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A basic object of study in multiplicative number theory are the arithmetic functions: functions ${f: {\bf N} \rightarrow {\bf C}}$ from the natural numbers to the complex numbers. Some fundamental examples of such functions include

• The constant function ${1: n \mapsto 1}$;
• The Kronecker delta function ${\delta: n \mapsto 1_{n=1}}$;
• The natural logarithm function ${L: n \mapsto \log n}$;
• The divisor function ${d_2: n \mapsto \sum_{d|n} 1}$;
• The von Mangoldt function ${\Lambda}$, with ${\Lambda(n)}$ defined to equal ${\log p}$ when ${n}$ is a power ${p^j}$ of a prime ${p}$ for some ${j \geq 1}$, and defined to equal zero otherwise; and
• The Möbius function ${\mu}$, with ${\mu(n)}$ defined to equal ${(-1)^k}$ when ${n}$ is the product of ${k}$ distinct primes, and defined to equal zero otherwise.

Given an arithmetic function ${f}$, we are often interested in statistics such as the summatory function

$\displaystyle \sum_{n \leq x} f(n), \ \ \ \ \ (1)$

the logarithmically (or harmonically) weighted summatory function

$\displaystyle \sum_{n \leq x} \frac{f(n)}{n}, \ \ \ \ \ (2)$

or the Dirichlet series

$\displaystyle {\mathcal D}[f](s) := \sum_n \frac{f(n)}{n^s}.$

In the latter case, one typically has to first restrict ${s}$ to those complex numbers whose real part is large enough in order to ensure the series on the right converges; but in many important cases, one can then extend the Dirichlet series to almost all of the complex plane by analytic continuation. One is also interested in correlations involving additive shifts, such as ${\sum_{n \leq x} f(n) f(n+h)}$, but these are significantly more difficult to study and cannot be easily estimated by the methods of classical multiplicative number theory.

A key operation on arithmetic functions is that of Dirichlet convolution, which when given two arithmetic functions ${f,g: {\bf N} \rightarrow {\bf C}}$, forms a new arithmetic function ${f*g: {\bf N} \rightarrow {\bf C}}$, defined by the formula

$\displaystyle f*g(n) := \sum_{d|n} f(d) g(\frac{n}{d}).$

Thus for instance ${1*1 = d_2}$, ${1 * \Lambda = L}$, ${1 * \mu = \delta}$, and ${\delta * f = f}$ for any arithmetic function ${f}$. Dirichlet convolution and Dirichlet series are related by the fundamental formula

$\displaystyle {\mathcal D}[f * g](s) = {\mathcal D}[f](s) {\mathcal D}[g](s), \ \ \ \ \ (3)$

at least when the real part of ${s}$ is large enough that all sums involved become absolutely convergent (but in practice one can use analytic continuation to extend this identity to most of the complex plane). There is also the identity

$\displaystyle {\mathcal D}[Lf](s) = - \frac{d}{ds} {\mathcal D}[f](s), \ \ \ \ \ (4)$

at least when the real part of ${s}$ is large enough to justify interchange of differentiation and summation. As a consequence, many Dirichlet series can be expressed in terms of the Riemann zeta function ${\zeta = {\mathcal D}[1]}$, thus for instance

$\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s)$

$\displaystyle {\mathcal D}[L](s) = - \zeta'(s)$

$\displaystyle {\mathcal D}[\delta](s) = 1$

$\displaystyle {\mathcal D}[\mu](s) = \frac{1}{\zeta(s)}$

$\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)}.$

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers ${{\bf N}}$, which form a rather complicated abelian semigroup with respect to multiplication (in particular the set of generators is the set of prime numbers). One can obtain a simpler analogue of the subject by working instead with the half-infinite interval ${{\bf N}_\infty := [1,+\infty)}$, which is a much simpler abelian semigroup under multiplication (being a one-dimensional Lie semigroup). (I will think of this as a sort of “completion” of ${{\bf N}}$ at the infinite place ${\infty}$, hence the terminology.) Accordingly, let us define a continuous arithmetic function to be a locally integrable function ${f: {\bf N}_\infty \rightarrow {\bf C}}$. The analogue of the summatory function (1) is then an integral

