into the World of Siegel's Continued Fractions
Continued Fractions: A Modern and Classical Journey into the World of Siegel's Continued Fractions bridges nineteenth and early twentieth-century classical results with entirely modern, transducer-based approaches. The book offers a unified treatment through the lens of Siegel's theory, presenting both foundational theory and cutting-edge research in a single cohesive volume.
The text is accessible to advanced undergraduates and graduate students in mathematics, while simultaneously serving as a research reference for specialists in number theory, Diophantine approximation, and theoretical computer science. No prior knowledge of continued fractions is assumed — the necessary background is developed carefully from first principles.
A distinctive feature is the treatment of the Zopf continued fraction [1; 2, 3, …] — a transcendental number whose continued fraction expansion exhibits remarkable algebraic properties. The authors develop its quadratic convergents from scratch and connect them to classical Siegel-type results via transducer machinery.
The book is divided into three main parts. The first establishes foundational theory: vector spaces of continued fractions, convergent sequences, and the Zopf framework. The second develops modern transducer-based machinery and its applications to Diophantine problems. The third presents advanced topics including gcd sequences, spectral properties, and connections to automata theory and cryptography.
Throughout, the authors take unusual care to motivate every definition and to connect classical tools with their modern counterparts. Proofs are given in full, and a rich collection of exercises accompanies each chapter.
Published by Springer / Birkhäuser (2025) and available via Springer. An errata sheet compiled by the authors is available on the .
Christopher Robin Havens & Carsten Elsner · March 2026
An interview exploring Carsten Elsner's four-decade pursuit of the irrationality of Euler's constant \(\gamma\), culminating in a single open conjecture about lattice basis reduction.
Euler's constant is defined by \[\gamma := \lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{1}{k} - \log n\right)\] and its regular continued fraction expansion begins \[\gamma = [0,1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,3\ldots]\,.\] Unlike \(e\) and \(\pi\), it is still unknown whether \(\gamma\) is rational or irrational — one of the most celebrated open problems in mathematics.
The beginning of this particular story takes place in Germany, over four decades ago — the year 1984, when Germany was a divided country. But behind the walls of the University of Hannover, Carsten Elsner was attending his very first lecture in number theory by Professor Rieger.
Carsten was curious and very pensive when it came to mathematical problems. While Prof. Rieger lectured on Euler's constant \(\gamma\) in the context of the asymptotic expansion of the harmonic series, Carsten was drawn towards its mysteries in the same way I had been drawn into the mysteries of the Zopf number \(\mathfrak{z} := [1,2,3,\ldots]\).
Carsten, can you tell me about some of the discussions with your professors on this topic and how they reacted to your goals of solving this problem?
In 1984, I was a student focused solely on studying — not research, let alone tackling problems unsolved for centuries. Nevertheless, these problems were naturally fascinating. Only a few years had passed since Roger Apéry proved the irrationality of \(\zeta(3)\) in 1978. My mentor, Prof. Rieger, had discussed Apéry's proof with him personally and drew my attention to Alf van der Poorten's article "A proof that Euler missed" in The Mathematical Intelligencer (1979).
In seminars, I met Paul Erdős, who encouraged me to prove the irrationality of \(\sum_{n=0}^{\infty} \frac{1}{n!+1}\), and also Alan Baker in person. But I did not dare discuss a possible approach to proving the irrationality of \(\gamma\) with either of them — at the time, I felt they were way above my level as a student.
Later, shortly before completing my studies in 1987, I studied Roth's proof of the approximation of real-algebraic numbers by rationals. My mentor then handed me a copy of Paul Appell's 1926 paper Sur la nature arithmétique de la constante d'Euler, saying: "Perhaps you can correct the proof!" Unfortunately, Appell's proof contained a fundamental flaw, as he himself noted a few days later.
Professor Rieger seemed supportive — enough to challenge you to correct the erroneous proof of Paul Appell. What amazes me is that Appell's proof had been read by mathematicians for decades before you saw it. The fact that \(\gamma\) is still not proven rational or irrational, even after Appell came so close, is a testament to its difficulty.
At this point, we have progressed another 4 decades from the time Carsten began pursuing this problem, and he has whittled it down to a single inequality. Carsten, can you describe a timeline of any critical milestones in your progress?
