Refining the Riemann Prime Counting Function via Chebyshev Substitution and Spectral Analysis

Introduction

The distribution of prime numbers remains one of the most profound mysteries in mathematics, sitting at the intersection of number theory and complex analysis. The Prime Number Theorem establishes that the number of primes less than x, denoted by pi(x), is asymptotically equivalent to the logarithmic integral li(x). However, li(x) is a smooth approximation that fails to capture the intricate, oscillatory nature of the prime counting function. To bridge this gap, research often turns to the explicit formulas derived from the zeros of the Riemann zeta function.

The source paper arXiv:hal-01071210 provides a rigorous numerical and theoretical exploration of these discrepancies. By calculating li(x) and pi(x) at extreme scales—specifically powers of 10 ranging from 10^26 to 10^50—the analysis highlights the persistent gap between smooth approximations and the reality of prime distribution. A central theme of this research is the potential improvement of the Riemann prime counting function, Ri(x), through a radical departure from traditional methods: the substitution of the variable x with the Chebyshev function psi(x).

Mathematical Background

To understand the contributions of arXiv:hal-01071210, we must define the core mathematical objects involved. The primary function pi(x) counts primes less than or equal to x. The Riemann prime counting function, Ri(x), is defined via the Gram series:

Ri(x) = 1 + sum_{n=1 to infinity} (log x)^n / (n * zeta(n+1) * n!)

This series converges rapidly and provides the smooth part of the prime distribution. However, the true pi(x) is recovered only when including the sum over the non-trivial zeros rho of the zeta function. The relationship is expressed as pi(x) approximately equal to Ri(x) minus the sum over zeros of Ri(x^rho).

The second Chebyshev function, psi(x), is defined as the sum of the von Mangoldt function over integers up to x. Its explicit formula links it directly to the zeros of the zeta function: psi(x) = x - sum(x^rho / rho) - log(2*pi). Because psi(x) contains information about the zeros, it serves as a powerful candidate for refining other prime-related approximations.

Main Technical Analysis

High-Precision Evaluation and Discrepancy Sequences

The data in arXiv:hal-01071210 lists values of li(10^n) for n up to 50. For instance, at 10^26, li(x) is approximately 1.699 x 10^24. The significance of these values lies in the comparison with pi(x). The differences li(x) - pi(x) are not monotonic; they exhibit oscillations predicted by the Riemann explicit formula.

The paper provides a sequence of differences that represent the failure of the smooth approximation to track the local density of primes. This sequence, identified as A215663 in the OEIS, demonstrates how the sum of terms Ri(x^rho) dominates the error in the intermediate range. These fluctuations are characteristic of Littlewood oscillations, where the error changes sign infinitely often, though the first sign change occurs at an extremely large value of x.

The Chebyshev Substitution Hypothesis

A provocative claim in arXiv:hal-01071210 is that Ri(x) can be improved by replacing the variable x with the Chebyshev function psi(x). Traditionally, psi(x) is a weighted version of pi(x). However, the author suggests that if we perform the account of periodic terms globally, we might find that pi(x) is approximated more accurately by Ri(exp(psi(x))) or similar structures.

Mathematically, this implies that the information contained in the jumps of psi(x) is sufficient to correct the smoothness of Ri(x). Since psi(x) contains the sum over the zeros in its explicit formula, substituting it into Ri(x) effectively injects the zero-distribution information back into the prime counting function without requiring the explicit summation over thousands of imaginary parts of rho.

Novel Research Pathways

1. The Psi-Substitution Transform

The proposal to replace x with psi(x) requires a formal operator theory. One could define a transform T such that T[f](x) = f(psi(x)). Research should investigate whether there exists a fixed point or an invariant measure under this transform that corresponds to the prime-generating function. If Ri(psi(x)) converges to pi(x) faster than Ri(x), it would imply that the fluctuations in psi(x) are perfectly synchronized with the fluctuations in pi(x), supporting the Riemann Hypothesis.

2. Spectral Analysis of Error Sequences

The sequence of differences Ri(10^i) - pi(10^i) contains information about the low-frequency components of the zeta zeros. Applying a Discrete Fourier Transform (DFT) to the sequence of 50 terms provided in the paper could reveal peaks corresponding to the imaginary parts of the first few Riemann zeros (gamma_1 approx 14.13, gamma_2 approx 21.02). This provides a numerical method to test the density of zeros near the critical line for very large x.

Computational Implementation

The following Wolfram Language code implements the Riemann prime counting function Ri(x) using the power series expansion and compares it against the standard logarithmic integral.

(* Section: Riemann Prime Counting Function via Gram Series *)
(* Purpose: Demonstrate the convergence of Ri(x) and its deviation from pi(x) *)

ClearAll[RiemannRiSeries, Discrepancy];

(* Define the Riemann Function Ri(x) using the entire function series *)
RiemannRiSeries[x_, terms_] := Module[{logX = Log[x]},
  1 + Sum[
    (logX^n) / (n * Zeta[n + 1] * Factorial[n]), 
    {n, 1, terms}
  ]
];

(* Calculate the error relative to LogIntegral *)
Discrepancy[x_] := LogIntegral[x] - RiemannRiSeries[x, 100];

(* Generate a table of values for powers of 10 *)
data = Table[
  {i, N[RiemannRiSeries[10^i, 150]], N[Discrepancy[10^i]]}, 
  {i, 1, 15}
];

Print[TableForm[data, 
  TableHeadings -> {None, {"n (10^n)", "Ri(10^n)", "li(x) - Ri(x)"}}]];

(* Plot the oscillatory nature of the difference *)
Plot[
  {RiemannRiSeries[x, 50] - LogIntegral[x]}, 
  {x, 2, 1000}, 
  PlotStyle -> Thick, 
  PlotLabel -> "Oscillation of Ri(x) - li(x)", 
  AxesLabel -> {"x", "Difference"}
]

The investigation of arXiv:hal-01071210 reveals that the Riemann prime counting function, while a massive improvement over li(x), still requires a global correction to match the true distribution of primes. The data provided for values up to 10^50 highlights the magnitude of the oscillatory terms governed by the zeros of the zeta function. The most promising avenue for further research is the formalization of the Chebyshev substitution Ri(psi(x)).

References

• arXiv:hal-01071210: Riemann prime counting function Ri(x) and its improvement.
• Riemann, B. (1859). "Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse."
• OEIS Sequence A215663: Difference between the Riemann prime counting function and the actual number of primes.
• arXiv:hal-00863138: Counting primes in residue classes.