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Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights
Abstract
We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by s,u∈[0,1/2]. The error is Og(Y/(log Y)2), uniformly for Y1/5≤ R≤ Y1/3, including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.
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BibTeX
@misc{xiao2026selberg,
title = {{Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights}},
author = {Yixiu Xiao and Hongze Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24248},
eprint = {2609.24248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24248},
}
RIS
TY - UNPB
TI - Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights
AU - Yixiu Xiao
AU - Hongze Li
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24248
UR - https://arxiv.org/abs/2609.24248
AB - We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by s,u∈[0,1/2]. The error is O_g(Y/(log Y)^2), uniformly for Y^1/5≤ R≤ Y^1/3, including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.
ER -