On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function
Abstract
An explicit formula for the prime-counting function π(x), usually attributed to Riemann and von Mangoldt, is prominently stated as the equation π(x)=R(x)−∑ρR(xρ), where the sum runs over all zeros ρ of the Riemann ζ-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, ΣRT(x):=∑0<|ℑ m(ρ)|≤ T R(xρ) converge as T→∞. Writing Θ:=sup{ℜ e(ρ): ζ(ρ)=0, 0<ℜ e(ρ)<1}, for what has recently been called ``Riemann's constant'', we prove that, for every fixed x>1 and every θ<Θ, the sums ΣRT(x) are not O(Tθ). As a consequence, limsupT→∞|ΣRT(x)|=∞ and ∑ρR(xρ) diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result -- and hence divergence -- for the contribution of the trivial zeros to ∑ρR(xρ). In both cases the exponent Θ is proved to be sharp.
Related
- Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis
- On the counting function of square-full numbers
- Explicit height bounds on modular polynomials for the elliptic j-invariant, cube root of j, and Weber modular function f
- Explicit bounds for Dickman's function
- Explicit Exponential Sum Estimates and Approximate Functional Equations for the Zeta Function
BibTeX
@misc{grobner2026divergence,
title = {{On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function}},
author = {Harald Grobner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.02713},
eprint = {2609.02713},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02713},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function
AU - Harald Grobner
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.02713
UR - https://arxiv.org/abs/2609.02713
AB - An explicit formula for the prime-counting function π(x), usually attributed to Riemann and von Mangoldt, is prominently stated as the equation π(x)=R(x)−∑_ρR(x^ρ), where the sum runs over all zeros ρ of the Riemann ζ-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, ΣR_T(x):=∑_0<|ℑ m(ρ)|≤ T R(x^ρ) converge as T→∞. Writing Θ:=sup{ℜ e(ρ): ζ(ρ)=0, 0<ℜ e(ρ)<1}, for what has recently been called ``Riemann's constant'', we prove that, for every fixed x>1 and every θ<Θ, the sums ΣR_T(x) are not O(T^θ). As a consequence, limsup_T→∞|ΣR_T(x)|=∞ and ∑_ρR(x^ρ) diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result -- and hence divergence -- for the contribution of the trivial zeros to ∑_ρR(x^ρ). In both cases the exponent Θ is proved to be sharp.
ER -