Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)^k

Abstract

We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for (ζ'/ζ)^k on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function Λ^*k in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of Λ^*k over monic polynomials in 𝔽_q[T]; we denote its leading coefficient by δ_k(c), where the total trace degree and matrix size grow with a fixed ratio c. Under a weaker shifted moments conjecture and RH, we recover this δ_k(c) as the leading coefficient of the variance of short-interval sums of Λ^*k over the integers. It is the counterpart of γ_k(c), the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of ζ(s)^k replaced by those of (−ζ'/ζ)^k.

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BibTeX RIS

BibTeX

@misc{baluyot2026almost,
  title = {{Almost primes in short intervals and averages of long Dirichlet polynomial approximations to $(ζ'/ζ)^k$}},
  author = {Siegfred Baluyot and John Brian Conrey and Ayesha Irfan and Nina C. Snaith},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05000},
  eprint = {2611.05000},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05000},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)^k
AU  - Siegfred Baluyot
AU  - John Brian Conrey
AU  - Ayesha Irfan
AU  - Nina C. Snaith
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05000
UR  - https://arxiv.org/abs/2611.05000
AB  - We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for (ζ'/ζ)^k on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function Λ^*k in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of Λ^*k over monic polynomials in 𝔽_q[T]; we denote its leading coefficient by δ_k(c), where the total trace degree and matrix size grow with a fixed ratio c. Under a weaker shifted moments conjecture and RH, we recover this δ_k(c) as the leading coefficient of the variance of short-interval sums of Λ^*k over the integers. It is the counterpart of γ_k(c), the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of ζ(s)^k replaced by those of (−ζ'/ζ)^k.
ER  -

Abstract

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