Periods

The Greeks raised many problems. One of the best known is that of squaring the circle. Given a circle, construct a square of the same area. For the Greeks to construct meant to give a graphical algorithm. The basic operations included the use of ruler and compass. The problem is mentioned by Aristophanes in the year 414 B.C.; we can imagine that at that time they were looking for a solution, if they had dreamt that the construction might not exist they would have considered the problem intractable. How can one prove that no matter what one does one will never get to construct the square?

It was not until 1761 that Lambert proved that \(\pi\) was irrational, and conjectured that it would be transcendental, that is, that it does not satisfy any equation of the form \(P(x)=0\), \(P\) being a polynomial of integer coefficients. In 1837 Wantzel shows that if it were constructible it would be algebraic, there would be a polynomial with integer coefficients and such that \(P(x)=0\). These concepts opened the way to the proof of the impossibility of construction. In a relatively short time, in 1882, Lindemann succeeded in proving that \(\pi\) was a transcendental number and that therefore the problem of construction with ruler and compass was impossible.

This story shows us that even if a problem seems intractable, mathematics creates new concepts –tools– with which the solution can be found later on. The history of mathematics teaches us that this has happened not once, but a thousand times, with sometimes much more complicated and convoluted paths. All this is because today we are going to consider a problem that in some ways seems intractable and is closely associated with the problem of quadrature, or rather, with the nature of the number \(\pi\). 

The problem is a recent one but, as always, it has antecedents. In this case it was Euler who first glimpsed the problem. Euler’s works are nowadays accessible in The Euler Archive. We can find there the originals, usually in Latin, together with many English translations. The work in which Euler deals with our problem is E565, we cannot find the translation in The Euler Archive, but there is a translation in arXiv.  (arXiv contains many others, currently 67 works by Euler are translated in arXiv).

In this work Euler considers that there are transcendent functions that easily enter into the analysis by means of the integral expressions, for example $$\log x=\int_1^x\frac{dt}{t},\qquad \arctan x=\int_0^x\frac{dt}{1+x^2}$$ Even when we want to obtain the length of the conic sections we arrive at the elliptic integrals $$\int R(x,\sqrt{(a+bx^2)(c+dx^2)})\,dx$$ Alongside these quantities there are others, generally the sum of series that resist being represented by integrals, and he gives as an example the series $$1+\frac{1}{3}+\frac{1}{7}+\frac{1}{15}+\frac{1}{31}+\frac{1}{63}+\cdots= \sum_{n=1}^\infty\frac{1}{2^n-1}.$$

Definition of periods

The number \(\pi\) and the integrals considered by Euler for the length of the conics appear as periods of the circular and elliptic functions. The idea of periods is extended to other algebraic varieties. The concept of periods was considered by Grothendieck. Grothendieck’s point of view in his theory of motives is to define a category to construct a Galois theory. The periods should have conjugates in which the group would act. Grothendieck conjectured that any polynomial relation between periods would have a motivic origin. All this would lead me into a field I do not know enough about. I am only explaining this to show that the simplest definition we will give has very deep notions behind it. It was Zagier and Kontsevich in 2001 who gave a simple definition, popularising the subject. 

Definition. A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients over domains defined by polynomial inequalities with rational coefficients. 

For example \(\pi\), \(\log2\) and \(\pi/(2\sqrt{2})\) are periods $$\pi=\int_{x^2+y^2\le 1}\,dx\,dy=\int_{-\infty}^{+\infty}\frac{dx}{1+x^2}, \quad \log 2=\int_1^2\frac{dx}{x},\quad \int_0^\infty \frac{dx}{1+x^4}=\frac{\pi}{2\sqrt{2}}$$ But it seems that number \(e\) is not. Or at least no one has managed to write an (algebraic) integral that takes this value.

It can be proved that in the definition we can change “rational function” and “rational coefficients” to “algebraic function” and “algebraic coefficients”. This does not change the comprehensiveness of the definition.

Periods build a ring, for example if $$\omega_1=\int_{A}f(x)\,dx_1\,\cdots\,dx_n,\qquad \omega_2=\int_{B}g(y)\,dy_1\,\cdots\,dy_m,$$ then, by Fubini’s Theorem, $$\omega_1\omega_2=\int_{A\times B}f(x)g(y)\,dx_1\,\cdots\,dx_n\,dy_1\,\cdots\,dy_m.$$

I became interested in mathematics at age 10 – 11, mainly because of the influence of my brother. Several books at popular level were very inspiring. Also, my brother was subscribed to the famous monthly “Kvant” magazine containing many wonderful articles on mathematics and physics addressed to high-school kids, sometimes explaining even new results or unresolved problems.  Also, I used to take part in Olympiads at various levels and was very successful.

