[Paper Review] On the Order Estimates for Specific Functions of $ζ(s)$ and its Contribution towards the Analytic Proof of The Prime Number Theorem

Subham De|arXiv (Cornell University)|Aug 28, 2023

Meromorphic and Entire FunctionsMathematics3 citations

TL;DR

This paper provides a comprehensive analytic proof of the Prime Number Theorem (PNT) by establishing precise order estimates for key functions of the Riemann zeta function ζ(s), particularly focusing on the logarithmic derivative −ζ′(s)/ζ(s). It employs contour integration, complex analysis, and the Riemann-Lebesgue Lemma to show that ψ(x) ∼ x as x → ∞, which implies π(x) ∼ x/log x, thereby completing the analytic proof of PNT via the non-vanishing of ζ(s) on the line Re(s) = 1.

ABSTRACT

This article provides a proof of the famous extit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second extit{Chebyshev Function} $ψ(x)$. We shall be extensively using complex analytic techniques in addition to certain meromorphic properties of the extit{Reimann Zeta Function} $ζ(s)$ and its extit{Analytic Continuation Property} a priori using Riemann's Functional Equation in order to establish our desired result.

Motivation & Objective

• To provide a rigorous analytic proof of the Prime Number Theorem (PNT) using the Riemann zeta function and its properties.
• To establish sharp order estimates for |ζ(s)|, |ζ′(s)|, |1/ζ(s)|, and |ζ′(s)/ζ(s)| near the line Re(s) = 1.
• To demonstrate the non-vanishing of ζ(s) on Re(s) = 1, a critical step in the analytic proof of PNT.
• To derive the asymptotic behavior ψ(x) ∼ x using contour integration and the Riemann-Lebesgue Lemma.
• To connect the behavior of ψ₁(x) to the zeta function via integral representations and justify the shift of the integration contour to Re(s) = 1.

Proposed method

• Derive a contour integral representation of ψ₁(x)/x² using the function h(s) = [−ζ′(s)/ζ(s) − 1/(s−1)] / [s(s+1]) for Re(s) > 1.
• Apply Cauchy’s Theorem to shift the contour of integration from Re(s) = c > 1 to Re(s) = 1, showing the integrals over horizontal segments vanish as T → ∞.
• Establish uniform bounds for |h(s)| on Re(s) = 1 using known estimates |ζ′(s)/ζ(s)| ≤ M(log T)^9 for Re(s) ≥ 1 and |t| ≥ e.
• Prove absolute integrability of |h(1+it)| over ℝ, enabling application of the Riemann-Lebesgue Lemma.
• Use the Riemann-Lebesgue Lemma to deduce that ψ₁(x)/x² → 1/2 as x → ∞, leading to ψ₁(x) ∼ x²/2.
• Leverage the relation between ψ₁(x) and ψ(x) to conclude ψ(x) ∼ x, which is equivalent to the Prime Number Theorem.

Experimental results

Research questions

• RQ1 How can precise order estimates for ζ(s) and its logarithmic derivative be used to prove the non-vanishing of ζ(s) on Re(s) = 1?
• RQ2 What is the role of contour integration and analytic continuation in deriving the asymptotic behavior of the Chebyshev function ψ(x)?
• RQ3 How does the convergence of ∫|h(1+it)|dt enable the application of the Riemann-Lebesgue Lemma to deduce ψ₁(x) ∼ x²/2?
• RQ4 What is the connection between the analytic behavior of h(s) and the equivalence of ψ(x) ∼ x to the Prime Number Theorem?
• RQ5 Can the analytic proof of PNT be completed using only order estimates and contour integration, without relying on elementary methods?

Key findings

• The function h(s) = [−ζ′(s)/ζ(s) − 1/(s−1)] / [s(s+1)] is holomorphic and absolutely integrable on Re(s) = 1, with |h(1+it)| ≤ M(log t)^9 / t^2 for |t| ≥ e.
• The integral ∫_{−∞}^{∞} |h(1+it)| dt converges, which allows the application of the Riemann-Lebesgue Lemma to the Fourier-type integral representation of ψ₁(x)/x².
• As x → ∞, the integral representation implies ψ₁(x) ∼ x²/2, which leads to ψ₁(x) = x²/2 + o(x²).
• The relation between ψ₁(x) and ψ(x) via integration by parts yields ψ(x) ∼ x as x → ∞, which is equivalent to the Prime Number Theorem.
• The non-vanishing of ζ(s) on Re(s) = 1 is essential and is established via the analyticity of −ζ′(s)/ζ(s) − 1/(s−1) at s = 1.
• The proof demonstrates that π(x) ∼ x/log x follows from ψ(x) ∼ x, completing the analytic proof of the Prime Number Theorem.