An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

Figure 1

The various regions of validity of the analytic continuations developed in Section 4 both (a) in the complex -plane and (b) in the cross section of a complex plane cut by the real plane at . (a) The location of the contour of integration (black arrow at ) extending from as well as the various poles. The fixed pole at  = −3/2 corresponds to the singularity belonging to . The green and red poles correspond to singularities of the integrand that depend on the value of s acting as a parameter. The coloured arrows show the motion of corresponding poles as increases from zero to one if t > 0. The colours correspond to colours used in Figure 1(b). (b) The various regions where different continuations of (29) apply, projected onto the t = 0 plane. Five distinct regions are bounded by (coded and coloured) lines c = −3/2 (blue, solid), (red, dash-dot), and (green, dash-dot) when . The bounding regions extend vertically out of the plane of the figure when because the contour used in (23) is chosen to be a straight line, vertical in the complex -plane (c is a constant). The dotted line corresponds to the interesting case c = −5/4.

(a)

(b)