Patrick Moss
25/10/1947–17/3/2024

Pat looked at the $p$-part of a sequence for each prime $p$, giving rise to the notion of ‘local’ realizability: A sequence is realizable locally at a prime $p$ if the sequence of $p$-parts is itself a realizable sequence. It is easy to see that a sequence that is locally realizable at every prime is realizable, but Pat showed that the converse also holds if the sequence is realized by an endomorphism of a locally nilpotent group (‘nilpotently realizable’).

This raised the question of understanding the property of being realized by a group automorphism more generally. If $a$ is realized by a group automorphism then in addition to the purely combinatorial congruence conditions it must be a divisibility sequence. Pat quickly found simple examples to show that far more is needed: Indeed, the sequence

which is realized by the permutation $(12345)(6)$ on the set $\{1,2,\dots,6\}$ is both a divisibility sequence and a linear recurrence sequence, but it cannot be realized by a group automorphism: There is no group of order $6$ with an automorphism that cycles its non-identity elements. What is clear about the sequence (1) is that both its $2$-part and its $3$-part are not realizable. Pat’s result [16, Th. 3.2.11] that nilpotent realizability (realizability by an automorphism of a nilpotent group) is equivalent to everywhere locally nilpotent realizability gives an explanation for this example, but the problem of characterising algebraic realizability combinatorially remains open, even for the restricted class of linear recurrent divisibility sequences. These have been classified by Bézivin, Pethő, and van der Poorten [2] but which of them are algebraically realizable is unclear.

Pat developed an extraordinary insight into the congruence conditions (D) for realizability, and this contributed greatly to a series of conversations between Graham Everest, Pat, and me about how sampling along a subsequence of times might preserve or not preserve the realizability property. Pat quickly established a fundamental ‘powers’ result: If $(a_{n})$ is realizable then $(a_{n^{k}})$ is realizable for any $k\geqslant 1$ [16, Th. 2.2.2]. This raised the question of understanding what kind of time-changing maps $\mathbb{N}\to\mathbb{N}$ in place of the map $n\mapsto n^{k}$ also have this property, but it was some years before any of us were able to return to it.

In 2018 Sawian Jaidee visited me in Leeds, and we realised that no polynomials apart from the monomials already identified could have this realizability along subsequences property. We contacted Pat, and pieced together the following way of thinking about these ‘time changes’ that preserve realizability—or, equivalently, act as symmetries of the space of dynamical zeta functions.

Pat had shown that the map $n\mapsto n^{k}$ lies in $\mathscr{P}$ for any $k\in\mathbb{N}$, and our argument complemented this. We also used the construction of certain families of non-polynomial time changes to show that the monoid $\mathscr{P}$ has the following properties (see [8]):

• •

  the only polynomials in $\mathscr{P}$ are the monomials, and

• •

  the monoid $\mathscr{P}$ is nonetheless uncountable.

The first result says that algebraically $\mathscr{P}$ is very small; the second that when viewed as a set $\mathscr{P}$ is very large.

This circle of questions is still actively studied, and recent work includes a complete description of a set of (topological) generators for $\mathscr{P}$ by Jaidee, Byszewski, and the author [7] and a geometric-combinatorial construction of the map realizing $(a_{n^{2}})$ by Grzegorz Graff and Jacek Gulgowski (in progress).

Pat also showed in [16, Cor. 2.1.6] that the terms of a realizable sequence may themselves be raised to polynomial powers in the following sense: If $h\in\mathbb{N}[x]$ and $(a_{n})$ is realizable, then $(a_{n}^{h(n)})$ is also realizable. This is unexpected, and its full ramifications have not been explored further. What is expected is that polynomials are essentially the only functions with this property.

Yash Puri and I had started our work by noticing that the classical Fibonacci sequence

is not realizable. After the work on the monoid $\mathscr{P}$ with Sawian Jaidee, Pat and I continued to correspond and he pointed out that the sequence $(5F_{n^{2}})$ is realizable. This was an extraordinary thing to notice, particularly as Pat did not use computers to test ideas numerically, and of course it is a very rapidly growing sequence. The observation once again proved to be a productive one. First, this led naturally to the notion of almost realizability: A sequence $(a_{n})$ is almost realizable if there is a constant $C$ such that $(Ca_{n})$ is realizable. This notion became important later in work of Miska [15] on the Stirling numbers.

The multiplier needed for a given $k\geqslant 1$ in this result is somewhat mysterious, as it is a priori impossible to compute. It is by definition the least common multiple of the denominators appearing in an infinite sequence of rational numbers of the form $\frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)S^{(2)}(d+k-1,k)$. If in calculating the first few terms (or indeed the first few thousand terms) this least common multiple reaches $(k-1)!$ then this certainly is the minimal multiplier needed. If—as is usually the case—this does not happen, then it is not clear at any stage if the minimal value has really been computed. While some suggestions and speculations were included in [15], there is at this stage very little knowledge about the general term of the sequence of multipliers $(C_{k})$, which begins

Going back to the Fibonacci sequence, Pat and I were able to show that $(F_{n^{k}})$ is almost realizable if $k$ is even, and is not almost realizable if $k$ is odd. Moreover, the constant needed for any even power is $5$, the discriminant of the Fibonacci sequence [17].

The second natural direction of travel starting with Pat’s result about the Fibonacci sequence sampled along the squares is to ask how prevalent this phenomenon is, and this has more or less been worked out in joint work with Florian Luca [13]. The details of this result are cumbersome, but it is in essence a direct strengthening of Pat’s result on the Fibonacci sequence, and the proof uses Binet’s formula for the terms of a linear recurrence sequence directly to replace the detailed knowledge of congruence properties for the Fibonacci sequence.

Pat also showed realizability for several sequences arising in arithmetic, two of which seemed particularly interesting. The ‘Euler numbers’ $(E_{n})$ are defined by the formal relation

By extending the classical Kummer congruences for the Euler numbers, Pat was able to show the following, recovering a result of Juan Arias de Reyna [1].

The ‘Bernoulli numbers’ $(B_{n})$ are similarly defined by the formal relation

By making ingenious use of classical congruences due to Adams, von Staudt-Clausen, and Kummer, Pat was able to show something quite unexpected. Write

with $\gcd(\tau_{n},\beta_{n})=1$ and $\tau_{n},\beta_{n}\geqslant 1$ for $n\geqslant 1$.

Pat also showed that the numerator sequence is realizable (but cannot be algebraically realizable).

Indeed, he was able to show precisely how the failure of algebraic realizability occurs. Recall that a prime $p$ is called irregular if it divides the class number of the $p$th cyclotomic field $\mathbb{Q}(\zeta_{p})$ where $\zeta_{p}$ denotes a primitive $p$th root of unity.