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

Figure 2

The individual terms composing the integrand of (66) as a function of the integration variable , using for several values of t. Each curve has been scaled by a (smooth) factor to separate them for easier viewing; otherwise, all of the curves coincide to a good degree of approximation, except near . (a) An illustration of the integrand at several values of ; the arrows mark the points  = t/2 that being the location of the peak of the pole term contained in for each of the curves at each individual value of t. (b) An illustration of the integrand at several values of ; the arrows mark the points  = t/2 that being the location of the zero of the pole term contained in for each of the curves at each individual value of t.

(a)

(b)