In the Soviet Union, some schools had special classes for gifted children, with an additional four hours per week devoted to extra-curricular education (usually in mathematics or physics) taught by university students who had passed through the same system themselves. At age 13 – 15 I was attending such a school in Moscow,…(Kontsevich)

The environment is important. Until now, Spain has never had an environment like this. However, thanks to projects such as Estalmat, something is changing…

A little more about periods

Periods form a ring \(\mathcal P\). All algebraic numbers are periods, they are units in \(\mathcal P\). It does not seem that \(P\) is a field. For example it is thought that \(\frac1\pi\) is not a period. So we could talk about divisibility. The values of \(\zeta(s)\) in the natural numbers are periods. In particular \(\zeta(2n)=(-1)^{n+1}\frac{(2\pi)^{2n}B_{2n}}{2\cdot (2n)!}\) are periods divisible by \(\pi^{2n}\). On the other hand, we do not know if the numbers \(\zeta(2n+1)\) are multiples of \(\pi^{2n+1}\), that is, we do not know if \(\pi^{-2n-1}\zeta(2n+1)\) is a period.

It is speculated that we can define conjugates between the periods, which would be permuted by a Galois group of the periods. Perhaps one day we will be able to clarify the relation of the numbers \(\zeta(2n+1)\) to each other by means of this group.

The logarithms of all the algebraic numbers are periods. So are the values of the \(L\) functions on the integers. Many of the interesting numbers are. More information on this question can be found in the original work of Kontsevich and Zagier quoted at the end. 

Proving that a particular number is not a period is difficult in the present state. But we assume that \(e\), \(\frac{1}{\pi}\) or \(\gamma\) –Euler’s constant– are not. Proving that any of these is not a period would be a major advance in the understanding of periods. Kontsevich and Zagier raise the question of constructing a number that is not a period. It is clear that they exist since the set of periods is numerable. This question has been solved by Yoshinaga by means of a diagonal procedure and using computability techniques. 

Another interesting question is whether we can provide an algorithm which, given two periods, proves that they are the same or different, as the case may be. All these questions seem as difficult to us now as the problem of squaring the circle must have seemed to the Greeks in their time.

The main conjecture about periods

The representation of a period by an integral \(\int_A f(x)\,dx\) is not unique. (Recall that we assume \(A\subset \mathbf{R}^n\) a set defined by inequalities between polynomials with algebraic coefficients and the logical connectives and \(f(x)\) a rational function with algebraic coefficients). 

The conjecture is presented by Kontsevich and Zagier in the most innocent way: A widely held belief, based on a judicious combination of experience, analogy and wishful thinking. (That is: to convince oneself that the world is the way one wants it to be and that things will turn out the way one wishes them to turn out). It is the following conjecture:

Conjecture. If a period has two representations, we can go from one to the other by means of the following four basic operations: linearity of the integrand, linearity of the domain of integration, algebraic changes of variable, and Stokes’ theorem with chains and algebraic forms.

These rules are:

1. \(\int_A(f(x)+g(x))\,dx=\int_A f(x)\,dx+\int_A g(x)\,dx\).
2. \(\int_{A\cup B}f(x)\,dx=\int_Af(x)\,dx+\int_B f(x)\,dx\) if \(A\cap B\) has null measure.
3. \(\int_{\varphi(A)}f(x)\,dx=\int_A f(\varphi(x))\cdot|\det \varphi'(x)|\,dx\) being \(\varphi\) an injective algebraic function of \(A\) in \(\varphi(A)\).
4. \(\int_{\partial C}\omega=\int_C d\omega\). Being the form \(\omega\) and the chain \(C\) algebraic.

In other words, we do not expect any miraculous coincidence of two integrals of algebraic functions which will be not possible to prove using three simple rules. This conjecture, which is similar in spirit to the Hodge conjecture, is one of the central conjectures about algebraic independence and transcendental numbers and is related to many of the results and ideas of modern arithmetic algebraic geometry and the theory of motives.

(Kontsevich y Zagier)

Let us recall that the Hodge conjecture is one of the millennium problems and this conjecture is also very close to the Grothendieck conjecture. Nevertheless, my first impression as an analyst is that the conjecture surprises me a lot. Although I have to say that what I have read afterwards and knowing the stature of the mathematicians proposing it deserves a lot of respect. However, I think my doubts will help to understand the scope of the conjecture. So I will expose myself to the risk of appearing naïve by exposing them.

