Talk at University of York

On April 20, 2024 By Robert G. C. SmithIn Physics

Thanks to the nice people at University of York for inviting me to talk as part of their mathematical physics seminar series. The title of my talk was, Introducing $\eta$ regularisation. My slides have been posted here.

Numberphile video about our recent paper: Does -1/12 protect us from infinity?

On February 16, 2024February 16, 2024 By Robert G. C. SmithIn Physics

Tony has filmed a video for Numberphile (embedded below) discussing some of the ideas in our recent paper that connects analytic number theory, divergent series, and divergent integrals in perturbative quantum field theory (I also wrote a bit about our paper here). Tony does a brilliant job at elucidating some of the beautiful results of our first paper, including a discussion about one of my absolute favourite regulators in relation to the astonishing formula $\lim \limits_{N \rightarrow \infty} \sum \limits_n n e^{-n/N} \cos(n/N) = -1/12$ . It became a bit of a joke between us, as I would always label this “the astonishing regulator” in our working notes. In fact, I think in the first draft of our paper this was the title I gave the relevant section. I think it is so beautiful. There are many interesting things to say about this particular choice of $\eta$ going beyond the immediate observation of its nature as a damped oscillator, which includes a related subclass of other remarkable cutoff functions. I look forward to sharing some bits in a future post.

In the video, Tony also shares a few comments describing our excitement about ongoing and future research. And yes, indeed, there is a sniff of something stringy ;)

Tao’s smoothed asymptotics

On February 8, 2024February 16, 2024 By Robert G. C. SmithIn Mathematics, Physics

In a recent paper, we explored possible connections between analytic number theory and the study of divergent series and the ultra-violet regularisation of loop integrals in perturbative quantum field theory. This project started quite innocently but eventually turned into a research programme that we’re really quite excited about, with many more interesting results and connections to be reported in our second paper and beyond. On the number theory side, an early motivation for our work was an expository article on smoothed asymptotics by Terence Tao. This article was first published on his blog; it was also included in his book Compactness and Contradiction, which is a collection of loosely related articles that explore the connections between finitary and infinitary mathematics. I personally prefer not to view his exposition on smoothed asymptotics in isolation of the book, because, although somewhat blurry, I sometimes think that maybe there is more to do with Tao’s definition of a compactness and contradiction argument from a physicist’s point of view, not least in relation to our ongoing work but also more generally in terms of the idea of constructing a bridge between the finitary (or quantitative) and the infinitary (or qualitative). But this is speculative, and I do not want to distract.

In this post, I’ll review some basics of Tao’s smoothed asymptotics and also make one or two comments about our paper.

Consider the sums of powers written in terms of their discrete partial sums:

$\displaystyle 1 + 2 + 3 + ... + n = \frac{1}{2}n(n+1) = \frac{1}{2}n + \frac{1}{2}n^2, \ \ \ \ \ (1)$

$\displaystyle 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{1}{6}n(n+1)(2n + 1) = \frac{1}{6} n + \frac{1}{2}n^2 + \frac{1}{3}n^3, \ \ \ \ \ (2)$

$\displaystyle 1^3 + 2^3 + 3^3 + ... + n^3 = \frac{1}{4}n^2(n+1)^2 = \frac{1}{4}n^2 + \frac{1}{2}n^3 + \frac{1}{4}n^4. \ \ \ \ \ (3)$

These three discrete sums (1–3), which have been represented as closed formulae, can be written as the general sum of powers ${\sum \limits_{n=1}^k n^i}$ with ${i}$ a fixed integer. When thought of in this way, we can compare the partial sums with their corresponding (infinite) divergent series. I will also include the case for ${i=0}$ :

$\displaystyle 1 + 1 + 1 + 1 + \dots = \sum \limits_{n=1}^{\infty} n^0, \ \ \ \ \ (4)$

$\displaystyle 1 + 2 + 3 + 4 + \dots = \sum \limits_{n=1}^{\infty} n, \ \ \ \ \ (5)$

$\displaystyle 1^2 + 2^2 + 3^2 + 4^2 + \dots = \sum \limits_{n=1}^{\infty} n^2, \ \ \ \ \ (6)$

$\displaystyle 1^3 + 2^3 + 3^3 + 4^3 + \dots = \sum \limits_{n=1}^{\infty} n^3. \ \ \ \ \ (7)$

As mentioned in our recent paper in relation to these formulae: the primary issue with such divergent series appears in the transition from partial sums to infinity. This was most notably exposed by Ramanujan, and is made explicit in the use of Tao’s methodology.

