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....

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...

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara’s quantity A(`), for ` = p with p prime and n > 3 odd. A modular interpretation of the towers is given as well.

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently...

The alpha-like globin gene cluster in rabbits contains embryonic zeta-globin genes, an adult alpha-globin gene, and theta-globin genes of undetermined function. The basic arrangement of genes, deduced from analysis of cloned DNA fragments, is 5'-zeta 0-zeta 1-alpha 1-theta 1-zeta 2-zeta 3-theta 2-3'. However, the pattern of restriction fragments containing zeta- and theta-globin genes varies am...

In this empirical study, results show that recently developed software pitch-trackers can be as fast as some current commercial hardware pitch-trackers. Since latency was the most crucial factor in the performability of the ZETA system, comparative analyses were made of the ZETA and the AXON interfaces, as well as the fiddle~ object, a real-time audio analysis tool in the Opcode’s MAX/MSP envir...

We generalize the theory of p-adic geometric zeta functions of Y. Ihara and K. Hashimoto to the higher rank case. We give the proof of rationality of the zeta function and the connection of the divisor to group cohomology, i.e. the p-adic analogue of the Patterson conjecture. Introduction. In [13] and [14] Y. Ihara defined geometric zeta functions for the group PSL2. This was the p-adic counter...

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the KodairaNéron classification of the special fibers of elliptic curves as determined by Tate’s algorithm.