Skip to content
DraconiticResearch
Search Sources API
186,631 studies31 sourcesUpdated 49 min ago
PreprintarXiv licencearXiv

Explicit bounds for the prime number theorem

  • Andrew Fiori
  • Mikko Jaskari
Open at sourcePDFDOICite

Abstract

We study the problem of finding tight explicit bounds on sums over zeros of L-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for xgeqslant21 we have |ψ(x)−x| < 0.239 x exp(−0.1982767 (1+logloglog(x)/(15loglog(x)))log(x)3/5loglog(x)−1/5). Combined with the zero-free half-plane ℜ(s)>7/8 announced by OpenAI in October 2026, the same methods give |ψ(x)−x| < 0.04934 x3/4log(x)2 + 0.02636425 x7/8 for xgeqslant 21.

Related

  1. On a variant of the Piatetski-Shapiro prime number theorem
    Linji Long, Jinjiang Li, Min Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Density Bounds in Hua's Theorem for Prime Squares
    Michael T. Lacey, Hamed Mousavi, Yaghoub Rahimi +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
    Pierre Tchamitchian
    Journal of Number TheoryPeer-reviewedCC BYFull textGeometry & topologyHAL
  4. On the hyperbolic prime number theorem
    Alisa Sedunova
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. The Boolean Prime Ideal Theorem and Vitali Sets
    Jindrich Zapletal
    arXivPreprintCC BY 4.0Full textLogic & foundationsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fiori2026explicit,
  title = {{Explicit bounds for the prime number theorem}},
  author = {Andrew Fiori and Mikko Jaskari},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.23222},
  eprint = {2609.23222},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23222},
}
RIS
TY  - UNPB
TI  - Explicit bounds for the prime number theorem
AU  - Andrew Fiori
AU  - Mikko Jaskari
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.23222
UR  - https://arxiv.org/abs/2609.23222
AB  - We study the problem of finding tight explicit bounds on sums over zeros of L-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for xgeqslant21 we have |ψ(x)−x| < 0.239 x exp(−0.1982767 (1+logloglog(x)/(15loglog(x)))log(x)^3/5loglog(x)^−1/5). Combined with the zero-free half-plane ℜ(s)>7/8 announced by OpenAI in October 2026, the same methods give |ψ(x)−x| < 0.04934 x^3/4log(x)^2 + 0.02636425 x^7/8 for xgeqslant 21.
ER  -