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arxiv: 2503.09519 · v2 · pith:RZTWU42Pnew · submitted 2025-03-12 · 🧮 math.NT · cs.NA· math.NA

Simple and accurate approximations to the Riemann zeta function

Alexey Kuznetsov This is my paper

Pith reviewed 2026-05-22 23:53 UTC · model grok-4.3

classification 🧮 math.NT cs.NAmath.NA
keywords Riemann zeta functionRiemann-Siegel formulaapproximationsGaussian quadraturecritical striphigh-precision computationzeta derivative
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The pith

Approximations compute the Riemann zeta function to high precision by pairing the Riemann-Siegel sum with an elementary-function remainder that uses precomputed Gaussian quadrature coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a practical method for evaluating the Riemann zeta function at high precision inside the critical strip and other vertical lines. It keeps the main oscillatory sum from the Riemann-Siegel formula but replaces the remainder with a short expression built from elementary functions and a fixed list of coefficients. Those coefficients are computed once by Gaussian quadrature. The same construction supplies an approximation for the derivative of zeta. Numerical checks across many points confirm that the resulting errors stay within the target bounds.

Core claim

The central claim is that the remainder after the main sum in the Riemann-Siegel formula for zeta(s) admits a simple, accurate representation that uses only elementary functions together with a modest number of precomputed coefficients obtained by Gaussian quadrature; the resulting hybrid formula therefore permits high-precision computation of both zeta(s) and zeta'(s) throughout the critical strip.

What carries the argument

The remainder-term approximation in the Riemann-Siegel formula, constructed from elementary functions and Gaussian-quadrature coefficients.

If this is right

  • High-precision values of zeta(s) become available throughout the critical strip from a formula whose non-sum part contains only elementary functions.
  • The same remainder approximation supplies a corresponding formula for the derivative zeta'(s).
  • The method extends without change to other vertical strips Re(s) = constant.
  • Numerical evidence presented in the paper shows that the observed errors remain below the expected thresholds over wide ranges of t.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Implementations in arbitrary-precision arithmetic libraries could adopt the precomputed coefficients as a fixed table, reducing per-evaluation cost.
  • The approach may shorten the inner loop when locating zeros of zeta to very high precision.
  • Because the quadrature coefficients are independent of s, the same table could be reused for related functions sharing a similar functional equation.

Load-bearing premise

The remainder term in the Riemann-Siegel formula can be accurately approximated using only elementary functions and precomputed coefficients obtained via Gaussian quadrature while preserving the high precision needed for computations inside the critical strip.

What would settle it

A side-by-side comparison at any point s = 1/2 + it (t large) in which the absolute difference between the new approximation and an independent high-precision reference exceeds the stated error bound would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2503.09519 by Alexey Kuznetsov.

Figure 1
Figure 1. Figure 1: The numbers λp,j and |ωp,j | for p ∈ {20, 40}. that reduces the complexity of evaluating ζ(1/2 + it) at a single point to O(t 4/13+o(1)), compared to the standard Riemann-Siegel formula’s complexity of O(t 1/2 ). A simpler version of Hiary’s algorithm [9] achieves complexity O(t 1/3+o(1)) with minimal memory requirement. Another algorithm due to Hiary [10] produces Riemann-Siegel-type approximations (requi… view at source ↗
Figure 2
Figure 2. Figure 2: The values of ∆p(t) for p ∈ {10, 20, 30, 40, 50}. The black dots on plot (b) correspond to values of ∆p(tn). The choice of 100 points in our definition of ∆p is arbitrary and not critical. Using 50 or 200 points would produce nearly identical numerical results. When focusing on the critical strip, we simplify notation and write ∆p(t) = ∆p(t; 0, 1). To compute ∆p(t; a, b), we require benchmark values of ζ(s… view at source ↗
Figure 3
Figure 3. Figure 3: The values of log [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The values of |ζ(1/2 + it) − ζp(1/2 + it)| for p ∈ {3, 5} and t close to 1010. The gray vertical lines show the locations of tn = 2πn2 for 39894 ≤ n ≤ 39897. • |ζ10(s) − ζ(s)| < 10−15 when t > 250 and |ζ10(s) − ζ(s)| < 10−20 when t > 6000; • |ζ20(s) − ζ(s)| < 10−30 when t > 350 and |ζ20(s) − ζ(s)| < 10−50 when t > 65000; • |ζ50(s) − ζ(s)| < 10−100 when t > 4000; On [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The values of ∆(1) p (t) for p ∈ {10, 20, 30, 40, 50}. Next, we examine the accuracy of our approximations for very large values of t. On [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The values of ∆p(t; 1/2, 2). The top curve corresponds to p = 8 implemented in double precision, the middle curve in gray color (the bottom curve) correspond to p = 8 and (respectively, p = 12) impemented in quadruple precision. was chosen because ζ(s) does not grow too rapidly as t → +∞ in this strip, unlike in any strip where Re(s) < 1/2. The results of these computations are presented in [PITH_FULL_IMA… view at source ↗
Figure 7
Figure 7. Figure 7: The values of |ζ(1/2 + iγ30; h)| (circles) and exp(− √π 2 × 1 h ) (gray line), where 1 h is on the x-axis. compute the benchmark values of ζ ′ (s) (needed for the results on [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The graph of B(t). where the points {yk}0≤k≤4p+1 are equally spaced in the interval [−1, 1]: yk := −1 + 2k 4p + 1 . Thus, we arrive at the following problem: we seek 2p + 1 complex numbers {ωp,j}0≤j≤p and {xp,j}1≤j≤p such that Z R e −2πx2+2πθykx cosh(πθx) dx = ωp,0 + X p j=1 ωp,je −πx2 p,j h e 2πθykxp,j + e −2πθykxp,j , (16) for all k = 0, 1, . . . , 4p + 1. To simplify the above problem, we note that the … view at source ↗
Figure 9
Figure 9. Figure 9: The values of |Hp(y) − H(y)| for −1 ≤ y ≤ 1 and p ∈ {3, 5}. It remains to explain the pattern in the error ζp(1/2 + it) − ζ(1/2 + it) that we observed on [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops approximations to the Riemann zeta function by augmenting the main sum of the Riemann-Siegel formula with an elementary-function approximation to the remainder term, where the approximation employs precomputed coefficients obtained via Gaussian quadrature. Analogous approximations are supplied for the derivative, and the claims are supported by extensive numerical evidence for accuracy inside the critical strip and other vertical strips.