The easiest way to see this issue is to note that the formulae (1–3) are special cases of Faulhaber’s formula

$\displaystyle \sum_{n=1}^N n^z=\frac{1}{z+1} \sum_{n=0}^z \binom{z+1}{n} B_n N^{z-n+1}, \ \ \ \ \ (8)$

where ${z}$ is a positive integer and ${B_n}$ are the Bernoulli numbers (we follow the convention ${B_1= 1/2}$ ). In the limit ${N \rightarrow \infty}$ Faulhaber’s formula breaks down.

On the other hand, we know that we can compute the divergent series (4–7) using the Riemann zeta function. The Riemann zeta function ${\zeta(s)}$ is defined in the region ${Re(s) > 1}$ by the absolutely convergent series

$\displaystyle \zeta(s) = \sum \limits_{n=1}^{\infty} \frac{1}{n^s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \dots. \ \ \ \ \ (9)$

After analytic continuation one has for the sums of powers ${\zeta(0) = -1/2}$ , ${\zeta(-1) = -1/12}$ , and so on, so that in general we can define a direct relation with the Bernoulli numbers

$\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}. \ \ \ \ \ (10)$

And so, if one formally applies (9) we obtain

$\displaystyle \sum \limits_{n=1}^{\infty} n^0 = 1 + 1 + 1 + \dots = -1/2, \ \ \ \ \ (11)$

$\displaystyle \sum \limits_{n=1}^{\infty} n = 1 + 2 + 3 + \dots = -1/12, \ \ \ \ \ (12)$

$\displaystyle \sum \limits_{n=1}^{\infty} n^2 = 1 + 4 + 9 + \dots = 0, \ \ \ \ \ (13)$

and so on for each integer value ${i}$ in the sum ${n^i}$ , which evidences the general pattern

$\displaystyle \sum \limits_{n=1}^{\infty} n^s = 1 + 2^s + 3^s + \dots = -\frac{B_{s+1}}{s+1}. \ \ \ \ \ (14)$

What we obtain therefore is a collection of bizarre, if not altogether absurd, formulae. In the classical sense of summation they do not appear in any way coherent or reasonable, with one obvious reason being that we have positive summands appearing to equate to some negative or zero value.

As Tao clearly demonstrates in his article, if one tries to inspect the partial sums of these divergent series there is no obvious relationship with the constant values appearing in (11–13). From Faulhaber’s formula, if we apply it to (14) we obtain

$\displaystyle S_s(N) = \sum \limits^N_{n=1} n^s = \frac{1}{s+1}N^{s+1} + \frac{1}{2}N^s + \frac{s}{12}N^{s-1} + ... + B_s N, \ \ \ \ \ (15)$

which resembles very little of the right-hand side of the formulae (11–13). One of the main issues here has to do with the discrete nature of the partial sum, particularly in the transition to infinite series. When one uses partial sums to sum an infinite series ${\sum \limits_{n=1}^{\infty} a_n}$ , one is imposing a sharp truncation of that series at some finite ${N}$ . So what one is really doing is modifying the infinite series with a step function such that for ${f : \mathbb{R} \rightarrow \mathbb{R}}$ it can be written

$\displaystyle f(x) = \sum \limits_{n=1}^{\infty} a_n \theta \left(\frac{n}{N}\right) = \sum \limits_{n=1}^{N} a_n, \ \ \ \ \ (16)$

where

$\displaystyle \theta(x)=\begin{cases} 1 & x \leq 1 \\ 0 &x>1 \end{cases}. \ \ \ \ \ (17)$

For example, consider the divergent series (4). When looking at the ordinary partial sums and plotting them as a function of ${N}$ one will obtain a step graph, in which there exist jump discontinuities at integer values ${N}$ (for example, see this Wiki).

From a physics point of view, I think there is reasonable grounds to ask why this should be considered the most natural way to treat infinite (divergent) series. There are physical examples we might draw on to ask, why not a smooth cutoff instead?

