Abscissa Of Convergence Research Articles

Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Abstract Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $ , we prove that any $\Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ . More generally, we prove that for a Zariski dense $\theta $ -transverse subgroup $\Gamma <G$ , any $(\Gamma , \psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ , provided the $\psi $ -Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

General multiple Dirichlet series from perverse sheaves

We give an axiomatic characterization of multiple Dirichlet series over the function field Fq(T), generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζG(k)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G(k)^{-1}$$\\end{document} at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}. We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} groups, where C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} is a class of finite groups with prescribed composition factors. We prove that every real number a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\ge 1$$\\end{document} is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} are rational functions in p-s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{-s}$$\\end{document} if G is virtually abelian. For finite groups G we calculate ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} using the rational representation theory of G.

A universal lower bound for certain quadratic integrals of automorphic L–functions

Let π be a cuspidal unitary representation od GL(m,A) where A denotes the ring of adèles of Q. Let L(s,π) be its L-function. We introduce a universal lower bound for the integral ∫−∞+∞|L(12+it,π)12+it−s|2dt where s is equal to 0 or is a zero of L(s) on the critical line. In the main text, the proof is given for m≤2 and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series L(s,π). In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.

Random Dirichlet series with [formula omitted]-stable coefficients

Let P be a set of non-negative real numbers, we consider a class of random Dirichlet series D(s)=∑p∈Pξpps with symmetric α-stable (0<α⩽2) coefficients. Real zero point problem of D is studied firstly. We prove that, if Z(τz)<∞, then with a positive probability there are no real zero points in the interval [τz/α,∞), while if Z(τz)=∞, for any ɛ>0, almost surely D has infinite number of real zeros in the interval s=(τz/α,τz/α+ɛ). Here, Z(s)=∑p∈P1ps and τz is its abscissa of convergence (see Section 1). In the proof, we find that lim sups→1/α+D(s)=∞ almost surely under the condition Z(τz)=∞. Finally, to get more asymptotic information for D(s), under P=N, some asymptotic properties for D(s) compared with Z(αs) are explored as s→1/α+.

On the value distribution of general Dirichlet series

On the value distribution of general Dirichlet series

On the domain of convergence of general Dirichlet series with complex exponents

Let $(\lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=\sum\limits_{n=0}^{+\infty} a_ne^{z\lambda_n}$, $z\in\mathbb{C}$, we denote $G_{\mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $\mu(z,F)=\max\big\{|a_n|e^{\Re(z\lambda_n)} : n\geq 0\big\}$, respectively. It is well known that $G_\mu(F), G_a(F)$ are convex domains.&#x0D; Let us denote $\mathcal{N}_1(z):=\{n : \Re(z\lambda_n)&gt;0\}$, $\mathcal{N}_2(z):=\{n : \Re(z\lambda_n)&lt;0\}$ and \[\alpha^{(1)}(\theta) :=\varliminf\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_1(e^{i\theta})}}\frac{-\ln|a_n|}{\Re(e^{i\theta}\lambda_n)},\qquad \alpha^{(2)}(\theta) :=\varlimsup\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_2(e^{i\theta})}}\frac{-\ln|a_n|}{\Re(e^{i\theta}\lambda_n)}.\] Assume that $a_n\to 0$ as $n\to +\infty$. In the article, we prove the following statements.&#x0D; $1)$ If $\alpha^{(2)}(\theta)&lt;\alpha^{(1)}(\theta)$ for some $\theta\in [0,\pi)$ then \[\big\{te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\big\}\subset G_\mu(F)\] as well as \[\big\{te^{i\theta} : t\in (-\infty,\alpha^{(2)}(\theta))\cup (\alpha^{(1)}(\theta),+\infty)\big\}\cap G_\mu(F)=\emptyset.\]&#x0D; $2)$ $G_\mu(F)=\bigcup\limits_{\theta\in [0,\pi)}\{z=te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\}.$&#x0D; $3)$ If $h:=\varliminf\limits_{n\to +\infty}\frac{-\ln |a_n|}{\ln n}\in (1,+\infty)$, then \[\Big(\frac{h}{h-1}\cdot G_a(F)\Big)\supset G_\mu(F)\supset G_c(F).\] If $h=+\infty$ then $G_a(F)=G_c(F)=G_\mu(F)$, therefore $G_c(F)$ is also a convex domain.

