Explicit Bounds for Chebyshev Functions and the Nicolas Criterion for the Riemann Hypothesis

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1. Introduction

The distribution of prime numbers is a cornerstone of analytic number theory, primarily governed by the properties of the Riemann zeta function ζ(s). The paper arXiv:hal-00666154v1 by Jean-Louis Nicolas contributes to the tradition of explicit inequalities, providing refined bounds for the Chebyshev functions and their relationship to the Riemann Hypothesis (RH). Specifically, it investigates the difference between the second Chebyshev function ψ(x) and the first Chebyshev function θ(x), utilizing these estimates to probe the Nicolas criterion for RH.

A central theme in this research is that error terms in prime number theory are constrained by the nontrivial zeros of ζ(s). Under the assumption of RH, where all nontrivial zeros lie on the critical line Re(s) = 1/2, the oscillatory part of prime-counting functions is significantly restricted. The source paper exploits this relationship to derive concrete inequalities involving primorial integers (N_k) and the Euler totient function φ(n). This article synthesizes these mathematical structures, explores their spectral properties, and proposes novel research pathways for verifying RH through arithmetic oscillations.

2. Mathematical Background

The fundamental objects of study are the Chebyshev functions:

θ(x): The sum of log p for all primes p ≤ x.
ψ(x): The sum of the von Mangoldt function Λ(n) for all n ≤ x, which includes powers of primes.

The difference ψ(x) - θ(x) represents the contribution of higher prime powers (p², p³, etc.) and is given by the sum of θ(x^1/m) for m ≥ 2. Under RH, Schoenfeld-type bounds suggest that the error term S(x) = θ(x) - x is approximately of the order x^1/2 log² x. The paper arXiv:hal-00666154v1 refines this by establishing that ψ(x) - θ(x) ≥ x^1/2 for x ≥ 121.

Another key structure is the primorial N_k, defined as the product of the first k primes. The Nicolas criterion relates RH to the growth of the ratio N_k/φ(N_k). The paper defines a step function f(x) such that for p_k ≤ x < p_k+1, f(x) = e^γ log log(N_k) φ(N_k)/N_k, where γ is the Euler-Mascheroni constant. The deviation of this function from 1 is captured by a coefficient c(N_k), which exhibits unbounded oscillation if RH is false, but remains tightly controlled if RH is true.

3. Main Technical Analysis

3.1. Explicit Estimates for ψ(x) - θ(x)

The paper leverages lower bounds on the difference between Chebyshev functions to control the influence of prime powers in the explicit formula. A primary derivation shows that ψ(x) - θ(x) ≥ x^1/2 + x^1/3 - T(x^1/2) - T(x^1/3), where T(x) is a bound on the error in the prime number theorem. Under RH, the paper demonstrates that the sum of these error terms divided by x^1/3 is bounded by a constant (approximately 0.86157), ensuring the positivity of the difference beyond a small threshold.

3.2. Integral Transforms and Smoothing

To convert pointwise bounds into global estimates, the analysis utilizes integral transforms denoted as K(x), J(x), and F_z(x). These transforms act as kernels that emphasize large-scale behavior while damping local fluctuations. For instance, the transform F_1/2(x) is shown to be sandwiched between 2/(x^1/2 log x) and a similar term with higher-order logarithmic corrections. These expansions allow the author to propagate estimates from the distribution of primes to the values of f(x) and c(N_k).

3.3. Monotonicity and Dyadic Decomposition

A striking aspect of the technical proof is the use of dyadic decomposition to prove monotonicity of error functions. By examining the difference H(2^j) - H(2^j+1), the paper establishes that for j ≥ 20, the function behaves predictably. The difference is lower-bounded by 2^-j/3 [(1 - 2^-1/12) 2^j/4 - 2^2/3], which is positive for all sufficiently large j. This monotonicity is crucial for verifying inequalities across infinite ranges by checking only a finite number of discrete points.

3.4. The Oscillation of c(N_k)

The coefficient c(N_k) acts as a normalized measure of how the primorial totient ratio differs from its Mertens heuristic. The paper establishes that c(N_k) is governed by a zero-sum term W(x), which represents the contribution of the nontrivial zeros of the zeta function. Under RH, W(x) is bounded, leading to an envelope for c(N_k) roughly between e^γ(2 - β) and e^γ(2 + β). However, the paper also proves that c(n) has a limit inferior of negative infinity and a limit superior of positive infinity, reflecting the deep oscillatory nature of prime distribution.

