The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena

Abstract

In this chapter, we provide a geometric reformulation of the Riemann Hypothesis in terms of a natural inverse spectral problem for fractal strings. After stating this inverse problem in Section 7.1, we show in Section 7.2 that its solution is equivalent to the nonexistence of critical zeros of the Riemann zeta function on a given vertical line. This was done earlier in [LapMal-2], but now we use the point of view of complex dimensions and the explicit formulas of Chapter 4. Then, in Section 7.3, we extend this characterization to a large class of zeta functions, including all the number-theoretic zeta functions for which the extended Riemann Hypothesis is expected to hold.