Beyond Weyl Sums: Decoding Pseudo-Polynomial Phases and Their Links to the Critical Line

Executive Summary

The research presented in arXiv:1606.03049 establishes a powerful analytic framework for controlling exponential sums and distributional statistics associated with pseudo-polynomial sequences. By refining the Weyl-van der Corput method and introducing iterated difference operators, the author provides precise bounds on the discrepancy of sequences involving non-integer powers and polynomial perturbations. The central technical contribution is the management of Type II bilinear sums, which are essential for extracting cancellation in oscillatory phases. This methodology is deeply connected to the Riemann Hypothesis (RH), as it targets the same fundamental bottleneck: the quantitative cancellation in sums resembling Dirichlet polynomials. This analysis suggests that the paper's machinery can be adapted to investigate the distribution of primes and the vertical spacing of zeta zeros, offering a pathway to bridge the gap between deterministic sequence distribution and the critical behavior of the zeta function.

Introduction

A central challenge in analytic number theory is the estimation of exponential sums, which serve as the primary tool for understanding the distribution of arithmetic sequences. The work in arXiv:1606.03049 focuses on sequences of the form f(n) = n^c + P(n), where P is a polynomial and c is a non-integer. These "pseudo-polynomial" sequences exhibit complex oscillatory behavior that mirrors the phase structures encountered in the study of the Riemann zeta function, ζ(s).

The Riemann Hypothesis remains the most significant conjecture regarding the distribution of primes, asserting that all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2. Proving this requires establishing near-optimal cancellation in sums involving the von Mangoldt function, which can be decomposed into Type I and Type II bilinear forms. The techniques developed in the source paper for bounding the discrepancy R(H) and managing iterated differences are precisely the tools needed to refine these prime-weighted sums.

This article provides a technical bridge between the methods of arXiv:1606.03049 and the analytic territory of the zeta function. We demonstrate how the paper's iterated difference operators and bilinear sum estimates can be repurposed to study Dirichlet polynomials and propose novel research directions that hybridize pseudo-polynomial technology with the explicit formulas of prime number theory.

Mathematical Background

To understand the connection to the Riemann Hypothesis, we must first formalize the key objects defined in arXiv:1606.03049.

The Discrepancy Function and Fourier Truncation

The paper defines a correlation measure R(H) that quantifies the difference between the distribution of a sequence and its expected uniform behavior. This is achieved through a Fourier-truncated approximation of characteristic functions. The bound for R(H) is expressed as a sum of exponential sums weighted by a truncation parameter H. This approach mirrors the Erdos-Turan philosophy: discrepancy is controlled by the magnitude of exponential sums at frequencies up to H. For the zeta function, this is analogous to approximating the function with a finite Dirichlet polynomial and bounding the remainder.

Iterated Difference Operators

The core analytic engine of the source paper is the iterated difference operator Delta. For a function g, the operator is defined through an integral representation involving higher-order derivatives. This operator allows the author to reduce the complexity of the phase function f(n), effectively "exposing" the cancellation properties of its derivatives. In the context of the Riemann Hypothesis, this is similar to the Weyl differencing used to estimate the growth of ζ(1/2 + it) in the critical strip.

Type I and Type II Bilinear Structures

The paper identifies two types of sums, S1 and S2, which are prototypes for the bilinear decompositions used in prime number theory. S2, in particular, involves coefficients that resemble divisor functions and smooth perturbations. The ability to bound these sums determines the strength of the resulting distribution theorems. As we will see, these structures are the key to connecting the paper's results to the primes.

Main Technical Analysis

Bilinear Estimates and the Lindelof Hypothesis

One of the most profound aspects of arXiv:1606.03049 is the treatment of Type II sums. The author derives bounds of the form |S2|² which involve a double sum over pairs of variables y1 and y2. This "squaring out" technique reduces the problem to bounding a one-dimensional exponential sum with a differenced phase. The paper demonstrates that if the derivatives of the phase satisfy certain growth conditions, the sum exhibits significant cancellation.

This is directly relevant to the Lindelof Hypothesis, which posits that ζ(1/2 + it) grows slower than any power of t. The Lindelof Hypothesis is a consequence of RH and can be reframed as a statement about the cancellation of exponential sums with the phase t log n. By treating log n as a smooth perturbation g(n) and applying the paper's derivative estimates, one can derive new bounds on the growth of the zeta function in short intervals.

Spectral Properties and Zero Spacing

The source paper also investigates the minimum distance between the fractional parts of the sequence and a fixed value xi. This is a problem of Diophantine approximation. The resulting bounds, such as those for the minimum of ||xi(n^c + n^k)||, provide a quantitative measure of how evenly the sequence is spread. In the study of the Riemann Hypothesis, the spacing of zeta zeros is believed to follow the GUE (Gaussian Unitary Ensemble) distribution. The paper's methods for proving equidistribution at explicit polynomial rates offer a new way to probe the statistics of these zeros by mapping the zeta phase to a pseudo-polynomial model.

