[Paper Review] Some identities for the Riemann zeta-function II

Aleksandar Ivić|arXiv (Cornell University)|Jun 11, 2005

Analytic Number Theory Research8 references3 citations

TL;DR

This paper establishes new integral identities connecting the Riemann zeta function ζ(s) to iterated fractional part integrals φₙ(x), generalizing the classical identity ζ(s)/s = −∫₀^∞ {x}x^{-1-s}dx. Using Mellin transforms and Parseval-type formulas, it derives exact representations for ζⁿ(s)/(-s)ⁿ and |ζ(σ+it)|²ⁿ/(σ²+t²)ⁿ, linking zeta moments to L² norms of φₙ(x), with applications to the Riemann Hypothesis and asymptotic expansions via the gamma function.

ABSTRACT

Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and Müntz's identity.

Motivation & Objective

• To generalize the classical identity ζ(s)/s = −∫₀^∞ {x}x^{-1-s}dx to higher powers of ζ(s) using iterated integrals of the fractional part function.
• To establish a connection between the n-th power of the zeta function and the Mellin transform of iterated fractional part functions φₙ(x), extending the classical framework.
• To derive explicit integral representations for the moments |ζ(σ+it)|²ⁿ/(σ²+t²)ⁿ using the L² norm of φₙ(x), enabling new approaches to the Riemann Hypothesis.
• To analyze the asymptotic behavior of the iterated functions φₙ(x) and use them to derive asymptotic expansions for integrals involving e^{-x/T} and ζ(s).
• To apply Müntz’s identity with a Gaussian kernel f(x) = e^{-πx²} to derive a new spectral identity involving the zeta function and the Jacobi theta function.

Proposed method

• Define iterated functions φₙ(x) recursively via φ₁(x) = {x} and φₙ(x) = ∫₀^∞ {u}φₙ₋₁(x/u)(du/u) for n ≥ 2.
• Use Mellin transform techniques and the Parseval identity for Mellin transforms to relate ζⁿ(s)/(-s)ⁿ to ∫₀^∞ φₙ(x)x^{-1-s}dx.
• Apply the residue theorem to the integral ∫_{c-i∞}^{c+i∞} ζ(s)/s · T^s Γ(s) ds to derive asymptotic expansions for ∫₀^∞ {x}/x · e^{-x/T} dx as T → ∞.
• Use the functional equation of the Jacobi theta function θ(z) = ∑_{n=-∞}^∞ e^{-πn²z} to transform the integral representation of the zeta moment identity.
• Apply the functional equation of ζ(s) to show invariance under σ → 1−σ and derive a symmetric form of the moment identity in terms of uθ(u²)−1−u.
• Use the known Mellin transform of e^{-πx²} and the Müntz operator Pf(x) to derive the identity involving ∫ |ζ(σ+it)Γ(σ/2 + it/2)|² dt.

Experimental results

Research questions

• RQ1 Can the classical identity ζ(s)/s = −∫₀^∞ {x}x^{-1-s}dx be generalized to higher powers of ζ(s) via iterated integrals of the fractional part?
• RQ2 What is the asymptotic behavior of the iterated functions φₙ(x) defined by repeated convolution with {u}/u?
• RQ3 How can the L² norm of φₙ(x) be related to the moments |ζ(σ+it)|²ⁿ/(σ²+t²)ⁿ via Mellin transform techniques?
• RQ4 Can asymptotic expansions for ∫₀^∞ {x}/x · e^{-x/T} dx be derived using the residue theorem and the Mellin transform of ζ(s)/s?
• RQ5 What spectral identities emerge when applying Müntz’s formula with the Gaussian kernel f(x) = e^{-πx²}?

Key findings

• The identity ζⁿ(s)/(-s)ⁿ = ∫₀^∞ φₙ(x)x^{-1-s}dx holds for 0 < σ < 1 and all integers n ≥ 1, generalizing the classical case n=1.
• The asymptotic behavior of φₙ(x) is φₙ(x) = x/(n−1)! · logⁿ⁻¹(1/x) + O(x logⁿ⁻²(1/x)) as x → 0⁺, and φₙ(x) = O(logⁿ⁻¹(x+1)) for x ≥ 1.
• The moment identity 1/(2π)∫_{-∞}^∞ |ζ(σ+it)|²ⁿ/(σ²+t²)ⁿ dt = ∫₀^∞ φₙ²(x)x^{-1-2σ} dx holds for 0 < σ < 1 and all n ≥ 1.
• The integral ∫₀^∞ {x}/x · e^{-x/T} dx has the asymptotic expansion ½ log T − ½γ + ½ log(2π) + ∑_{m=1}^M ζ(1−2m)/((2m−1)!(1−2m)) T^{1−2m} + O_M(T^{-1−2M}) as T → ∞.
• For f(x) = e^{-πx²}, the identity ∫_{-∞}^∞ |ζ(σ+it)Γ(σ/2 + it/2)|² dt = 2π^{σ+1} ∫₀^∞ (uθ(u²)−1−u)² u^{2σ−3} du holds for 0 < σ < 1.
• The resulting integral converges absolutely at both 0 and ∞, with convergence at ∞ due to exponential decay and at 0 due to θ(1/u²) = 1 + O(e^{-u^{-2}}).