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Short sums of the Möbius function over function fields
Abstract
We develop a new method which allows us to estimate, with a power saving, sums of the M{ö}bius function over function fields 𝔽p(X). The sums run over monic polynomials with all other coefficients in short intervals, of length below the Pólya--Vinogradov threshold p1/2. A classical result of L. Stickelberger (1898) reduces this question to bounding character sums with polynomial discriminants which are inhomogeneous multivariate polynomials. Variants of Burgess's method with non-linear shifts {\it {à} la carte} are introduced to go beyond the Pólya--Vinogradov range, combined with a stratification result of J. Xu (2020) for complete character sums.
Related
- Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function
- Primes and the Möbius function over quadratic polynomials
- Chowla Conjecture for the truncated Möbius function over 𝔽q[t]
- Discorrelation of the Möbius function with linear phases in short intervals
- A1 - Coulomb branches over finite fields and Selberg Character sums
BibTeX
@misc{fouvry2026short,
title = {{Short sums of the Möbius function over function fields}},
author = {Etienne Fouvry and Igor E. Shparlinski and Ping Xi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28822},
eprint = {2610.28822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28822},
}
RIS
TY - UNPB
TI - Short sums of the Möbius function over function fields
AU - Etienne Fouvry
AU - Igor E. Shparlinski
AU - Ping Xi
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28822
UR - https://arxiv.org/abs/2610.28822
AB - We develop a new method which allows us to estimate, with a power saving, sums of the M{ö}bius function over function fields 𝔽_p(X). The sums run over monic polynomials with all other coefficients in short intervals, of length below the Pólya--Vinogradov threshold p^1/2. A classical result of L. Stickelberger (1898) reduces this question to bounding character sums with polynomial discriminants which are inhomogeneous multivariate polynomials. Variants of Burgess's method with non-linear shifts {\it {à} la carte} are introduced to go beyond the Pólya--Vinogradov range, combined with a stratification result of J. Xu (2020) for complete character sums.
ER -