By H. Gordon Garbedian

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Chapter 6 Explicit Formulae This chapter discusses the approach to the superzeta functions (introduced in the previous chapter) by means of Explicit Formulae. In number theory, Explicit Formulae designate certain summation formulae connecting the primes and the Riemann zeros [55, Chap. IV]. As we may also refer to explicit formulae in their generic sense of closed-form results, we will specifically use capital initials to distinguish that number-theoretical meaning. We focus on just one type of Explicit Formula, the Guinand–Weil form [42, 43, 111], which accommodates general test functions and appears closest to the Poisson summation formula [50].

Then quite lately, the third family Z was proved a meromorphic function of its argument, independently by Hirano, Kurokawa and Wakayama [51, Sect. 3] (as Z(w, x) ≡ Z(w | −x)) and by us [106, Sect. 2]. , over zeros of more general zetaand L-functions than Riemann’s, their even smaller bibliography will be given in Chap. 10, which is devoted to this extension. So, our topic of “superzeta functions” has benefited from some seminal contributions. Yet, these are scarce (a dozen, spread over three families and nearly one century), and almost each of them treats an isolated case under a single aspect: typically meromorphic continuation (often just qualitatively), or particular instances of special values.

2) (plus our basic analytical continuation formula for a zeta function over the Riemann zeros in Chap. 7), require some control over the growth of ζ /ζ in the complex plane. The problematic region is clearly about the critical strip, where the unknown Riemann zeros lie; if Re x > x0 for some x0 > 1 then ζ /ζ will be trivially bounded (and the symmetrical half-plane {Re x < 1 − x0 } is thereupon controlled by applying the Functional Equation to ζ /ζ: we will skip this). 4 Basic Bounds on ζ /ζ 37 Without loss of generality, we may now fix x0 = 2 (the customary choice), and only consider the upper half-plane (by conjugation symmetry).

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Albert Einstein: Maker of Universes by H. Gordon Garbedian


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