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- W55282006 abstract "AbstractIn view of the central symmetry of the set of Riemann zeros, (rho leftrightarrow 1 - rho) which crucially reflects the functional equation (3.26), there is no clear reason to pick either the ρ themselves (with (rho equiv frac{1}{2} pm itau_{k}, {rm Re} tau_{k} > 0)) or the (rho(1 - rho) = frac{1}{4} + tau_{k}^{2}) as the basic set upon which to build zeta functions over the Riemann zeros. This is a nonlinear remapping freedom as discussed in Sect. 1.1, which leads to generalized zeta functions of several kinds. This chapter gives an introductory overview of the various possibilities. In decreasing order of analytical tractability, and in our older notation to be soon discarded, we may define: Functions of first kind: (sum_{rho}(x - rho)^{-s}, qquad {rm Re} s > 1;) Functions of second kind: (sumlimits_{k = 1}^{infty}(tau_{k}^{2} + v)^{-sigma},qquad mathfrak{R}sigma > frac{1}{2};) Functions of third kind: (sumlimits_{k = 1}^{infty}(tau_{k}^{2} + tau)^{-s},qquad {rm Re} s > 1.) In all three kinds the argument (in exponent) is the principal variable, the other one a shift parameter. Each parametric family starts as an analytic function of its argument in its indicated half-plane of convergence, due to (4.7) or (4.26), but it will extend to a meromorphic function in the entire complex plane (of its argument), with a computable singular part. The domain of each shift parameter is left as large as possible: simply, none of the quantities to be raised to a power should lie on the cut (mathbb{R}_{-}); this will hardly matter with the fairly small parameter values that we will emphasize.KeywordsMeromorphic FunctionZeta FunctionCentral SymmetryShift ParameterRiemann HypothesisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
- W55282006 created "2016-06-24" @default.
- W55282006 creator A5064940143 @default.
- W55282006 date "2009-10-24" @default.
- W55282006 modified "2023-09-27" @default.
- W55282006 title "Superzeta Functions: an Overview" @default.
- W55282006 doi "https://doi.org/10.1007/978-3-642-05203-3_5" @default.
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