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The distribution of prime numbers has remained one of the most significant challenges in mathematics since Riemann's seminal 1859 paper. While the Prime Number Theorem provides the asymptotic density of primes, the fine-grained fluctuations in the prime-counting function, pi(x), are governed by the non-trivial zeros of the Riemann zeta function. Traditional approximations, such as the logarithmic integral li(x) and the Riemann prime counting function R(x), rely on smooth analytical models. However, the source paper arXiv:hal-00627233v3 introduces a fundamentally different approach by incorporating the discrete jumps of the Chebyshev function psi(x) into the counting model.
By defining a refined function eta_n(x) that utilizes the second Chebyshev function as its argument, the authors Michel Planat and Patrick Sole demonstrate a significant reduction in the maximum error of prime counting. This article analyzes the mathematical mechanics of this approach, specifically focusing on the "jumps" at prime powers and the identification of Chebyshev primes. We explore how these discrete structures provide a microlocal probe into the explicit formula, potentially offering a more direct path toward understanding the critical line of the Riemann zeta function.
The foundation of this analysis rests on the relationship between the prime counting function pi(x) and the second Chebyshev function psi(x). Recall that psi(x) is defined as the sum of the von Mangoldt function Lambda(n) for all n less than or equal to x. Unlike the smooth variable x, psi(x) is a step function that increases by log p at every prime power p^k. The Riemann prime counting function R(x) is traditionally expressed as the sum of (mu(n)/n) * li(x^(1/n)) for n from 1 to infinity, where mu is the Mobius function.
The innovation presented in arXiv:hal-00627233v3 is the construction of the approximation function:
eta_N(x) = sum_{n=1 to N} (mu(n)/n) * li[psi(x)^(1/n)] - pi(x)
This construction is significant because it replaces the smooth input x with the arithmetic input psi(x). Under the Riemann Hypothesis (RH), the difference psi(x) - x is bounded by O(x^1/2 * log^2 x). By using psi(x) within the logarithmic integral, the function eta_N(x) effectively "pre-conditions" the approximation with the oscillatory data of the primes themselves, dampening the fluctuations that usually lead to higher error margins in R(x).
The function li[psi(x)] is a piecewise constant function with discontinuities at primes and prime powers. The source paper defines the jump at a prime p as j_psi(p) = li[psi(p)] - li[psi(p-1)]. Since the jump in psi(x) at x=p is exactly log p, the jump in the composite function is approximately (log p) / log(psi(p)). Because psi(p) is approximately p, this ratio stays near 1. However, the exact value depends on the error term E(x) = psi(x) - x.
A prime p is classified as a Chebyshev prime if j_psi(p) is less than 1. This condition is equivalent to psi(p) being greater than p, meaning the prime distribution has locally "surpassed" its expected average. The paper identifies that these jumps are controlled by an infinite sequence of primes, which relates directly to Littlewood's theorem on the oscillations of pi(x) - li(x).
The technical superiority of eta_N(x) is evident in the numerical data provided in arXiv:hal-00627233v3. For ranges up to x = 10^5, the maximum error for the eta function with N=3 is significantly smaller than the error for the standard Riemann function. For instance, at x < 10,000, the maximum error for eta_3 is approximately 1.118, whereas the maximum error for R(x) - pi(x) is 6.174. This suggests that the inclusion of the Chebyshev function's jump data captures the "phase" of prime distribution more accurately than the smooth variable x.
Furthermore, the paper explores jumps at prime powers p^l. It is observed that the jump of the function eta_N(x) at x = p^l - 1 tends toward 1/l as p approaches infinity. This property aligns with the weighted counting of prime powers in the von Mangoldt function and suggests that the eta function naturally incorporates the logarithmic weights required for high-precision prime counting.
One promising direction is the application of Fourier analysis to the sequence of jumps Delta_eta(p) = eta_N(p) - eta_N(p-1). If the Riemann Hypothesis is true, the oscillatory component of psi(x) is a sum over the imaginary parts of the zeta zeros. Consequently, the sequence of jump deviations should exhibit spectral peaks at frequencies corresponding to the heights of the zeros on the critical line. Researchers could use windowed Fourier transforms on the prime-index sequence to detect these frequencies, providing a computational method for zero-detection that bypasses direct zeta function evaluation.
The density of Chebyshev primes (where j_psi(p) < 1) may be linked to the pair correlation of the zeros of the zeta function. Under the GUE (Gaussian Unitary Ensemble) hypothesis for zero spacing, the oscillations of psi(x) - x follow specific statistical patterns. Investigating whether Chebyshev primes cluster or repel could provide a new arithmetic test for the Montgomery pair correlation conjecture. Specifically, one could analyze the distribution of gaps between successive Chebyshev primes to see if they reflect the spectral gap of the zeta zeros.
(* Section: Jump Analysis of Refined Prime Counting Functions *)
(* Purpose: Implement eta_N(x) and analyze jump discontinuities at primes *)
(* Define the Chebyshev psi function *)
chebyshevPsi[x_] := Sum[MangoldtLambda[n], {n, 1, Floor[x]}];
(* Refined prime counting function eta_N(x) as defined in hal-00627233v3 *)
etaFunction[N_, x_] := Module[{sum = 0, n},
For[n = 1, n <= N, n++,
sum += (MoebiusMu[n]/n) * LogIntegral[chebyshevPsi[x]^(1/n)]
];
sum - PrimePi[x]
];
(* Calculate jump discontinuities at prime arguments *)
jumpAtPrime[N_, p_] := etaFunction[N, p] - etaFunction[N, p - 1];
(* Identify Chebyshev primes: primes where li[psi(p)] - li[psi(p-1)] < 1 *)
chebyshevJump[p_] := LogIntegral[chebyshevPsi[p]] - LogIntegral[chebyshevPsi[p - 1]];
identifyChebyshevPrimes[maxLimit_] := Module[{primes, chebList},
primes = Prime[Range[PrimePi[maxLimit]]];
chebList = Select[primes, chebyshevJump[#] < 1 &];
chebList
];
(* Example: Compare eta_3 error with Gauss and Riemann errors at x=5000 *)
errorComparison[x_] := Module[{gauss, riemann, refined},
gauss = N[LogIntegral[x] - PrimePi[x]];
riemann = N[Sum[LogIntegral[x^(1/n)]/n, {n, 1, 10}] - PrimePi[x]];
refined = N[etaFunction[3, x]];
{gauss, riemann, refined}
];
Print["Chebyshev Primes up to 200: ", identifyChebyshevPrimes[200]];
Print["Errors {Gauss, Riemann, Refined} at 5000: ", errorComparison[5000]];
The analysis of eta_N(x) and the Chebyshev primes offers a compelling refinement to the theory of prime distribution. By leveraging the discontinuous nature of the Chebyshev function, the source paper arXiv:hal-00627233v3 demonstrates that we can achieve tighter error bounds and a clearer view of the local oscillations in the prime counting function. The most promising avenue for future research lies in the spectral analysis of jump sequences, which may allow for a direct mapping between prime-index discontinuities and the zeros of the zeta function on the critical line. Continued exploration of these arithmetic jumps will likely yield deeper insights into the structural constraints imposed by the Riemann Hypothesis.