$\displaystyle \int_1^x f(t)\ dt,$

and similarly the analogue of (2) is

$\displaystyle \int_1^x \frac{f(t)}{t}\ dt.$

The analogue of the Dirichlet series is the Mellin-type transform

$\displaystyle {\mathcal D}_\infty[f](s) := \int_1^\infty \frac{f(t)}{t^s}\ dt,$

which will be well-defined at least if the real part of ${s}$ is large enough and if the continuous arithmetic function ${f: {\bf N}_\infty \rightarrow {\bf C}}$ does not grow too quickly, and hopefully will also be defined elsewhere in the complex plane by analytic continuation.

For instance, the continuous analogue of the discrete constant function ${1: {\bf N} \rightarrow {\bf C}}$ would be the constant function ${1_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, which maps any ${t \in [1,+\infty)}$ to ${1}$, and which we will denote by ${1_\infty}$ in order to keep it distinct from ${1}$. The two functions ${1_\infty}$ and ${1}$ have approximately similar statistics; for instance one has

$\displaystyle \sum_{n \leq x} 1 = \lfloor x \rfloor \approx x-1 = \int_1^x 1\ dt$

and

$\displaystyle \sum_{n \leq x} \frac{1}{n} = H_{\lfloor x \rfloor} \approx \log x = \int_1^x \frac{1}{t}\ dt$

where ${H_n}$ is the ${n^{th}}$ harmonic number, and we are deliberately vague as to what the symbol ${\approx}$ means. Continuing this analogy, we would expect

$\displaystyle {\mathcal D}[1](s) = \zeta(s) \approx \frac{1}{s-1} = {\mathcal D}_\infty[1_\infty](s)$

which reflects the fact that ${\zeta}$ has a simple pole at ${s=1}$ with residue ${1}$, and no other poles. Note that the identity ${{\mathcal D}_\infty[1_\infty](s) = \frac{1}{s-1}}$ is initially only valid in the region ${\mathrm{Re} s > 1}$, but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at ${1}$, and so one can define ${{\mathcal D}_\infty[1_\infty]}$ in this region also.

In a similar vein, the logarithm function ${L: {\bf N} \rightarrow {\bf C}}$ is approximately similar to the logarithm function ${L_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, giving for instance the crude form

$\displaystyle \sum_{n \leq x} L(n) = \log \lfloor x \rfloor! \approx x \log x - x = \int_1^\infty L_\infty(t)\ dt$

of Stirling’s formula, or the Dirichlet series approximation

$\displaystyle {\mathcal D}[L](s) = -\zeta'(s) \approx \frac{1}{(s-1)^2} = {\mathcal D}_\infty[L_\infty](s).$

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure ${\frac{dt}{t}}$: given two continuous arithmetic functions ${f_\infty, g_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, one can define their convolution ${f_\infty *_\infty g_\infty: {\bf N}_\infty \rightarrow {\bf C}}$ by the formula

$\displaystyle f_\infty *_\infty g_\infty(t) := \int_1^t f_\infty(s) g_\infty(\frac{t}{s}) \frac{ds}{s}.$

Thus for instance ${1_\infty * 1_\infty = L_\infty}$. A short computation using Fubini’s theorem shows the analogue

$\displaystyle D_\infty[f_\infty *_\infty g_\infty](s) = D_\infty[f_\infty](s) D_\infty[g_\infty](s)$

of (3) whenever the real part of ${s}$ is large enough that Fubini’s theorem can be justified; similarly, differentiation under the integral sign shows that

$\displaystyle D_\infty[L_\infty f_\infty](s) = -\frac{d}{ds} D_\infty[f_\infty](s) \ \ \ \ \ (5)$

again assuming that the real part of ${s}$ is large enough that differentiation under the integral sign (or some other tool like this, such as the Cauchy integral formula for derivatives) can be justified.

