Statistical independence in mathematics–the key to a Gaussian law

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

Lacunarity of Han–Nekrasov–Okounkov q-Series

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms $$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$ F a , b , c ( z ) : = η ( 24 a z ) a η ( 24 a c z ) b - a η ( 24 z ) , defined in terms of the Dedekind $$\eta $$ η -function, for integers $$a,c \ge 1$$ a , c ≥ 1 , where $$b \ge 1$$ b ≥ 1 is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when $$a = c = 1$$ a = c = 1 . Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the $$F_{a,b,1}(z)$$ F a , b , 1 ( z ) which are lacunary. Here, we consider all c and show that for $${a \in \{1,2,3\}}$$ a ∈ { 1 , 2 , 3 } , there are infinite families of lacunary series. However, for $$a \ge 4$$ a ≥ 4 , we show that there are finitely many triples (a, b, c) such that $$F_{a,b,c}(z)$$ F a , b , c ( z ) is lacunary. In particular, if $$a \ge 4$$ a ≥ 4 , $$b \ge 7$$ b ≥ 7 , and $$c \ge 2$$ c ≥ 2 , then $$F_{a,b,c}(z)$$ F a , b , c ( z ) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).

Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions

This work builds upon previous studies of centered polygonal lacunary functions by presenting proofs of theorems showing how rotational and dihedral mirror symmetry manifest in these lacunary functions at the modulus level. These theorems then provide a general framework for constructing other lacunary functions that exhibit the same symmetries. These investigations enable one to better explore the effects of the gap behavior on the qualitative features of the associated lacunary functions. Further, two renormalized products of centered polygonal lacunary functions are defined and a connection to Ramanunjan’s triangular lacunary series is made via several theorems.

On Domains of Convergence of Multidimensional Lacunary Series

This paper is devoted to multidimensional analogues of the Fabry and P˙olya theorems on lacunary series. Domains of convergence of lacunary Hartogs series and series in homogeneous polynomials are studied in this paper. Analogues of the Fabry and P˙olya theorems for such series are given and domains of convergence of these series are described. Results of the work develop well-known result of J. Siciak on the domain of convergence of the lacunary series with respect to homogeneous polynomials