Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 20 min ago
PreprintCC BY 4.0arXiv

Sums of von Mangoldt convolutions via trace-sum averages over U(N)

  • Ayesha Irfan
Open at sourcePDFDOICite

Abstract

We study piecewise polynomial functions δk(c) that appear in the conjectured asymptotics for the variance of short-interval sums of von Mangoldt convolutions Λ*k. By a theorem of Kuperberg and Lalín, a unitary matrix integral involving a sum of trace products governs the large-q limit of the function-field analogue of this variance. We identify δk(c) as the leading-order coefficient of this integral when the total trace degree and matrix size grow with a fixed ratio c. Through an asymptotic analysis of two lattice-point representations for the matrix integral, we obtain explicit forms of δk(c). A finite spline expansion and a contour-integral representation expressing δk(c) as the inverse Laplace transform of a two-weight block Hankel determinant are also obtained.

Related

  1. On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices
    Ayesha Irfan
    arXivPreprintCC BY 4.0Full textMathematical physicsProbabilityarXiv
  2. Odd order cases of Chowla's averages over arbitrary rectangles
    Qiang Huo, Zhengxing Lian, Kangbo Ouyang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryDynamical systemsarXiv
  3. Rigidity of Averages over the Two Largest Prime Factors
    Dijia Chen
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Matrix Kloosterman sums and product-trace estimates for semisimple algebras
    Xuejun Guo, Chen Lin, Chenhao Tang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. The distribution and the structure of the maximum of partial sums in families of trace functions
    Kilian Lebreton
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{irfan2026sums,
  title = {{Sums of von Mangoldt convolutions via trace-sum averages over $U(N)$}},
  author = {Ayesha Irfan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33953},
  eprint = {2609.33953},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33953},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sums of von Mangoldt convolutions via trace-sum averages over U(N)
AU  - Ayesha Irfan
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33953
UR  - https://arxiv.org/abs/2609.33953
AB  - We study piecewise polynomial functions δ_k(c) that appear in the conjectured asymptotics for the variance of short-interval sums of von Mangoldt convolutions Λ^*k. By a theorem of Kuperberg and Lalín, a unitary matrix integral involving a sum of trace products governs the large-q limit of the function-field analogue of this variance. We identify δ_k(c) as the leading-order coefficient of this integral when the total trace degree and matrix size grow with a fixed ratio c. Through an asymptotic analysis of two lattice-point representations for the matrix integral, we obtain explicit forms of δ_k(c). A finite spline expansion and a contour-integral representation expressing δ_k(c) as the inverse Laplace transform of a two-weight block Hankel determinant are also obtained.
ER  -