Perhaps an example of Tao’s notion of post-rigour, by which it is meant that from the foundations of mathematical rigour one may allow the guidance of good intuition, I see the idea of smoothed asymptotics as creating a conceptual bridge of sorts. Instead of considering the ill-behaved discrete series ${\sum \limits_{n=1}^N a_n}$ , it can be replaced so that the convergence of a series is defined through the limit

$\displaystyle \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1}^{\infty} a_n \eta \left(\frac{n}{N}\right), \ \ \ \ \ (18)$

where ${\eta : [0, \infty) \rightarrow \mathbb{R}}$ equals the characteristic function of the interval ${[0,1]}$ . As a bounded function of compact support, we shall call any ${\eta}$ a cutoff function satisfying the conditions

$\displaystyle \eta(x)=\begin{cases} 1 & x \in [0,1] \\ 0 &x>1 \end{cases}. \ \ \ \ \ (19)$

One can ask if it is possible to consider other possible cutoffs ${\eta}$ , and, indeed, this is precisely what we do. In fact, we find that ${\eta}$ can be any Schwartz function so long that the smooth cutoff is normalised to ${\eta(0) = 1}$ . (It is also possible, as Tao notes, that the assumption of compact support in the interval ${[0,1]}$ can be extended to a more general case, which can be deduced by redefining the ${N}$ parameter). We show therefore that in extending Tao’s methodology we can define an infinite class of smooth cutoff functions ${\eta}$ , and a lot of time has since been spent studying what might be described as special subclasses of such cutoff functions!

The method of smooth asymptotics also has a lovely relation with the Euler-Maclaurin formula, which, for me, is one of the most beautiful formulae in modern mathematics. This was beautifully elucidated by Tao. We see that the Euler-Maclaurin formula gives a lot of information that helps one understand the results of smoothed asymptotic expansions, and also serves to define a rather lovely relation between smoothed sums and the Riemann zeta function (I will discuss much of this in another post, because I want to build the picture first from a derivation of the E-M formula as I think this gives some nice intuition about smoothed sums). We can also prove a lot of nice properties about ${\eta}$ without the Euler-Maclaurin formula, using many of the basic tools from analysis. For instance, earlier I mentioned convergence of smoothed sums. Recall the definition of series convergence:

$\displaystyle \sum \limits_{n=1}^{\infty} a_n = s \ \text{iff} \ \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1}^{N} a_n = s \ \ \ \ \ (20)$

with ${s \in \mathbb{R}}$ . In the language of smoothed sums, we can equivalently define convergence of series as

$\displaystyle \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1}^{\infty} a_n \left(\frac{n}{N}\right). \ \ \ \ \ (21)$

A convergent series ${\sum a_n}$ of non-negative terms is absolutely convergent, since ${\sum a_n}$ and ${\lvert \sum a_n \rvert}$ are the same. Using the fact that ${\eta}$ is bounded, since there is an ${M > 0}$ such that ${\mid \eta(x) \mid < M}$ for all ${x}$ , and the fact that ${\eta}$ is continuous, one can show that for the absolutely convergent sum ${\sum \limits_{n=1}^{\infty} a_n}$ it also true ${\lim \limits_{N \rightarrow \infty} \sum \limits_{n=1}^{\infty} a_n \eta \left(\frac{n}{N}\right) = s}$ . One can similarly prove the case where ${\sum \limits_{n=1}^{\infty}a_n}$ is conditionally convergent.

***

From a physics perspective, this general idea of using a smooth cutoff function is not necessarily new. Often when dealing with divergent integrals in QFT, one approach is to introduce some sort of function to smear or smooth interactions. This can be thought of formally in the space of distributions. In a more direct and physically intuitive manner, a textbook example is how one might simply consider in place of a sharp momentum cutoff some smooth cutoff function ${\rho(p, \Lambda)}$ in Euclidean space. This smooth cutoff is then defined such that ${\lim_{p \rightarrow \infty} \rho(p, \Lambda) = 0}$ when ${\lambda}$ is fixed. Or, when ${p}$ is fixed, ${\lim_{\Lambda \rightarrow \infty} \rho(p, \Lambda) = 1}$ . An example of an explicit choice of cutoff is ${\rho = \exp(-p^2 / \Lambda^2)}$ . Another intuitive example from physics can be found in Joe Polchinski’s great work on renormalisation group flow equations, where a simple smooth regulator function is introduced. Similarly, in the heat-kernel approach to QFT and its rigorous mathematical formulation, one finds motivation for the use of a smooth cutoff function. In more general scenarios, such as quantum gravity, the reluctance of string theory to travel deep into the UV is a direct manifestation of the smoothing or smearing effect of the string length scale (as seen in a study of modular invariance of the worldsheet).

