Asymptotic properties of zeros of Riemann zeta function

2025/07/09 by de Reyna, Juan Arias, Meyer, Yves
#30D99 #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 52C23

paper · doi:10.48550/arxiv.2507.07253

Abstract

We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let (z_k)_{k∈ ℕ} be the sequence of nontrivial zeros of ζ(s) with positive imaginary part. We write z_k= 1/2+iτ_k (RH says that these τ_k are all real). Then the sequence (τ_k)_{k∈ ℕ}, satisfies the following asymptotic relation ∑_k∈ℕ(2x)/(x²+τ_k²)≃ \frac12log(x)/(2π)+∑_n=1^∞ (a_n)/(xⁿ), x→ +∞ where a_2n+1=2^-2n-2(8-E_2n), a_2n=(1-2^-2n+1)B_2n/(4n). Are there other sequences (α_k)_{k∈ ℕ}, of real or complex numbers enjoying this property? These problems are addressed in this note.

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