The Riemann Zeta Function

Abstract

The surface is a graph of the reciprocal of the absolute value of the Riemann zeta function ζ (s). The spikes correspond to the zeros on the critical line ½ + iy. Recall that the global behavior of π(x), the prime distribution function, is well approximated by Riemann’s smooth function R(x) (discussed in Chapter 2). More delicate information about π(x), such as the local distribution of the primes, is determined by the location of the zeros of ζ. If π 0(x) denotes the function that agrees with π(x) except at prime numbers p, where its value is π(p)− ½, then modifying R(x) with correction terms for the zeros of ζ yields the function π 0(x) exactly. The lower graph is obtained by adding 100 correction terms to R(x), one for each of the first hundred zeros on the critical line. The dots are the points (p, π 0(p)), p prime. The convergence to the step function can be clearly seen: the graph comes close to all the prime points and has several near-horizontal segments.