Riemann Zeros and Factorizations of the Zeta Function

Abstract

This is the second half of our review on the basic properties of ζ(x). The theory of entire functions of finite order (cf. Sect. 2.1) applies to Riemann’s Ξ function in a classic way [26, Sect. 12] [89, Appendix 5]. We first bound ζ(x) and the trivial factor $\mathbf{G}^{-1}(x)(x - 1)$ separately in the half-plane $\{{\rm Re} x \geq \frac{1}{2}\}$. Applying the Euler–Maclaurin formula (1.14) to f(u) = u ^−x with Re x > 1 and K = 1, K ^′ = +∞, yields (with $\{u\} \stackrel{\rm def}{=}$ the fractional part of u here)

$$\zeta(x) = \frac{1}{x - 1} + \frac{1}{2} - x \int_{1}^{\infty} B_1 (\{u\})u^{-x-1} {\rm d}u;$$

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but as the right-hand side converges and defines an analytic function for Re x > 0, it analytically continues ζ(x) to this half-plane. The integral is bounded by $\int_{1}^{\infty} \frac{1}{2}u^{-{\rm Re} x - 1} {\rm d}u = (2 {\rm Re}\,x)^{-1} \leq 1\ {\rm if\ Re}\,x \geq \frac{1}{2}$, hence as x → ∞ in the latter half-plane, the bound ζ(x) = O(|x|) holds.