the previous identity) we obtain the claim. \Box

The first two of Mertens’ theorems (1), (2) are relatively easy to prove, and imply the third theorem (3) except with {\gamma} replaced by an unspecified absolute constant. To get the specific constant {\gamma} requires a little bit of additional effort. From (4), one might expect that the appearance of {\gamma} arises from the refinement

\displaystyle  \zeta(s) = \frac{1}{s-1} + \gamma + O(|s-1|) \ \ \ \ \ (7)

that one can obtain to (6). However, it turns out that the connection is not so much with the zeta function, but with the Gamma function, and specifically with the identity {\Gamma'(1) = - \gamma} (which is of course related to (7) through the functional equation for zeta, but can be proven without any reference to zeta functions). More specifically, we have the following asymptotic for the exponential integral:

Proposition 3 (Exponential integral asymptotics) For sufficiently small {\epsilon}, one has

\displaystyle  \int_\epsilon^\infty \frac{e^{-t}}{t}\ dt = \log \frac{1}{\epsilon} - \gamma + O(\epsilon).

A routine integration by parts shows that this asymptotic is equivalent to the identity

\displaystyle  \int_0^\infty e^{-t} \log t\ dt = -\gamma

which is the identity {\Gamma'(1)=-\gamma} mentioned previously.

Proof: We start by using the identity {\frac{1}{i} = \int_0^1 x^{i-1}\ dx} to express the harmonic series {H_n := 1+\frac{1}{2}+\ldots+\frac{1}{n}} as

\displaystyle  H_n = \int_0^1 1 + x + \ldots + x^{n-1}\ dx

or on summing the geometric series

\displaystyle  H_n = \int_0^1 \frac{1-x^n}{1-x}\ dx.

Since {\int_0^{1-1/n} \frac{1}{1-x} = \log n}, we thus have

\displaystyle  H_n - \log n = \int_0^1 \frac{1_{[1-1/n,1]}(x) - x^n}{1-x}\ dx;

making the change of variables {x = 1-\frac{t}{n}}, this becomes

\displaystyle  H_n - \log n = \int_0^n \frac{1_{[0,1]}(t) - (1-\frac{t}{n})^n}{t}\ dt.

As {n \rightarrow \infty}, {\frac{1_{[0,1]}(t) - (1-\frac{t}{n})^n}{t}} converges pointwise to {\frac{1_{[0,1]}(t) - e^{-t}}{t}} and is pointwise dominated by {O( e^{-t} )}. Taking limits as {n \rightarrow \infty} using dominated convergence, we conclude that

\displaystyle  \gamma = \int_0^\infty \frac{1_{[0,1]}(t) - e^{-t}}{t}\ dt.

or equivalently

\displaystyle  \int_0^\infty \frac{e^{-t} - 1_{[0,\epsilon]}(t)}{t}\ dt = \log \frac{1}{\epsilon} - \gamma.

The claim then follows by bounding the {\int_0^\epsilon} portion of the integral on the left-hand side. \Box

Below the fold I would like to record how Proposition 2 and Proposition 3 imply Theorem 1; the computations are utterly standard, and can be found in most analytic number theory texts, but I wanted to write them down for my own benefit (I always keep forgetting, in particular, how the third of Mertens’ theorems is proven).

Read the rest of this entry »

Archives