Significance. If the numerical tests confirm that the remainder approximation preserves high precision across the tested ranges, the method supplies a lightweight, elementary-function-based route to high-precision zeta evaluations after a one-time precomputation step. This could be useful for large-scale numerical work in analytic number theory.

major comments (2)
  1. [Numerical evidence and remainder approximation sections] The central claim that the Gaussian-quadrature remainder approximation achieves the precision needed inside the critical strip rests on numerical evidence alone; the manuscript should supply explicit a-posteriori error bounds or a comparison of the approximated remainder against the exact remainder (computed via the full Riemann-Siegel formula) for representative values of t up to at least 10^6.
  2. [Numerical evidence section] No baseline comparisons (e.g., against the plain Riemann-Siegel formula truncated at the same number of terms, or against other known remainder approximations) are described; without such controls it is impossible to quantify the practical gain in accuracy or speed.
minor comments (2)
  1. The abstract states that the approximations enable 'high-precision computation' but does not quantify the achieved absolute or relative error; adding concrete figures (e.g., 10^{-12} for |t| < 1000) would clarify the scope.
  2. Notation for the precomputed quadrature coefficients should be introduced once and used consistently; a short table listing the coefficients for the lowest-order approximations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments below and will update the manuscript with additional numerical comparisons as requested.

read point-by-point responses

  1. Referee: [Numerical evidence and remainder approximation sections] The central claim that the Gaussian-quadrature remainder approximation achieves the precision needed inside the critical strip rests on numerical evidence alone; the manuscript should supply explicit a-posteriori error bounds or a comparison of the approximated remainder against the exact remainder (computed via the full Riemann-Siegel formula) for representative values of t up to at least 10^6.

    Authors: We agree that direct comparisons against the exact remainder would strengthen the claims. In the revised manuscript we will add tables showing the approximated remainder versus the exact remainder (via the full Riemann-Siegel formula) at representative t values up to 10^6, together with the observed maximum absolute errors in those tests. revision: yes

  2. Referee: [Numerical evidence section] No baseline comparisons (e.g., against the plain Riemann-Siegel formula truncated at the same number of terms, or against other known remainder approximations) are described; without such controls it is impossible to quantify the practical gain in accuracy or speed.

    Authors: We accept that baseline comparisons are needed to quantify improvement. The revised version will include accuracy and timing comparisons against the plain Riemann-Siegel formula truncated after the same number of main-sum terms, and, where feasible, against other published remainder approximations. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper starts from the established Riemann-Siegel formula (an external, pre-existing result) and augments it with an independent approximation to the remainder term via Gaussian quadrature on elementary functions. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain. The construction is parameter-light and the accuracy claims rest on direct numerical verification rather than internal re-derivation. This is the normal case of a self-contained numerical method paper.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are described beyond reliance on the standard Riemann-Siegel formula and Gaussian quadrature as background techniques.

free parameters (1)
  • precomputed coefficients
    Coefficients obtained via Gaussian quadrature for the remainder approximation; their specific values and fitting process are not detailed in the abstract.

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Reference graph

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