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

Moments of the Riemann Zeta Function and Eisenstein Series I

Notes Relating to Newton Series for the Riemann Zeta Function

This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably ...

Improving Multi-Step Prediction of Learned Time Series Models

Most typical statistical and machine learning approaches to time series modeling optimize a singlestep prediction error. In multiple-step simulation, the learned model is iteratively applied, feeding through the previous output as its new input. Any such predictor however, inevitably introduces errors, and these compounding errors change the input distribution for future prediction steps, break...

Riemann zeta function and the best constants of five series ofSobolev inequalities

We clarified the variational meaning of the special values ζ(2M) (M = 1, 2, 3, · · · ) of Riemann zeta function ζ(z). These are essentially the best constants of five series of Sobolev inequalities. In the background, we consider five kinds of boundary value problem (periodic, antiperiodic, Dirichlet, Neumann, Dirichlet-Neumann) for a differential operator (−1) (d/dx) . Green functions for thes...

On Some Conjectures and Results for the Riemann Zeta-function and Hecke Series

We investigate the pointwise and mean square order of the function Z2(s), where Zk(s) = ∫∞ 1 |ζ(12 + ix)|2kx−s dx, k ∈ N. Three conjectures involving Z2(s) and certain exponential sums of Hecke series in short intervals are formulated, which have important consequences in zeta-function theory. A new order result for Z2(s) is obtained, and the function Zk(s) is discussed. 1. Spectral theory and ...