My reasons for doubting the conjecture

This is a relatively recent personal story. I have always liked definite integrals. I have consulted many times the classical tables of Gradshteyn and Ryzhik (GR in the following). This is why I was quite intrigued by an article by Moll published in the AMS Notices in 2010. Moll tells the story of how he and his student Boros stopped at an entry in the GR tables GR $$I:=\int_0^\infty\frac{dx}{(1+x^2)^{3/2}\sqrt{\varphi(x)+\varphi(x)^{1/2}}}=\frac{\pi}{2\sqrt{6}}$$ where $$\varphi(x)=1+\frac{4x^2}{3(1+x^2)^2}.$$ After futile attempts, they proved that the formula was incorrect by the simple means of calculating both members to exactly 6 decimal places. This raises two problems: what is the actual value of the integral, and can we modify the integrand slightly to get an integral which has a value of \(\frac{\pi}{2\sqrt{6}}\)?

In the last days of December 2017, I managed to express the Moll integral by means of two elliptic integrals. In the form \(I=\alpha I_1+\beta I_2\) where, being \(\Delta(x)=\sqrt{(1-x^2)(1-\frac13x^2)}\) and \(k=2-\sqrt{3}\) we have $$\alpha=\frac{\sqrt{3}-1}{\sqrt{2}},\quad \beta=-\frac{1}{\sqrt{2}},$$ y $$I_1=\int_0^1\frac{dx}{(1-kx^2)\Delta(x)},\quad I_2=\int_0^{\sqrt{k}}\frac{dx}{\Delta(x)}.$$ Note that \(\alpha\), \(\beta\), \(I\), \(I_1\) and \(I_2\) are all periods. According to the Kontsevich-Zagier conjecture there should exist algebraic transformations which prove the relation \(I=\alpha I_1+\beta I_2\). My proof published in Ramanujan Journal is about 20 pages long, of which pages 9 to 20 are algebraic transformations. But the first steps proving $$I=\int_0^1\frac{dx}{\sqrt{1+\frac{16}{3}x^2(1-x)^2+\sqrt{1+\frac{16}{3}x^2(1-x)^2}}}.$$ are of a very analytical nature. By serially developing the integrand and integrating term by term, you get to see$$I=\sum_{n=0}^\infty(-1)^n\frac{1}{2^{2n}}\Bigl({2n\atop n}\Bigr)\sum_{k=0}^\infty \frac{(-1)^k}{k!}\frac{\Gamma(\frac{n+1}{2}+k)}{\Gamma(\frac{n+1}{2})}2^{2k}\frac{2^{2k}}{3^k}\frac{(2k)!(2k)!}{(4k+1)!}.$$ This series is not absolutely convergent, so the order of the sums is important. Then, using the expression of the function \(\Gamma\), we arrive at $$I=\int_0^\infty\Bigl(\sum_{n=0}^\infty\frac{(-1)^n}{2^{2n}}\Bigl({2n\atop n}\Bigr)\frac{t^{\frac{n-1}{2}}}{\Gamma(\frac{n+1}{2})}\Bigr)\Bigl(\sum_{k=0}^\infty \frac{(-1)^k}{k!}2^{2k}\frac{(2k)!(2k)!}{(4k+1)!}\frac{4^kt^k}{3^k}\Bigr)e^{-t}\,dt,$$ whose integrand is not algebraic. Next we express each factor as an integral, to obtain a triple integral. $$I= \frac{1}{2\pi i}\int_0^\infty\int_H\int_0^1\frac{e^{tz}}{\sqrt{z+\sqrt{z}}} e^{-\frac{16}{3}u^2(1-u)^2t}e^{-t}\,du\,dz\,dt.$$ The integral on \(z\) along a \(H\) contour of Hankel type. Cauchy’s theorem of complex variable and Fubini’s theorem (both allowed by the Kontsevich-Zagier rules, but here with non-algebraic integrands) lead us to the integral we said.

All these transformations should be replaced by transformations that only handle algebraic functions!

Why am I surprised by this conjecture? I am used to calculating definite integrals and the four rules of Kontsevich and Zagier seem to me to forget other very important ones, in particular the series developments. And the restriction to use only algebraic functions seems to me too demanding. 

I must not leave the history of our integral here. Shortly after calculating the integral as the sum of two elliptic integrals, Peter Blaschke noted that

$$I_0:=\int_0^\infty\frac{dx}{(1+x^2)^{3/2}\sqrt{\varphi(x)+\varphi(x)^{3/2}}}=\frac{\pi}{2\sqrt{6}}.$$ This was the integral that should have appeared in the GR tables. We have all seen those old badly printed books where an exponent \(3/2\) is not properly distinguished from \(1/2\).

Ayoub’s work

The most important results in the direction of the main Kontsevich-Zagier conjecture are to be found in an article by Joseph Ayoub. First Ayoub writes the Kontsevich-Zagier conjecture in a surprising form, which I will now describe.