4. Novel Research Pathways

4.1. Refined Zero-Sum Modeling

Future research could replace the uniform bound β with an explicit truncated sum over the first several thousand zeta zeros. By modeling W(x) as a finite trigonometric sum in log x and bounding the tail error using zero-density estimates, researchers could obtain much sharper, x-dependent envelopes for c(N_k). This would allow for a more granular verification of the Nicolas criterion in ranges where computational power currently reaches its limits.

4.2. Extreme Value Theory for Primorial Deviations

While the paper provides global limits for c(n), focusing specifically on the subsequence of primorials remains a fertile ground. One could apply probabilistic models (random phase approximations) to the explicit formula to predict the frequency and magnitude of near-extremal values of c(N_k). This would help determine if the primorial sequence is sufficient to capture the full oscillatory power of the zeta zeros or if other highly composite structures are required.

4.3. Extension to Generalized Riemann Hypothesis (GRH)

The machinery developed in arXiv:hal-00666154v1 can be extended to Dirichlet L-functions. By replacing the standard totient function with character-weighted Euler products, one could derive "Nicolas-type" criteria for primes in arithmetic progressions. This would link the distribution of zeros for L(s, χ) to the growth of generalized primorial structures, providing a unified framework for explicit inequalities across various L-functions.

5. Computational Implementation

The following Wolfram Language code calculates the primorial totient step function and compares its deviation to an oscillatory proxy derived from the first 50 nontrivial zeros of the Riemann zeta function.

Wolfram Language

(* Section: Primorial Totient and Zeta-Zero Proxy Analysis *)
(* Purpose: Compare arithmetic deviations with zeta-zero oscillations *)

Module[
  {kmax = 30, primes, primorials, fVal, gammaEuler, zeros, zIm, Wproxy, data},

  gammaEuler = EulerGamma;
  primes = Prime[Range[kmax]];
  
  (* Build primorials N_k iteratively *)
  primorials = FoldList[Times, 1, primes];

  (* Define f(p_k) step function *)
  fVal[k_] := Module[{N = primorials[[k + 1]]},
    Exp[gammaEuler] * Log[Log[N]] * (EulerPhi[N]/N)
  ];

  (* Extract imaginary parts of the first 50 nontrivial zeros *)
  zeros = ZetaZero[Range[1, 50]];
  zIm = Im[zeros];

  (* Define a proxy for the zero-sum term W(x) *)
  Wproxy[x_] := Total[Cos[zIm * Log[x]] / zIm^2];

  (* Generate comparison data *)
  data = Table[
    {k, primes[[k]], fVal[k] - 1, Wproxy[primes[[k]] ]},
    {k, 3, kmax}
  ];

  (* Display results *)
  Print["Table of k, Prime p_k, Deviation f(p_k)-1, and Zero Proxy W(p_k)"];
  Print[Grid[Prepend[data, {"k", "p_k", "f(p_k)-1", "W_proxy"}], Frame -> All]];

  (* Visualization *)
  Print[ListLinePlot[data[[All, {1, 3}]], PlotLabel -> "Deviation f(p_k)-1"]];
]

6. Conclusions

The estimates provided in arXiv:hal-00666154v1 establish a rigorous framework for understanding the interplay between prime power contributions and the zeros of the Riemann zeta function. By refining the bounds for ψ(x) - θ(x) and analyzing the resulting oscillations in the Nicolas criterion, the research clarifies the conditions under which arithmetic inequalities become equivalent to the Riemann Hypothesis. The most promising avenue for future investigation lies in the synthesis of explicit zero-sum terms with probabilistic models of primorial deviations, potentially offering a definitive computational path toward verifying the critical line hypothesis.

7. References

arXiv:hal-00666154v1: J.-L. Nicolas, "Explicit estimates for the Chebyshev function ψ(x) - θ(x)".
L. Schoenfeld, "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II," Mathematics of Computation (1976).
J.-L. Nicolas, "Petites valeurs de la fonction d'Euler," Journal of Number Theory (1983).