Perhaps one of the oldest examples, as referenced in Matthew D. Schwartz’s wonderful textbook on QFT and the standard model, comes from the calculation of the Casimir effect. In his original paper, Casimir showed that there is a way to calculate the force in a regulator independent way. In this approach the energy is defined as

$\displaystyle E(a) = \frac{\pi}{2} \sum \limits_n \frac{n}{a} f \left( \frac{n}{a\Lambda} \right), \ \ \ \ \ (22)$

where ${f(x)}$ is a generic cutoff function. It is shown that any sensible regulator will correctly give the Casimir force so long that ${\lim \limits_{x \rightarrow \infty} xf^{(i)}(x) = 0}$ and at the origin ${f(0) = 1}$ . In taking this approach it is possible to choose as a special case any one of the common regularisation schemes that satisfies the above conditions.

This raises the question whether there is a more general way to formulate such an idea for all QFTs. We show that $\eta$ regularisation is, indeed, incredibly general in the extended setting. In fact, in the context of perturbative QFT we can capture all of the common regularisation schemes as a particular choice of $\eta$ . What’s more, both Tao’s methodology and QFT exhibit regulator dependence of power law divergences, while the universal features of finite terms in Tao’s study of divergent series mirror conjectured universal features of logarithms in QFT. These are among a number of insights that provide intriguing hints at potentially something deeper. But most fascinatingly, we also found a surprising connection between the ${\eta}$ regularisation of divergent series in analytic number theory and the preservation of gauge invariance at one loop in a regularised quantum field theory! This result is the main feature of our first paper. We show in the calculation of the vacuum polarisation that cancellation of the quadratic divergences requires precisely enhanced regulators of order one, which goes back to the fact that for sum of the naturals the corresponding Mellin transform is vanishing $C[\eta]=0$ . It is really quite a beautiful result.

New paper – Smoothed asymptotics: from number theory to QFT

On January 23, 2024January 24, 2024 By Robert G. C. SmithIn Physics

The first of a series of papers co-authored with my supervisor, Tony Padilla, has today appeared on arXiv. In it we find a deeply surprising connection between analytic number theory, from which we define an extremely general class of smooth regulator functions, and the underlying symmetries of fundamental physics. Central to this story are the sums of integer powers, and on a purely number theoretic level we derive what I think are some astonishing formulae. But extending to quantum field theory is where the work starts to really get interesting. In particular, we show that for the sum of natural numbers this general class of $\eta$ regulators, which allow the infinite series of natural numbers to converge towards the seemingly absurd value of $-1/12$ , are intimately connected to the preservation of gauge invariance at one-loop in a wide class of non-abelian gauge theories!

Why might this be? I think it is clear from this paper that there is enough reason to ask whether there is some underlying fundamental structure – that maybe there is some deeper structure underlying the physics, particularly the way in which $\eta$ regularisation preserves the symmetries of the microscopic theory. Indeed, it turns out that generalising these observations and results has so far been like discovering a vein of interconnected concepts, and the further we mine the deeper it seems to go. The first batch of more general results will be presented in our second paper.

Of course, it is possible that at the end of the day we could end up eating red herring; but the connections we’ve observed between the maths and the physics has encouraged us to take what we’ve found seriously.

Here is the abstract of today’s submission:

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultra-violet regularisation scheme for loop integrals in quantum field theory which we call η regularisation. This allows us to reveal a surprising connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularisation of loop integrals in quantum field theory. In particular, we note that a method for regularising the series of natural numbers so that it converges to minus one twelfth inspires a regularisation scheme for non-abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarisation tensor. We also comment on a possible connection to Schwinger proper time integrals.

Theorem connecting partial sums and the values of the Riemann zeta function

On January 15, 2024January 18, 2024 By Robert G. C. SmithIn Mathematics

In the course of some studies relating to our first paper on analytic number theoretic sums and quantum field theory, I came across a simple yet seemingly not too well known theorem (as far as I have seen in the literature; however I do note the likeness to, and manner from which the result can be directly extrapolated from, Hans Rademacher’s 1973 book Topics in Analytic Number Theory). The theorem can be found in a short paper by Jan Minac from 1994. The main result gives a slightly different perspective on the relation between the values of the Riemann zeta function and the partial sums of the sums of integer powers.

