Riemann–von Mangoldt Approximation

• The Riemann–von Mangoldt approximation is a cornerstone of analytic number theory, providing explicit formulas that link prime-counting functions with the distribution of nontrivial zeta zeros.
• It refines classical prime-counting estimates through the use of Chebyshev functions and oscillatory corrections, enhancing numerical precision even at extreme scales.
• Extensions of the approximation establish deep ties with spectral analysis and quantum mechanics, offering alternative formulations related to the Riemann Hypothesis.

The Riemann–von Mangoldt approximation occupies a central position in analytic number theory, providing a precise quantification of both the distribution of prime numbers via the “prime-counting function” and the nontrivial zeros of the Riemann zeta function. The approximation consists of explicit formulas—both for the zero counting function and for arithmetic functions counting primes and prime powers—which are refined using oscillatory terms reflecting contributions from the critical zeros of $\zeta(s)$. Modern developments extend the classical formula in various directions: through improved prime-counting fits, ergodic and spectral interpretations, connections with quantum mechanics, and equivalent reformulations relating to the Riemann Hypothesis.

1. Classical Formula and Explicit Corrections

The original Riemann–von Mangoldt formula expresses the number of nontrivial zeros with $0 < \operatorname{Im}(s) < T$ as

$$N(T) = \frac{T}{2\pi} \log \left( \frac{T}{2\pi} \right) - \frac{T}{2\pi} + O(\log T).$$

For prime counting, Riemann introduced a function

$$\operatorname{Ri}^{(N)}(x) = \sum_{n=1}^N \frac{\mu(n)}{n}\operatorname{Li}\left(x^{1/n}\right),$$

where $\mu(n)$ is the Möbius function, $\operatorname{Li}$ is the logarithmic integral, and $N$ is large. The explicit formula, however, requires the subtraction of oscillatory corrections involving the critical zeros $\rho$ of $\zeta(s)$,

$$\pi(x) \sim \operatorname{Ri}^{(N)}(x) - \sum_{\rho}\operatorname{Ri}(x^\rho).$$

This sum decays asymptotically, but is responsible for the subtle fluctuations—the “thickening and thinning”—of the prime density as compared to a continuous interpolation like $\operatorname{li}(x)$ (Planat et al., 2014).

2. Chebyshev Function, von Mangoldt’s Explicit Formula, and Refined Fits

The Chebyshev function $\psi(x) = \sum_{p^k \le x} \log p$, summing over all prime powers, admits the von Mangoldt explicit formula: $$\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \frac{\zeta'}{\zeta}(0) - \frac{1}{2}\log (1-x^{-2}), \quad x>1,$$ making transparent the oscillatory contribution from each critical zero $\rho$. Leveraging this, recent refinements propose to replace the argument $x$ by $\psi(x)$ in the Riemann counting function, yielding the approximation

$\pi(x)\approx \operatorname{Ri}^{(3)}[\psi(x)],$

which numerically produces three to four new exact digits compared to $\operatorname{li}(x)$ even at extreme ranges such as $x=10^{20}$, using data from the first $2\times 10^6$ zeros (Planat et al., 2014). The Gram formula, with $\psi(x)$ as its argument, yields an equivalent fit.

3. Oscillatory Terms, Zero Distribution, and Equivalents to Riemann Hypothesis

Both $\psi(x)$ and the prime counting corrections are dominated by sums over the zeros $\rho$. The regularity of $\psi(x)$, as forced by the location of the zeros, directly impacts the sharpness of the approximations; the remaining error mirrors the “regularity” imposed by the Riemann Hypothesis (RH). For instance, modified Robin’s criterion asserts

$$\epsilon_{\psi(x)} = \operatorname{li}(\psi(x)) - \pi(x) > 0$$

if and only if RH holds, making the precise structure of the zero-induced corrections functionally equivalent to RH in analytic contexts.

4. Spectral and Quantum Perspectives

The connection to spectral theory and quantum mechanics is developed through both Fourier analysis and quantum Hamiltonian construction. For example, Fourier transforms of modified von Mangoldt functions—designed to “mark” zero positions—reveal that the nontrivial zeros (when viewed as a sequence of spikes) reconstruct via superpositions of harmonic waves, leading to periodic and spiral structures in the frequency domain (Csoka, 2017). This spectral regularity is reminiscent of the Hilbert–Pólya conjecture that the zeros correspond to eigenvalues of a (Hermitian) operator.

Quantum analogues, such as the Berry–Keating $xp$ Hamiltonian, have been “polymerized” to introduce a scale parameter $\mu_0$,

$p_\mu = \frac{1}{\mu_0}\sin(\mu_0 p),$

yielding discrete energy spectra and a semiclassical state count

$$N_{\text{poly}}(E) \sim \frac{E}{2\pi}\log\frac{E}{2\pi} + \text{corrections},$$

with corrections directly tied to $\mu_0$ and the energy, capturing fluctuation behavior analogous to that induced by the zeros in the classical formula (Berra-Montiel et al., 2016). Selfadjointness and spectral analysis of such operators reinforce the physical reality of the zero spectrum.

5. Extension to Arithmetic Functions, Twisted Sums, and the Selberg Class

The explicit formula and the Riemann–von Mangoldt approximation have been generalized to the generalized von Mangoldt functions $\Lambda_k(n)$ and to convolutions $\Lambda^{k}$, whose twisted sums uniformly approximate prime density and provide reformulations of RH (Banks et al., 2022): $\sum_{n\le x}\Lambda_k(n)n^{-iy}$ has a main term and an error $O(x^{1/2}(x+|y|)^\epsilon)$ if and only if RH holds. Further, for $a$-points of derivatives of the zeta and Selberg class L-functions, zero-density theorems and Riemann–von Mangoldt style counting formulas quantify the distribution of zeros and $a$-points across critical strips, frequently with error terms proportional to $O(\log T)$ (Onozuka, 2016, Sourmelidis et al., 2020).

6. Applications and Computational Implications

Practical ramifications include robust numerical evaluation of $\pi(x)$ with precision surpassing classical methods (by incorporating millions of zeros), the design of weighted summatory functions offering alternative formulations of the Riemann Hypothesis based on sign constancy or averaged inequalities (Suzuki, 11 Nov 2024), and improved bounds in exponential sums relevant to the quasi-Riemann Hypothesis, where error terms precisely reflect known zero-free regions and density results (Ren et al., 2022). In ergodic theory, cubic averages with von Mangoldt weights converge almost surely, reflecting random-like distribution properties in prime patterns (Abdalaoui et al., 2018).

7. Further Directions, Generalizations, and Open Problems

Beurling generalized numbers extend the explicit formula and the Riemann–von Mangoldt approximation to arithmetical semigroups, necessitating analytic continuation, zero-density, and explicit contour constructions accommodating more abstract “prime” systems (Révész, 2021). In quantum and spectral settings, the necessity of including a large number of zeros for sharp approximations (as supported by Heisenberg's inequality) underscores computational barriers and theoretical thresholds (Balanzario et al., 2023). Finally, chaotic dynamics derived from the Riemann–von Mangoldt structure offer insights into unpredictability and operator Hermiticity, further informing potential bridges to quantum models and the Hilbert–Pólya scenario (Rafik, 31 Mar 2024).

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The Riemann–von Mangoldt approximation thus forms the backbone for explicit prime counting, spectral analysis, quantum mechanical analogies, and alternative formulations of deep conjectures such as RH, while its extensions and refinements continue to drive developments across analytic number theory, computational mathematics, and related fields.