Analytic properties of zeta functions and subgroup growth

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For \alpha a real number and N a nonnegative integer, define s_N^\alpha(G) = sum_{n=1}^N a_n(G)/n^\alpha. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence \alpha(G) of \zeta_G(s) is a rational number and \zeta_G(s) can be meromorphically continued to Re(s)>\alpha(G)-\delta for some \delta >0. The continued function is holomorphic on the line \Re(s) = (\alpha)G except for a pole at s=\alpha(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_{N}(G) ~ c N^{\alpha(G)}(\log N)^{b(G)} s_{N}^{\alpha(G)}(G) ~ c' (\log N)^{b(G)+1} for N\rightarrow \infty .Comment: 41 pages, published version, abstract added in migratio

Non-PORC behaviour of a class of descendant pp-groups

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture

The great unknown: seven journeys to the frontiers of science

Ever since the dawn of civilization we have been driven by a desire to know—to understand the physical world and the laws of nature. The idea that there might be a limit to human knowledge has inspired and challenged scientists and functioned as a spur to innovation. Now, in this dazzling journey through seven great breakthroughs in our understanding of the world, Marcus du Sautoy invites us to consider the outer reaches of human understanding. Are some things beyond the predictive powers of science? Or are those thorny challenges our next breakthroughs? In 1900, Lord Kelvin—who gave the world telegraph cables and the Second Law of Thermodynamics—pronounced that there was “nothing new to be discovered in physics now.” Then came Einstein. Du Sautoy reminds us that again and again major breakthroughs were ridiculed and dismissed at the time of their discovery. He takes us into the minds of the greats and reveals the fraught circumstances of their discoveries. And he carries us on a whirlwind tour of everything from probability to particle physics, grounding his deeply personal exploration in simple concepts like the roll of dice, the notes of a cello, or how a clock measures time. At once exhilarating and accessible, The Great Unknown will challenge you to think in new ways about every aspect of the known world and will give you the tools to understand the riddles our most creative scientists are still struggling to solve

The Music Of the Primes

ill.;335 hal.; 25 c