Direct calculation shows that for any complex number ${\rho}$, one has

$\displaystyle \frac{1}{s-\rho} = D_\infty[ t \mapsto t^{\rho-1} ](s)$

(at least for the real part of ${s}$ large enough), and hence by several applications of (5)

$\displaystyle \frac{1}{(s-\rho)^k} = D_\infty[ t \mapsto \frac{1}{(k-1)!} t^{\rho-1} \log^{k-1} t ](s)$

for any natural number ${k}$. This can lead to the following heuristic: if a Dirichlet series ${D[f](s)}$ behaves like a linear combination of poles ${\frac{1}{(s-\rho)^k}}$, in that

$\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{(s-\rho)^{k_\rho}}$

for some set ${\rho}$ of poles and some coefficients ${c_\rho}$ and natural numbers ${k_\rho}$ (where we again are vague as to what ${\approx}$ means, and how to interpret the sum ${\sum_\rho}$ if the set of poles is infinite), then one should expect the arithmetic function ${f}$ to behave like the continuous arithmetic function

$\displaystyle t \mapsto \sum_\rho \frac{c_\rho}{(k_\rho-1)!} t^{\rho-1} \log^{k_\rho-1} t.$

In particular, if we only have simple poles,

$\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{s-\rho}$

then we expect to have ${f}$ behave like continuous arithmetic function

$\displaystyle t \mapsto \sum_\rho c_\rho t^{\rho-1}.$

Integrating this from ${1}$ to ${x}$, this heuristically suggests an approximation

$\displaystyle \sum_{n \leq x} f(n) \approx \sum_\rho c_\rho \frac{x^\rho-1}{\rho}$

for the summatory function, and similarly

$\displaystyle \sum_{n \leq x} \frac{f(n)}{n} \approx \sum_\rho c_\rho \frac{x^{\rho-1}-1}{\rho-1},$

with the convention that ${\frac{x^\rho-1}{\rho}}$ is ${\log x}$ when ${\rho=0}$, and similarly ${\frac{x^{\rho-1}-1}{\rho-1}}$ is ${\log x}$ when ${\rho=1}$. One can make these sorts of approximations more rigorous by means of Perron’s formula (or one of its variants) combined with the residue theorem, provided that one has good enough control on the relevant Dirichlet series, but we will not pursue these rigorous calculations here. (But see for instance this previous blog post for some examples.)

For instance, using the more refined approximation

$\displaystyle \zeta(s) \approx \frac{1}{s-1} + \gamma$

to the zeta function near ${s=1}$, we have

$\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s) \approx \frac{1}{(s-1)^2} + \frac{2 \gamma}{s-1}$

we would expect that

$\displaystyle d_2 \approx L_\infty + 2 \gamma$

and thus for instance

$\displaystyle \sum_{n \leq x} d_2(n) \approx x \log x - x + 2 \gamma x$

which matches what one actually gets from the Dirichlet hyperbola method (see e.g. equation (44) of this previous post).

Or, noting that ${\zeta(s)}$ has a simple pole at ${s=1}$ and assuming simple zeroes elsewhere, the log derivative ${-\zeta'(s)/\zeta(s)}$ will have simple poles of residue ${+1}$ at ${s=1}$ and ${-1}$ at all the zeroes, leading to the heuristic

$\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)} \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho}$

suggesting that ${\Lambda}$ should behave like the continuous arithmetic function

$\displaystyle t \mapsto 1 - \sum_\rho t^{\rho-1}$

leading for instance to the summatory approximation

$\displaystyle \sum_{n \leq x} \Lambda(n) \approx x - \sum_\rho \frac{x^\rho-1}{\rho}$

which is a heuristic form of the Riemann-von Mangoldt explicit formula (see Exercise 45 of these notes for a rigorous version of this formula).

