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, R1ps and R2ps, and prove that R=R1ps+R2ps, so that ζ=Σ1+R1ps+Σ2+R2ps. 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)=∑n=1mn−s+d̂1(m+1)−s+χ(s)∑n=1mns−1+d̂2χ(s)(m+1)s−1, with d̂1,d̂2 real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when σ=1/2 one has d1=d2 (equivalently d̂1=d̂2); 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.
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BibTeX
@misc{stahura2026riemannsiegel,
title = {{The Riemann-Siegel remainder as a fractional summand}},
author = {Paul Stahura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03728},
eprint = {2610.03728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03728},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Riemann-Siegel remainder as a fractional summand
AU - Paul Stahura
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.03728
UR - https://arxiv.org/abs/2610.03728
AB - 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)=∑_n=1^mn^−s+d̂_1(m+1)^−s+χ(s)∑_n=1^mn^s−1+d̂_2χ(s)(m+1)^s−1, with d̂_1,d̂_2 real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when σ=1/2 one has d_1=d_2 (equivalently d̂_1=d̂_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.
ER -