Multi-step prediction of zero series and gap series of Riemann zeta function

Abstract

This paper reveals the chaotic characteristic of the zero-gap series of Riemann zeta function from a new perspective of chaotic dynamics, and addresses the multi-step prediction of the zero series and gap series using the proposed EMD-ELM (empirical mode decomposition-extreme learning machine) method. Firstly, the chaotic time series analysis approach is utilized in the state space reconstruction of the gap series, and the chaotic characteristic parameters such as correlation dimension and maximal Lyapunov exponent are calculated. Numerical results indicate that the gap series possesses weak chaotic property, rather than pure random series. Subsequently, an effective time series forecasting method named as EMD-ELM based on the machine learning technique is proposed, so as to predict the zero series and gap series of Riemann zeta function. It is shown that the multi-step prediction of the zero series presents a fairly high accuracy while that of the gap series has an acceptable predicted accuracy, which implies that these series contains intrinsic order.