[Paper Review] Some sums over the non-trivial zeros of the Riemann zeta function

Jesús Guillera|arXiv (Cornell University)|Jul 22, 2013

Analytic Number Theory Research8 references3 citations

TL;DR

This paper derives new convergent asymptotic expansions and exact representations for arithmetic functions—such as the Mangoldt, Möbius, and Euler's totient functions—using sums over the non-trivial zeros of the Riemann zeta function. It establishes a smooth, invariant representation of the Mangoldt function under finite zero truncation, analogous to Landau's formula, and provides new identities under the Riemann Hypothesis, including a novel limit evaluation involving zeta derivatives and hyperbolic functions.

ABSTRACT

We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cramér, and also new representations for some arithmetical functions in terms of the non-trivial zeros.

Motivation & Objective

• To derive new convergent representations of arithmetic functions using sums over non-trivial zeros of the Riemann zeta function.
• To generalize Landau's formula for the Mangoldt function by introducing a smooth, invariant representation under finite zero truncation.
• To extend these results to the Möbius and Euler's totient functions under the Riemann Hypothesis and simplicity of zeros.
• To provide explicit asymptotic expansions related to Cramér's work, improving on earlier approximations.
• To establish a new limit evaluation involving zeta values and derivatives at critical zeros, assuming the Riemann Hypothesis.

Proposed method

• Derives a new identity for the Mangoldt function using a sum over non-trivial zeros with hyperbolic sine and cosine weights, involving a cotangent factor that captures the asymptotic limit as x→π⁻.
• Uses the functional equation and Hadamard product representation of the xi function to relate zeta zeros to arithmetic functions.
• Applies complex analysis techniques, including bounds on the gamma and zeta functions, to justify convergence and asymptotic behavior.
• Employs the Riemann-Siegel theta function and exact zero-counting formula to analyze the distribution of non-trivial zeros.
• Derives representations for Möbius and Euler's totient functions via analytic continuation and series manipulation of zeta-related Dirichlet series.
• Validates results numerically using SageMath with Odlyzko's database of zeta zeros, comparing computed sums to known arithmetic functions.

Experimental results

Research questions

• RQ1 Can a smooth, convergent representation of the Mangoldt function be constructed using non-trivial zeta zeros that remains invariant under finite truncation, similar to Landau's formula?
• RQ2 What is the precise asymptotic behavior of sums over non-trivial zeros involving sinh(xγ)/sinh(πγ) and trigonometric functions of log t, as x→π⁻?
• RQ3 How can the Möbius and Euler's totient functions be represented in terms of zeta zeros and their derivatives under the Riemann Hypothesis?
• RQ4 What new convergent asymptotic expansions can be derived that refine or extend Cramér’s work on the distribution of primes?
• RQ5 What is the value of the limit expression involving zeta values and derivatives at critical zeros, and how does it relate to arithmetic functions?

Key findings

• A new representation for the Mangoldt function is derived: Λ(t) = -4π√t lim_{x→π⁻} [cot(x/2) Σ_{γ>0} (sinh(xγ)/sinh(πγ)) cos(γ log t) ] under the Riemann Hypothesis.
• The formula for Λ(t) is smooth and invariant under finite truncation of the sum over zeros, unlike previous discrete sums.
• A new identity is established for Euler's totient function: φ(t) = 4π√t cot(x/2) Σ_{γ>0} [Re(ζ(-1/2+iγ)/ζ’(1/2+iγ)) (sinh(xγ)/sinh(πγ)) cos(γ log t) - Im(...) (cosh(xγ)/sinh(πγ)) sin(γ log t)] + lower-order terms.
• A novel limit evaluation is proven: lim_{x→π⁻} Σ_{γ>0} [sinh(xγ)/sinh(πγ)] ( (log 2)/√2 cos(γ log t) - Λ(t)/√t cos(γ log 2) ) = (log 2)/√2 (√t/2 - 1/(2(t²-1)√t)) - Λ(t)/√t (5√2)/12, assuming the Riemann Hypothesis.
• The paper provides a new convergent asymptotic expansion for the sum over non-trivial zeros that improves on Cramér’s earlier results.
• Numerical validation via SageMath using Odlyzko’s zeta zero database confirms the convergence and accuracy of the derived formulas for φ(t) and Λ(t) over the range t ∈ [2,26].