The origin of the Riemann zeta function ${\zeta(s)}$ goes back to the study of many-particle systems and how they may be encoded in a single partition function. In 1737, Euler proved the identity

$\displaystyle \prod \limits_p (1 - \frac{1}{p^s})^{-1} = \sum \limits_{n=1}^{\infty} = \zeta(s), \ \ \ \ \ (1)$

which holds for all real numbers ${s > 1}$ and, amazingly, refers to all prime numbers ${p}$ . After Euler’s introduction of ${\zeta (s)}$ over the real numbers, it was Bernhard Riemann in his seminal 1859 paper who took this formula as his starting point and extended it to a complex variable. In the process, he proved among other things the relation between its zeros and the distribution of prime numbers. From this comes the Riemann zeta function, which can be written as the sum of the reciprocals of the positive integers

$\displaystyle \zeta(s)=\sum \limits_{n=1}^{\infty}\frac{1}{n^{s}}=1+\frac{1}{2^s}+\frac{1}{3^s}+ \dots, \ \ \ \ \ (2)$

where ${s}$ is a complex number with ${Re(s) > 1}$ . For negative integers (and also zero), the Riemann zeta function can be related to the Bernoulli numbers ${\zeta(-s) = (-1)^s B_{s+1} / (s+1)}$ . Another remarkable fact about this, and about the Bernoulli numbers in general, is that all the values ${\zeta(-s)}$ are rational numbers: ${\zeta(-1) = - 1/12}$ , ${\zeta(-2) = 0}$ , ${\zeta(-3) = 1/120}$ , and so on.

From the definition of Faulhaber’s formula, heuristically there is a relation ${\sum \limits_{n=1}^{\infty} n^s = \frac{(-1)^s B_{s+1}}{s+1}}$ that can be deduced, which agrees with the value of the Riemann zeta function for negative integers ${s = -p >0}$ . However, in general, when one expands the partial sums for the respective sums of integer powers, there is no obvious relationship between the partial sum form and the values obtained by the zeta function as indicated in the above equality.

Interestingly, from Minac’s paper, we see that we can define the partial sums ${S_a (N) := \sum \limits_{n=1}^{N-1} n^a}$ such that they are expressible as polynomials in ${N}$ of degree ${a+1}$ with rational coefficients (again, see Rademacher’s 1973 book). One can then ask whether any connection between ${\zeta(-a)}$ and ${S_a(N)}$ can be made. This leads us to the main theorem of Minac’s paper, for which I also repeat the proof.

Theorem 1 : Consider the power sum ${S_a (N) = \sum \limits_{n=1}^{N-1} n^a}$ . For positive integers ${a}$ there exists the relation ${\zeta(-a) = \int_0^1 dx \ S_a (x)}$ .

Proof: Let ${B_k}$ be the kth Bernoulli number for ${k = 0,1,2,\dots}$ , and ${B_0(x) = 1}$ . For each ${m \in \mathbb{N}}$ we define the mth Bernoulli polynomial ${B_m (x) = \sum \limits_{k=0}^{m} \begin{pmatrix} m \\ k \\ \end{pmatrix} B_k x^{m-k}}$ . Importantly, we note ${B_m(1) = (-1)^m B_m = (-1)^m B_m(0)}$ , and the derivative of the Bernoulli polynomial takes the form ${d/dx \ B_m (x) = mB_{m-1}(x)}$ . Finally, we shall use the fact ${S_a(N) = \frac{B_{a+1}(N) - B_{a+1}(1)}{a+1}}$ , which is based on one of the most important properties of Bernoulli polynomials $B_m (x + 1) - B_m (x) = mx^{m-1}$ , as well as ${\zeta(-a) = (-1)^a \frac{B_{a+1}}{a+1} = - \frac{B_{a+1}}{a+1}}$ for ${a \in \mathbb{N}}$ .

Since ${S_a(N)}$ is polynomial in ${N}$ of degree ${a + 1}$ , the important observation is that we can exploit these basic properties of Bernoulli polynomials to establish the integration of the polynomial representation ${S_a(N)}$ of the partial sums from ${0}$ to ${1}$ . Let ${a \in \mathbb{N}}$ , then from the above it follows

$\displaystyle \int_0^1 dx \ S_a(x) = \int_0^1 \frac{B_{a+1}(x) - B_{a+1}(1)}{a+1}$

$\displaystyle = \frac{B_{a+2}(1) - B_{a+2}(0)}{(a+1)(a+2)} + (-1)^a \frac{B_{a+1}}{a+1} = 0 + (-1)^a \frac{B_{a+1}}{a+1} = \zeta(-a). \ \ \ \ \ (3)$

$\Box$

As an example, let us consider one of my favourite cases: the sum of natural numbers corresponding formally to ${zeta(-1)}$ . Theorem 1 gives

$\displaystyle 1 + 2 + 3 + 4 + \dots = \zeta(-1) = \int_0^1 dx \ \frac{x(x - 1)}{2}= \int_0^1 dx \ \left(\frac{x^2}{2} - \frac{x}{2}\right) = \frac{1}{6} - \frac{1}{4} = -\frac{1}{12}. \ \ \ \ \ (4)$

Very cool!