A note on Bohr’s theorem for Beurling integer systems

Given a sequence of frequencies {λn}n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\lambda _n\\}_{n\\ge 1}$$\\end{document}, a corresponding generalized Dirichlet series is of the form f(s)=∑n≥1ane-λns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(s)=\\sum _{n\\ge 1}a_ne^{-\\lambda _ns}$$\\end{document}. We are interested in multiplicatively generated systems, where each number eλn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{\\lambda _n}$$\\end{document} arises as a finite product of some given numbers {qn}n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{q_n\\}_{n\\ge 1}$$\\end{document}, 1<qn→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < q_n \\rightarrow \\infty $$\\end{document}, referred to as Beurling primes. In the classical case, where λn=logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _n = \\log n$$\\end{document}, Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {ℜs>θ}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Re s> \ heta \\}$$\\end{document}, then it actually converges uniformly in every half-plane {ℜs>θ+ε}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Re s> \ heta +\\varepsilon \\}$$\\end{document}, ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document}. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr’s condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr’s theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.

Composition operators and generalized primes

We study composition operators on the Hardy space H 2 \mathcal {H}^2 of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on H 2 \mathcal {H}^2 . To do that we extend our notions to a Hardy space H Λ 2 \mathcal {H}_{\Lambda }^2 of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.

Quasiperiodic Sets at Infinity and Meromorphic Extensions of Their Fractal Zeta Functions

We construct an interesting family of unbounded sets at infinity having a quasiperiodic structure in their geometry. We study these sets by analyzing their complex dimensions—a generalization of the classical Minkowski dimension, which are defined analytically—as poles of a suitably defined Lapidus zeta function at infinity introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation that relates it to the Lapidus zeta function at infinity much as in the classical setting of relative fractal drums. Furthermore, under suitable assumptions, we provide some general results about extending the fractal zeta functions at infinity beyond their abscissae of convergence. We also provide a sufficiency condition for Minkowski measurability as well as a bound from above for the upper Minkowski content directly from the corresponding zeta functions at infinity. We show that complex dimensions of quasiperiodic sets at infinity possess a quasiperiodic structure which can be either algebraic or transcendental. Furthermore, we use the quasiperiodic sets at infinity to construct a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set for which the abscissa of convergence of the associated fractal zeta function at infinity becomes its natural boundary.

Scaling a theorem of Harald Bohr

A outstanding result of H. Bohr shows that under a certain condition on the frequency λ=(λn) (preventing the λn's from getting too close too fast) every λ-Dirichlet series D=∑ane−λns converges uniformly on all half-planes [Re>σ] with σ>0, provided D is pointwise convergent on some half-plane and has a limit function extending to a bounded holomorphic function f on [Re>0]. Riesz summability plays an important role within the classical literature of general Dirichlet series - recall that a Dirichlet series D=∑ane−λns is said to be Riesz summable of order ℓ≥0 in s∈C, whenever the sequence of all Riesz means ∑λn<xan(1−λnx)ℓe−λns converges as x tends to infinity. Extending classical work of M. Riesz, we for a given ℓ≥0 isolate and study the class of those frequencies λ, which have the following property: If the Dirichlet series D=∑ane−λns is pointwise Riesz summable for some order on some half-plane and has a limit function extending to a holomorphic function f on [Re>0] with growth of order O((1+s)ℓ), then uniformly on all half-planes [Re>σ],σ>0, we have f(s)(1+s)ℓ=1(1+s)ℓlimx→∞⁡∑λn<xan(1−λnx)ℓe−λns. Among others, we show that this holds true if λ satisfies a well-known condition isolated by E. Landau, which recovers the classical bounded case ℓ=0.

Universality of general Dirichlet series with respect to translations and rearrangements

We give sufficient conditions for a general Dirichlet series to be universal with respect to translations or rearrangements.

On the degree of polynomial subgroup growth of nilpotent groups

Let N be a finitely generated nilpotent group. The subgroup zeta function \(\zeta _N^{\scriptscriptstyle \leqslant }(s)\) and the normal zeta function \(\zeta _N^{\scriptscriptstyle \lhd }(s)\) of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence \(\alpha _N^{\scriptscriptstyle \leqslant }\) and \(\alpha _N^{\scriptscriptstyle \lhd }\), also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions \(N\mapsto \alpha _N^{\scriptscriptstyle \leqslant }\) and \(N\mapsto \alpha _N^{\scriptscriptstyle \lhd }\) when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic \(\mathbb {C}\)-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields.

