# Riemann Zeros and Factorizations of the Zeta Function

• André Voros
Chapter
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 8)

## 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) = ux with Re x > 1 and K = 1, K = +∞, yields (with $$\{u\} \stackrel{\rm def}{=}$$ the fractional part of u here)
$$\zeta(x) = \frac{1}{x - 1} + \frac{1}{2} - x \int_{1}^{\infty} B_1 (\{u\})u^{-x-1} {\rm d}u;$$
( )
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.