[1] Ján Mináč, “A remark on the values of the Riemann zeta function,” Expositiones Mathematicae, 12: 459-462, 1994.

Notes from my talk at NCOG winter meeting

On December 16, 2023December 16, 2023 By Robert G. C. SmithIn Physics

On Friday I gave a talk as part of the Nottingham Centre of Gravity (NCOG) winter meeting. My slides can be found here. There is a bit more detail in these notes (as I had a bit more time to present), with some more results shared from what will be the first in our forthcoming series of papers.

It was my first in-person talk, and so naturally it was a big occasion for me. I think I did alright overall. There were one or two typos in my slides, which annoyed me greatly. I also received so many questions (much more than I was anticipating) it threw me off a bit, and then toward the end I felt pressed for time; so, admittedly, I ended up not hitting all the points I wanted to about our research. As one example, I really wanted to say more about some of the stringy connections being explored and about some of the challenges and observations we have so far faced on this front. I wanted to especially emphasise some examples we already have of stringy $\eta$ -regulators or those closely related, particularly in terms of a preliminary study of certain modular invariant regulators and their motivation, and how I think it is important to study more deeply how these behave, etc. in moduli space. I think it would have helped put more of a bow on the overall presentation. But given that it was my first in-person talk, I’ll take it as a learning experience as I prepare for the next one.

Despite some of these annoyances with myself, I am happy with the overall feedback I have received. There also seems to be a lot of interest in our work, which is exciting.

Notes from my talk at UK QFT XII

On November 20, 2023December 13, 2023 By Robert G. C. SmithIn Physics

From number theory to physics: Regularising QFTs

Last week I gave a talk at UK QFT XII. Here are my notes. It was a condensed talk, so the slides only touch on a few choice results focusing primarily in QFT (the middle part of the sandwich in my overview on slide 2). Next month I am giving a more extensive talk at The Nottingham Centre of Gravity, which will also give more time for stringy discussion. I’ll post my more extensive notes when the time comes.

UK QFT was hosted this year at the Institute of Physics. Although I gave my talk remotely, it was fitting that my first talk would be given at a conference sponsored by IoP (from whom I was awarded a PhD scholarship). It felt slightly serendipitous.

Anyway, I’m chuffed by the fact that my talk was well-recieved. This was the first time Tony and I shared any substantial bits from our research over the past year. We’ve got some really exciting results, which I look forward to sharing more about in the coming months as and when we start releasing our series of papers.

The discrete and the continuous

On September 17, 2023September 17, 2023 By Robert G. C. SmithIn Mathematics

I think it is interesting that at the heart of mathematics resides the distinction between the discrete and the continuous [1,2]. On the one hand, the field of discrete mathematics deals with countable data. Continuous mathematics, on the other hand, deals with uncountable data. So, when considering the former we are primarily interested in finite (or countable) objects, examples of which include the integers, finite series, or finite metric spaces. This is very different than the field of continuous mathematics, which is home to the techniques of algebraic theory and the powerful tools of calculus. Here we have the study of continuous functions (i.e., functions that have no gaps or discontinuities), measure theory, topological group theory, and algebraic geometry. This is also the world in which such mathematical objects as manifolds, real or complex numbers, and continuous metric spaces and topological spaces, are utilised.

In nature both languages are exhibited. It is therefore quite easy to motivate some sort of quest in search for a bridge between the discrete and continuous worlds – to discover formulae that relate discrete sums of numerical powers with integration, for example. Such a quest goes back to the Pythagoreans, Archimedes, and other ancients [1, 3]. The Pythagoreans famously mastered the practice of utilising discrete approximations to obtain continuous areas and volumes by the method of exhaustion. They viewed successive positive integers as triangular numbers, which, in modern notation, gives the famous formula for the sum of powers of positive integers. Furthermore, consider the following sums of powers, which we may write as discrete partial sums:

$\displaystyle 1 + 2 + 3 + ... + n = \frac{1}{2}n(n+1) = \frac{1}{2}n + \frac{1}{2}n^2, \ \ \ \ \ (1)$

$\displaystyle 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{1}{6}n(n+1)(2n + 1) = \frac{1}{6} n + \frac{1}{2}n^2 + \frac{1}{3}n^3, \ \ \ \ \ (2)$

$\displaystyle 1^3 + 2^3 + 3^3 + ... + n^3 = \frac{1}{4}n^2(n+1)^2 = \frac{1}{4}n^2 + \frac{1}{2}n^3 + \frac{1}{4}n^4. \ \ \ \ \ (3)$

