The Riemann Hypothesis and Inverse Spectral Problems

Abstract

In this chapter, we provide an alternative formulation of the Riemann hypothesis in terms of a natural inverse spectral problem for fractal strings. After stating this inverse problem in Section 9.1, we show in Section 9.2 that its solution is equivalent to the nonexistence of critical zeros of the Riemann zeta function on a given vertical line. This modifies and extends the earlier work of [LapMa1-2], but now we use the point of view of complex dimensions and the explicit formulas of Chapter 5. In Section 9.3, we then 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.KeywordsZeta FunctionComplex DimensionRiemann Zeta FunctionRiemann HypothesisInverse Spectral ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.