Derivative Control in Pseudo-Polynomial Phases

The paper provides precise asymptotics for the derivatives of the differenced phase. Specifically, it shows that the j-th derivative of the phase is proportional to (|y1 - y2|/y1) X^theta X1^-j. This allows for the application of the stationary phase method or van der Corput's lemma. These estimates are the fundamental building blocks of the zero-free region for the zeta function. Strengthening these derivative-based bounds directly translates to a wider zero-free region and a more precise Prime Number Theorem.

Novel Research Pathways

Pathway 1: Prime-Weighted Pseudo-Polynomial Adaptations

A promising research direction is to apply the paper's Type II technology to prime-weighted sums. By using Vaughan's identity or the Heath-Brown identity, the sum over primes of e(n^c + P(n)) can be decomposed into exactly the bilinear forms S1 and S2 studied in the paper. The goal would be to prove that these sums exhibit square-root cancellation, which would be an "RH-quality" result for this class of sequences. This would establish that pseudo-polynomial sequences are "orthogonal" to the primes, a property related to the Sarnak Conjecture.

Pathway 2: Local Taylor Expansions of the Zeta Phase

Another pathway involves treating the phase of the zeta function, t log n, as a pseudo-polynomial. For n in a short interval around M, log n can be expanded as a polynomial in (n-M)/M plus a smooth remainder. By applying the paper's Delta operator framework to this expansion, researchers can import the paper's discrepancy bounds directly into the study of ζ(1/2 + it). This could yield competitive exponent-pair bounds for the zeta function using the paper's integral formulas for the difference operators.

Computational Implementation

The following Wolfram Language code demonstrates the oscillatory behavior of the exponential sums studied in arXiv:1606.03049. It computes the magnitude of the sum for a pseudo-polynomial phase and visualizes its growth compared to the imaginary parts of the first few zeta zeros, illustrating the "spectral" nature of the frequencies.

(* Section: Pseudo-Polynomial Sums and Zeta Zeros *)
(* Purpose: Visualize the growth of S(h) and compare to Zeta frequencies *)

Module[{
  nMax = 1000, 
  c = 1.4, 
  k = 1, 
  hMax = 60, 
  f, S, data, zeros, zeroIm, p1, p2
},
  (* Define the phase function f(n) = n^c + n^k *)
  f[n_] := n^c + n^k;

  (* Compute the exponential sum S(h) = Sum e(h*f(n)) *)
  S[h_Integer] := Sum[Exp[2 Pi I h f[n]], {n, 1, nMax}];

  (* Generate data for |S(h)| *)
  data = Table[{h, Abs[S[h]]}, {h, 1, hMax}];

  (* Get imaginary parts of the first 15 Zeta zeros *)
  zeros = Table[ZetaZero[m], {m, 1, 15}];
  zeroIm = Im[zeros];

  (* Plot the magnitude of the exponential sum *)
  p1 = ListLinePlot[data, 
    PlotRange -> All, 
    AxesLabel -> {"h", "|S(h)|"}, 
    PlotLabel -> "Weyl Sum Magnitude for f(n) = n^1.4 + n",
    ImageSize -> 500,
    PlotStyle -> Blue];

  (* Overlay vertical lines at scaled Zeta zero locations *)
  p2 = Graphics[Table[{
    Red, Dashed, 
    Line[{{zeroIm[[m]]/5, 0}, {zeroIm[[m]]/5, Max[data[[All, 2]]]}}]
  }, {m, 1, Length[zeroIm]}]];

  Show[p1, p2]
]

Conclusions

The analytic framework presented in arXiv:1606.03049 provides essential tools for the modern study of exponential sums and their distribution. By mastering the interaction between polynomial and smooth phases through iterated differencing and bilinear sum estimates, the paper offers a robust methodology that is highly relevant to the Riemann Hypothesis. The most promising avenue for future research is the integration of these pseudo-polynomial bounds into prime-counting identities, which could yield unconditional improvements in our understanding of prime distribution. Furthermore, the spectral interpretation of these sums provides a diagnostic tool for investigating the spacing of zeta zeros. These steps represent a concrete path toward aligning deterministic sequence analysis with the deep arithmetic properties of the critical line.

References

• arXiv:1606.03049: On the distribution of some sequences involving the fractional part of n^c.
• H. L. Montgomery, "The pair correlation of zeros of the zeta function," 1973.
• E. C. Titchmarsh, "The Theory of the Riemann Zeta-Function," Oxford University Press.