The first relationship for the sum of the natural numbers is the one that goes back to the Pythagorean school in the 6th century BC. Archimedes of Syracuse (circa 287-212 BC), considered the greatest mathematician of antiquity, discovered the second relationship for the sum of the squares. The third relationship, for the sum of cubes, can be found in the work of Nicomachus of Gerasa (circa 60–120 AD), after which the identity ${\sum \limits_{k=1}^{n} k^3 = (\sum \limits_{k=1}^{n} k)^2}$ is named Nicomachus’ theorem. However, this formula, albeit without proof, is said to have appeared much early in a work by the Indian mathematician Aryabhata (476–550 CE) [4].

These three discrete sums, which we’ve represented as of closed formulae, can be written as the general sum of powers ${\sum \limits_{k=1}^{n} k^i}$ with ${i}$ a fixed natural number. The sum of powers in closed form is interesting to study in-itself; but where it gets really interesting is when we start to think about its infinite series. Indeed, such a curiosity has motivated some famous results. In modern times the formula for sums of integer powers was first given in generalisable form by Thomas Harriot (c. 1560-1621). Johann Faulhaber (1580-1635) also gave formulas for these sums up to the 17th power [1,2]. It was Pierre de Fermat who made the important when he argued that he could use what are called “figurate numbers” to capture generalised formulae for these discrete sums of numerical powers. It was then Blaise Pascal in his extensive treatise Potestatum Numericarum Summa (circa 1654), which produced the first explicit recursive formula for sums of powers in arithmetic progression using binomial coefficients. This work relates to what many will recognise as Pascal’s triangle. One of Pascal’s motivations was to define a recursive formula that could sum higher powers up to infinity.

Observe that in the three cases (1–3), the sum of the sth powers ${n^s}$ of the integers ${k}$ from 1 to ${n}$ is given, for ${s = 1, 2, 3,}$ by a polynomial in ${n}$ , of degree ${s+1 = 2, 3, 4}$ . Therefore let us define for ${s = 1,2,3,...}$

$\displaystyle S(s,n) := 0^s + 1^s + 2^s + ... + n^s = \sum \limits_{k=0}^{\infty} k^s. \ \ \ \ \ (4)$

It was a general expression for this sum ${S(n,s)}$ that Pascal defined in the form

$\displaystyle S(s,n) = \frac{1}{s+1} ((n+1)^{s+1} - \begin{pmatrix} s + 1 \\ s - 1 \\ \end{pmatrix} S(s-1, n) - ... - \begin{pmatrix} s + 1 \\ 1 \\ \end{pmatrix} S(1,n) - \begin{pmatrix} s + 1 \\ 0 \\ \end{pmatrix} S(0,n) - 1). \ \ \ \ \ (5)$

The next important contribution came in the 18th century, with the work of Jacob Bernoulli titled Ars Conjectandi. This is quite a magnificent treatise, which, in many ways, represents the beginnings of probability theory. Bernoulli conjectured the general pattern in the coefficients of the closed-form polynomial solution to the summation problem. The connection between probability theory and sums of powers is via the combinatorial counting numbers (for more on this history, see this note). He also introduced the all-important Bernoulli numbers into mathematics, which will review in a later note. To see a bit of this , consider Bernoulli’s reformulation of Pascal’s solution given below in a form commonly considered today

$\displaystyle S(s,n) = \frac{1}{s + 1}n^{s + 1} + \frac{\sigma^1_1}{1 !}n^s + \frac{\sigma^2_1}{2!}s n^{s-1} + ... + \frac{\sigma^s_{1}}{s!}s(s-1) ... 2n, \ \ \ \ \ (6)$

where ${\sigma^s_m}$ denote the coefficients. This first form of this formula can be more cleanly written in terms of combinatorial factors

$\displaystyle S(s,n) = \frac{1}{s + 1}(\begin{pmatrix} s+1 \\ 0 \\ \end{pmatrix}\sigma^0_1 n^{s+1} + \begin{pmatrix} s + 1 \\ 1 \end{pmatrix}n^s + \begin{pmatrix} s+1 \\ 2 \\ \end{pmatrix}\sigma^2_1 n^{s-1} + ... + \begin{pmatrix} s+1 \\ s \\ \end{pmatrix} \sigma^s_1 n). \ \ \ \ \ (7)$

From Bernoulli’s formula, we see how we may write the polynomials ${S(s,n)}$ in such a way that all coefficients ${\sigma^s_m}$ are expressed in terms of a single sequence of numbers [3]. These ${\sigma^s_m}$ follow from a well-known table of coefficients for the case ${s = 1,2,3,...}$ in which we’re only interested when ${m = 1}$ . When we study the recurrence relations associated with the formula (7) we find for fixed coefficients ${\sigma^s_1}$ the Bernoulli numbers (ignoring the discrepancy in the fact of (-1)); but again, we’ll save these special numbers for a post in their own right, as we will also study their generating function and deeper relation to the sums of powers.