I was never euphoric in my belief that I would reach the goal in the near future. However, it was frustrating when an approach failed, and motivating when I reached a milestone. Three such subgoals stand out:
Now there is still one missing piece: the proof of an inequality between the largest and smallest successive minima of a lattice. Numerical examples for the outstanding conjecture are encouraging, but they do not meet the required condition of very large parameter values, which would exceed the computational capabilities of computers. Another question also bothers me: no one has reviewed my work so far. Does it perhaps already contain fundamental errors that cannot be corrected? This uncertainty is indeed an emotional burden.
Well, you have put 40 years of thought and work into this problem, and it sounds like fresh minds would be extremely useful. Today we make a "call to arms" — a challenge to any mathematician out there.
Carsten has made incredible progress. He has boiled the question of the irrationality of \(\gamma\) down to a single inequality. The full mathematical paper, "Euler's Constant and Series Transformations with Bases of Balanced Lattices," is available for download above. In it, Carsten breaks the whole problem down and commences a near-complete proof — pending one remaining lemma (currently a conjecture) about a lattice inequality.
Your mission, should you choose to accept it:
If you can contribute meaningfully as a collaborator or reviewer, email Carsten directly at carsten.elsner73@gmail.com.
Carsten has formulated the missing piece precisely. The full statement is Lemma 13 (Third Fundamental Lemma; Conjecture) in the paper. In brief:
For infinitely many (very large) natural numbers \(N\), consider the \((N+1)\)-dimensional lattice \(\Lambda_N\) in \((N+2)\)-dimensional Euclidean space, defined as the solution lattice of a specific linear Diophantine equation. Let \(\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_{N+1}\) be the successive minima of \(\Lambda_N\) with respect to the maximum norm.
Minkowski's second fundamental theorem gives \[\lambda_1\lambda_2\cdots\lambda_{N+1} \leq d(\Lambda_N).\] The conjecture asserts that for the specific lattices arising in Elsner's construction, the basis vectors satisfy the additional inequality \[\sqrt{\lambda_{N+1}} < \lambda_1,\] meaning that the lengths of the basis vectors do not differ too much from one another in the maximum norm.
If this conjecture is true, then Euler's constant \(\gamma\) is irrational. Numerical verification (using the LLL algorithm) for small values \(N = 25, 31, 38\) confirms the inequality in all three cases; see Section 4.5 of the paper.
Carsten adds: "I am familiar with many analytical properties of my lattices, but I suspect that to prove the inequality \(\sqrt{\lambda_{N+1}} < \lambda_1\), additional arithmetic properties of the Diophantine equation generating the lattices must be taken into account. I am not a lattice expert, and at this point I am completely at a loss."
In my lifetime, significant conjectures have been proven: the irrationality of \(\zeta(3)\), the Bieberbach conjecture, Fermat's Last Theorem, the Poincaré conjecture. These have been magnificent mathematical milestones. If someone were to join me in completing the proof of the irrationality of Euler's constant, it would be a spectacular conclusion of my own mathematical career.
In 1997, I had a monument erected at the site of Ferdinand von Lindemann's birthplace in Hannover to remember his magnificent contribution to proving the transcendence of \(\pi\). What matters is not so much who achieves a significant result, but that we achieve it — in accordance with Hilbert's statement:
We must know — we will know.
Compiled by Christopher Robin Havens & Carsten Elsner. Only corrections considered relevant to readers are listed.
The first part of the proof is incorrect: the series mentioned do not exist. The beginning of the proof should be replaced as follows.
The last two equations on page 62 are incorrect. On page 63 in the same definition, the assignment direction is reversed.
An incorrect symbol appears in the conjecture statement.
The matrix in the proof has an incorrect top-right entry.
There is a missing \(p_k\) term in the second line after "from (3.13) to obtain".
The QED box (□) appears in the wrong location and should follow the final displayed equation ending with \(\dfrac{p_n \xi_{n+1} + p_{n-1}}{q_n \xi_{n+1} + q_{n-1}}\). The orphaned □ after equation (3.17) on the same page should be removed.
The predicate notation must be updated in both the surrounding text and the sum inside equation (3.73).
An incorrect symbol appears at the bottom of page 234 and continues into the same computation on page 235.
A missing subsubsection number for the section On the Approximation of z by Quadratic Convergents (p. 130) causes the title to be absent from the table of contents.
A section number is absent. The text "given in Sect. can be rewritten in" should reference Section 3.1.1.
The symbol \((\omega, \vartheta)\) should appear in the column headers below \(Q_1\) on pages 295 and 296.
Three unnecessary spaces appear after Umlaut characters in entries 21 and 22 of the bibliography. These should be removed.