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52) m−1 (−1) (log Δ∞ )(m) (y) = Z∞ (m, y) (m − 1)! (m = 1, 2, . 55) ≡ Z(m, y) if Z(s, y) is regular at s = m. 55) can be skipped as this makes no difference. However, Appendix B (devoted to the zeros of Selberg zeta functions) makes a brief incursion into μ0 = 2 territory, where that omission would cause fatal errors. 58) n=0 the sequence {μn } is as in Sect. 2, and 0 < θ ≤ 12 π. In linear spectral problems of physics, Θ(z) is the trace of the heat operator (“partition function”). Certain complex sequences {xk } can qualify as well [53, 60, 61, 90].

29) of the zeta-regularized form, Δ∞ (x) = exp −Z (0, x) , goes in the wrong direction for us: it precisely uses a zeta function that we aim at understanding (plus analytical continuation in totally abstract form). Fortunately, given an eligible sequence {xk } of order μ0 in the sense of Sect. 13), we now basically have a mechanical zeta-regularization rule. 47) μn ∈N / m=0 m (recalling that Hm = j=1 1/j are the harmonic numbers). 48) including additive constants (b0 ). 49) while all terms with bm are banned, namely bm xm for all m ∈ N, Consequently, the zeta-regularized form has to be, simply (cf.

11. Another unproved conjecture is that all zeros are simple, but we adopt the standard convention of counting all objects with their multiplicities if any. 9350616, . 3 Hadamard Products for Ξ(x) By the general theory of entire functions of finite order as referred to in Sect. 1, in the case of Ξ(x) of order 1, the zeros ρ satisfy ρ |ρ|−1−ε < ∞ if ε > 0, and Ξ(x) admits the Hadamard factorization 1− Ξ(x) = Ξ(0) e[log Ξ] (0) x ρ x x/ρ e ρ (∀x ∈ C). 3). Hence |ρ|−1−ε converges for ε > 0, diverges for ε = 0.