This formula by Bernoulli then takes us to the great Leonhard Euler. To finish the brief historical sketch, we come also to the modern story in the search for a way to make sense (formally and rigorously) of divergent series. It was in this time that Euler displayed his genius and completely broke with the view of his contemporaries, taking a radically new approach to the Basel problem [7]. The main task of the Basel problem was to find the reciprocal squares of the natural numbers ${\sum \limits_{i=1}^{\infty} 1/i^2 = \frac{1}{2} + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + ...}$ . It was first formulated by Gottfried Leibniz and the brothers Jakob and Johann Bernoulli \citep{Peng02b}. Euler’s genius was such that by refitting the problem into a new context, namely in using calculus to relate the discrete sum of an arbitrary function ${\sum \limits f(i)}$ to the antiderivative of the form ${\int_0^{n} \ dx \ f(x)}$ , he could prove the sum of reciprocal squares is exactly ${\pi^2 / 6}$ .

In solving the Basel problem, Euler would conclude a centuries long search for closed expressions for sums of numerical powers ${\sum \limits_{i=1}^{n} i^k \approx \int_0^n x^k \ dx}$ . Since the sum and the integral provide first approximations to each other – from elementary calculus, one will know that the sum can be interpreted as the total area of rectangles forming a step graph along some curve, while the integral, when evaluated between limits, can be interpreted as the area under the same curve – Euler could imagine a bridge connecting continuous and discrete mathematics.

The relation of the first approximation of a discrete sum and an integral that Euler derived was the initial step toward what, today, we know as the Euler-Maclaurin summation formula. The asymptotic nature of the Euler-Maclaurin formula has proven very important. Indeed, so important is this formula that we will spend much time studying it in future posts. (For a historical review of the development of some of Euler’s key ideas, including his motivation to study the problem of interpolation and the connection to expressing the sum of a series using an integral, see also [8]).

I really want to look at other ways the discrete and the continuous are found to engage in a beautiful interplay. Although parallel to my PhD research, it is something I have been thinking about a little bit and taking notes on. Such an investigation includes the study of sums, recurrences, and number theory, including things like generating functions, discrete probability, and asymptotic methods [5]. It also takes us to the heart of divergent series [6] and many important concepts, techniques, and results throughout physics. I even find it interesting that there are some people who speculate that these seemingly disconnected branches of study should be seen as part of a unity of methods and theorems that defines mathematics as a whole – that perhaps there is a more complete foundation of mathematics in which the distinction no longer exists. I’m not entirely sure about this, though I like the picture it represents. Moreover, I do think it is noteworthy that in mathematical physics – indeed, at the heart of quantum gravity – there seems to reside the very same analogous search for a bridge between the discrete. In the stringy picture, sometimes I like to imagine that given the complete non-perturbative theory – or at least a better understanding of brane quantisation – maybe this distinction will be found to fail altogether. It is such a cool thought.

References

[1] David J. Pengelley. Dances between continuous and discrete: Euler’s summation formula. 2019. 1, 2

[2] David J. Pengelley. Dances between continuous and discrete: Euler’s summation formula. 2019. 1, 2

[3] M. Santander. Sumas de potencias y series divergentes un panorama sobre sumaci´on de series, sumas suavizadas, n´umeros y polinomios de bernoulli, la f´ormula de euler-maclaurin y la funci´on de riemann. 2019. 1, 3

[4] C.B. Boyer and U.C. Merzbach. A History of Mathematics. Wiley, 2011. 1

[5] Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik. Concrete mathematics – a foundation for computer science. 1991. 4

[6] G.H Hardy. Divergent Series. Clarendon Press, Oxford, 1973. 4 [7] Leonhard Euler. On the sums of series of reciprocals [arXiv: math/0506415]. [8] Giovanni Ferraro. Some Aspects of Euler’s Theory of Series: Inexplicable Functions and the Euler–Maclaurin Summation Formula. Historia Mathematica, 1998, 25: 290-317.

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Tag: Analytic Number Theory