Let  \(\mathcal O\) be the algebra of functions \(f(z_1,\dots, z_n)\) analytic in \(\mathbb{D}^n=\{(z_1,\dots, z_n)\colon |z_j|\le 1\}\). Here \(n=0\), \(1\), \(2\),… is arbitrary. Of course a function of \(0\) variables is a constant and we can always consider a function \(f(z_1,\dots, z_n)\) as a function with \(m\) variables (with \(m>n\)) which simply does not depend on the last variables.

Now let \(\mathcal O_\text{alg}\) be the subalgebra formed by the functions of \(\mathcal O\) which are algebraic, i.e. satisfy a polynomial equation with polynomial coefficients \(p(z_1,\dots, z_n)\) with rational coefficients. 

We now consider an operator $$I\colon\mathcal O_\text{alg}\to \mathbf{C}$$ defined by transforming \(f=f(z_1,\dots, z_n)\) in $$I(f)=\int_{[0,1]^n}f(x_1,\dots,x_n)\,dx_1\,\cdots\,dx_n.$$ By the fundamental theorem of calculus this operator cancels out for functions of the form $$\frac{\partial}{\partial z_i}f(z_1,\cdots, z_i,\cdots,z_n)-f(z_1,\cdots, 1,\cdots,z_n)+f(z_1,\cdots, 0,\cdots,z_n).\qquad (1)$$

Ayoub’s form of the Kontsevich-Zagier Conjecture. If \(I(F)=0\), then \(f\) belongs to the vector space generated by functions of the form (1), with \(f\in\mathcal O_{\text{alg}}\).

That is, expressed in this form, the Kontsevich-Zagier conjecture needs no more than the fundamental theorem of calculus. Ayoub’s work focuses on proving a result analogous to this but which is no longer the conjecture. Ayoub states that the proof of the equivalence between the Kontsevich-Zagier conjecture and the previous statement is not easy and does not detail it. The difficulty of the new conjecture is again that we only allow algebraic \(f\) in (1). Ayoub shows a simple proof of the conjecture if we allow any functions. 

Proof with analytic functions: Suppose \(I(f)=0\) and \(f\) only depends on one variable, then we know that \(f\) admits a primitive on \(\mathbb D\) defined by $$F(z)=\int_0^z f(t)\,dt.$$ We have \(F(0)=0\) and by hypothesis \(F(1)=I(f)=0\) Therefore $$f(z)=\frac{\partial}{\partial z}F(z)-F(1)+F(0)$$ is a linear combination of functions as in (1).

By induction, if \(f\) depends on \(n\) variables with \(I(f)=0\), then we define $$g(z_1,\dots, z_n)=\int_0^{z_n}f(z_1,\dots, z_{n-1},t)\,dt$$ and we will have that  $$h=f(z_1,\dots, z_n)-\Bigl(\frac{\partial}{\partial z_n}g(z_1,\dots, z_n)-g(z_1,\dots, 1)+g(z_1,\dots, 0)\Bigr)$$ also satisfies \(I(h)=0\) (check) and only depends on \(n-1\) variables (check). By the induction hypothesis \(h\) will be a linear combination of functions of type (1). That same expression proves that \(f\) is combination of functions of type (1). ⬛

But of course the difficulty is that the primitives of algebraic functions are not algebraic. It is certain that if \(I(f)=0\) depends on \(n\) variables, it will be necessary to make use of functions of more variables to use only algebraic functions.

Para Saber más

Kontsevich and Zagier’s presentation of the periods is outstanding:

M. Kontsevich y D. Zagier, Periodos, Mathematics Unlimited – 2001 and beyond, Springer-Verlag, New York 2001, pp. 771–808.

Ayoub’s paper is highly technical, but the introduction, as in almost all the articles published in Annals, is very accessible:

J. Ayoub, Une version relative de la conjecture des périodes de Kontsevich-Zagier, Annals of Mathematics, 181 (2015) 915-992.

There are two brief notes on the concept of period,

S. Müller-Stach, What is… a period, Notices of the Amer. Math, Soc, 61 (2014) 898-899.

M. Waldschmidt,  Raconte-Moi… Une periode, Societé Mathématique de France.

Yoshinaga’s work with the construction of a computable but non-period number has been on the arXiv since 2008. I have found that it is often cited. It is somewhat surprising that it is not published yet, but it looks good and demonstrates a property of computability common to all periods:

M. Yoshinaga, Periods and elementary real numbers, arXiv 0805:0349.

My paper on the integral:

J. Arias de Reyna, True value of an integral in Gradshteyn and Ryzhik’s table, The Ramanujan Journal, 50 (2019) 551-571.

Periods are an active research topic related to the theory of motives. More information at:

J. Ayoub, Periods and the conjectures of Grothendieck and Kontsevich-Zagier.