Exercise 1 Go through some of the other explicit formulae listed at this Wikipedia page and give heuristic justifications for them (up to some lower order terms) by similar calculations to those given above.

Given the “adelic” perspective on number theory, I wonder if there are also ${p}$-adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as ${\sum_{n \leq x: n = a \hbox{ mod } p^j} f(n)}$. A key problem here is that there does not seem to be any good interpretation of the expression ${\frac{1}{t^s}}$ when ${s}$ is complex and ${t}$ is a ${p}$-adic number, so it is not clear that one can analyse a Dirichlet series ${p}$-adically. For similar reasons, we don’t have a canonical way to define ${\chi(t)}$ for a Dirichlet character ${\chi}$ (unless its conductor happens to be a power of ${p}$), so there doesn’t seem to be much to say in the ${q}$-aspect either.

In the previous set of notes, we studied upper bounds on sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$ for ${1 \leq M \leq N}$ that were valid for all ${t}$ in a given range, such as ${[T,2T]}$; this led in turn to upper bounds on the Riemann zeta ${\zeta(\sigma+it)}$ for ${t}$ in the same range, and for various choices of ${\sigma}$. While some improvement over the trivial bound of ${O(N)}$ was obtained by these methods, we did not get close to the conjectural bound of ${O( N^{1/2+o(1)})}$ that one expects from pseudorandomness heuristics (assuming that ${T}$ is not too large compared with ${N}$, e.g. ${T = O(N^{O(1)})}$.

However, it turns out that one can get much better bounds if one settles for estimating sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$, or more generally finite Dirichlet series (also known as Dirichlet polynomials) such as ${|\sum_n a_n n^{-it}|}$, for most values of ${t}$ in a given range such as ${[T,2T]}$. Equivalently, we will be able to get some control on the large values of such Dirichlet polynomials, in the sense that we can control the set of ${t}$ for which ${|\sum_n a_n n^{-it}|}$ exceeds a certain threshold, even if we cannot show that this set is empty. These large value theorems are often closely tied with estimates for mean values such as ${\frac{1}{T}\int_T^{2T} |\sum_n a_n n^{-it}|^{2k}\ dt}$ of a Dirichlet series; these latter estimates are thus known as mean value theorems for Dirichlet series. Our approach to these theorems will follow the same sort of methods used in Notes 3, in particular relying on the generalised Bessel inequality from those notes.

Our main application of the large value theorems for Dirichlet polynomials will be to control the number of zeroes of the Riemann zeta function ${\zeta(s)}$ (or the Dirichlet ${L}$-functions ${L(s,\chi)}$) in various rectangles of the form ${\{ \sigma+it: \sigma \geq \alpha, |t| \leq T \}}$ for various ${T > 1}$ and ${1/2 < \alpha < 1}$. These rectangles will be larger than the zero-free regions for which we can exclude zeroes completely, but we will often be able to limit the number of zeroes in such rectangles to be quite small. For instance, we will be able to show the following weak form of the Riemann hypothesis: as ${T \rightarrow \infty}$, a proportion ${1-o(1)}$ of zeroes of the Riemann zeta function in the critical strip with ${|\hbox{Im}(s)| \leq T}$ will have real part ${1/2+o(1)}$. Related to this, the number of zeroes with ${|\hbox{Im}(s)| \leq T}$ and ${|\hbox{Re}(s)| \geq \alpha}$ can be shown to be bounded by ${O( T^{O(1-\alpha)+o(1)} )}$ as ${T \rightarrow \infty}$ for any ${1/2 < \alpha < 1}$.

In the next set of notes we will use refined versions of these theorems to establish Linnik’s theorem on the least prime in an arithmetic progression.

Our presentation here is broadly based on Chapters 9 and 10 in Iwaniec and Kowalski, who give a number of more sophisticated large value theorems than the ones discussed here.