In semiclassical theories for chaotic systems, such as Gutzwiller’s periodic orbit theory, the energy eigenvalues and resonances are obtained as poles of a non-convergent series g(w) =∑n An exp(isnw). We present a general method for the analytic continuation of such a non-convergent series by harmonic inversion of the ‘time’ signal, which is the Fourier transform of g(w). We demonstrate the gen...

Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression.

Fix a prime number l. In this paper we prove a conjecture [16, p. 300], which Ihara attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. It is stated below. Similar techniques are also used to prove part of a conjecture of Goncharov [11, Conj. 2.1], also about the action of the absolute Gal...

Riemann Zeta Function 602 25.2 Definition and Expansions . . . . . . . . 602 25.3 Graphics . . . . . . . . . . . . . . . . . . 603 25.4 Reflection Formulas . . . . . . . . . . . . 603 25.5 Integral Representations . . . . . . . . . 604 25.6 Integer Arguments . . . . . . . . . . . . 605 25.7 Integrals . . . . . . . . . . . . . . . . . . 606 25.8 Sums . . . . . . . . . . . . . . . . . . . 606 25....

Resistive pulse sensors, RPS, are allowing the transport mechanism of molecules, proteins and even nanoparticles to be characterized as they traverse pores. Previous work using RPS has shown that the size, concentration and zeta potential of the analyte can be measured. Here we use tunable resistive pulse sensing (TRPS) which utilizes a tunable pore to monitor the translocation times of nanopar...

Turing encountered the Riemann zeta function as a student, and developed a life-long fascination with it. Though his research in this area was not a major thrust of his career, he did make a number of pioneering contributions. Most have now been superseded by later work, but one technique that he introduced is still a standard tool in the computational analysis of the zeta and related functions...

An pa, bq-Dyck path P is a lattice path from p0, 0q to pb, aq that stays above the line y “ a b x. The zeta map is a curious rule that maps the set of pa, bq-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of P and zeta of P conjugate is enough to recover P . Our method begets an area-preserv...

Some basic results for Dirichlet series ψ with positive terms via log-convexity properties are pointed out. Applications for Zeta, Lambda and Eta functions are considered. The concavity of the function 1/ψ is explored and, as a main result, it is proved that the function 1/ζ is concave on (1,∞) . As a consequence of this fundamental result it is noted that Zeta at the odd positive integers is b...

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Let G be a profinite group. We denote by rn(G) the number of isomorphism classes of irreducible n-dimensional complex continuous representations of G (so that the kernel is open in G). Following [20], we call rn(G) the representation growth function of G. If G is a finitely generated profinite group, then rn(G) < ∞ for every n if and only if G has the property FAb (that is, H/[H,H] is finite fo...