The Riemann-Siegel remainder as a fractional summand

Abstract

Starting from the Riemann-Siegel decomposition of $ζ(s)$ given in Siegel's 1932 paper, we introduce a change of variable, $t=I(T)$, that replaces Siegel's coupled pair "imaginary part $t$ / summation index $m$" with one real index $T$. We then split Siegel's single remainder integral $R$ into two exact pieces, $R_{1ps}$ and $R_{2ps}$, and prove that $R=R_{1ps}+R_{2ps}$, so that $ζ=Σ_1+R_{1ps}+Σ_2+R_{2ps}$. Our central observation is that each of these remainders is nothing more than one additional, fractional partial summand appended to its Dirichlet sum: $ζ(s)=\sum_{n=1}^{m}n^{-s}+\hat{d}_1(m+1)^{-s}+χ(s)\sum_{n=1}^{m}n^{s-1}+\hat{d}_2χ(s)(m+1)^{s-1}$, with $\hat{d}_1,\hat{d}_2$ real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when $σ=\frac{1}{2}$ one has $d_1=d_2$ (equivalently $\hat{d}_1=\hat{d}_2$); this fact is formally verified in Lean. Also, with this rescaling the remainder terms are nearly periodic in $T$ with period one, converging to a fixed waveform in the fractional part of $T$. This decomposition of Siegel's $R$ was discovered through experimental mathematics using a spiral visualization of the partial sums, described later in the paper. We also discuss a number of other observations, including what we call the yin yang curves, the zero counting function, and ovals of equal length leg loci.