Fourier Interpolation with Zeros of Zeta and L-Functions

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other L-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp’s abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

A Jentzsch-Theorem for Kapteyn, Neumann and General Dirichlet Series

Comparing phase plots of truncated series solutions of Kepler’s equation by Lagrange’s power series with those by Bessel’s Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch’s classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

The Dirichlet-type Laplace transforms

We show that it is possible to define extensions of the Laplace transform that use a general Dirichlet series as a kernel. These transforms, denoted by DLTs, further generalize those, considered in previous papers, in which the kernels were related to Laguerre-type exponentials or Bell polynomials. Computational techniques, exploiting expansions in Laguerre polynomials, and using Tricomi's method, have been considered. Since it turns out that the transforms considered are obtained as linear combinations of ordinary Laplace transforms, it is also possible to define an approximation of the relevant inverse transforms. Numerical experiments, performed with the algebra program Mathematica, show that the introduced technique is fast and efficient.

An extension of Maass theory to general Dirichlet series

In this paper we establish the modular relation for Maass forms to the effect that the Fourier–Whittaker expansion and the ramified functional equation are equivalent, i.e. the RHB (Riemann–Hecke–Bochner) correspondence. This arises from the new standpoint that Maass' procedure is one of the ways leading to the Fourier–Whittaker expansion similar to the situation that various representations of the modified Bessel function have provided such vast amount of different methods including the Hardy transform and the beta-transform leading to the Fourier-Bessel expansion from opposite directions. Naturally, we may abridge the gap between Hecke Eisenstein series and Epstein zeta-functions and the Chowla–Selberg integral formula and Ramanujan–Guinand formula are immediate corollaries, whose remark is due to the referee.

Holomorphic functions of finite order generated by Dirichlet series

Given a frequency \(\lambda = (\lambda _n)\) and \(\ell \ge 0\), we introduce the scale of Banach spaces \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) of holomorphic functions f on the open right half-plane \([{{\,\mathrm{Re}\,}}> 0]\), which satisfy (A) the growth condition \(|f(s)| = O((1 + |s|)^\ell )\), and (B) have a \(\lambda\)-Riesz germ, i.e. on some open subset and for some \(m \ge 0\) the function f coincides with the pointwise limit (as \(x \rightarrow \infty\)) of the so-called \((\lambda ,m)\)-Riesz means $$\begin{aligned} \sum_{\lambda _n < x} a_n e^{-\lambda _n s}\left( 1-\frac{\lambda _n}{x}\right) ^m ,\,x >0 \end{aligned}$$of some \(\lambda\)-Dirichlet series \(\sum a_n e^{-\lambda _n s}\). Reformulated in our terminology, an important result of Riesz shows that in this case the function f for every \(k >\ell\) is the pointwise limit of the \((\lambda ,k)\)-Riesz means of D on \([{{\,\mathrm{Re}\,}}> 0]\). Our main contribution is an extension showing that ’after translation’ every bounded set in \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) is uniformly approximable by all its \((\lambda ,k)\)-Riesz means of order \(k>\ell\). This follows from an appropriate maximal theorem, which turns out to be at the very heart of a seemingly interesting structure theory of the Banach spaces \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\). One of the many consequences is that \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) basically consists of those holomorphic functions on \([{{\,\mathrm{Re}\,}}>0]\), which have a \(\lambda\)-Riesz germ and are of finite uniform order \(\ell\) on \([{{\,\mathrm{Re}\,}}>0]\). To establish all this and more, we need to reorganize (and to improve) various aspects and keystones of the classical theory of Riesz summability of general Dirichlet series as invented by Hardy and M. Riesz.

Representation growth of compact special linear groups of degree two

We study the finite-dimensional continuous complex representations of SL2 over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is 1, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of C[SL2(Z/(22rZ))] and C[SL2(F2[t]/(t2r))] are not isomorphic for any r>1. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of SL2(F2[t]/(t2r)) and SL2(Z/22rZ) for 1≤r≤3.

Regenerative properties of the linear Hawkes process with unbounded memory

Regenerative properties of the linear Hawkes process are proved under minimal assumptions on the transfer function, which may have unbounded support. For this, an original construction of the Hawkes process as a functional of a Poisson point process is derived from the immigration-birth representation, and the independence properties of the Poisson point process are exploited to exhibit regeneration times which are anticipative and not even measurable w.r.t. the Hawkes process. The regeneration time is interpreted as the renewal time at zero of an M/G/∞ queue, which yields a formula for its Laplace transform. When the transfer function has exponential moments, we stochastically dominate the cluster length by exponential random variables with computable parameters. This provides explicit bounds on the Laplace transform of the regeneration time in terms of simple integrals or of special functions, which yields an explicit negative upper-bound on its abscissa of convergence. The power of the regenerative properties is showcased by being applied to long-time asymptotic results for a class of sliding window statistical estimators, using coupling and sample-path decomposition techniques.