Mean Estimates for Short Polynomial Exponential Sums over Primes
Abstract
Let n≥2 be a fixed integer, and let K,x,y be positive integers satisfying 2≤ K≤ y<x. Consider a polynomial f(u)=αun+αn−1un−1+⋯+α1u+α0 with real coefficients. We study the quantity Vn(K,x,y) = ∑k=1K | ∑x−y<p≤ xe(kf(p)) |, e(t)=e2πit, where the inner sum is over primes. For α=a/q+θ/q2, a∈ℤ, q∈ℕ, (a,q)=1, |θ|≤1, we establish the bound Vn(K,x,y) &≪ KyBiggl[ 1/√(Klog(2y))+ min{ Δ1/2n (log(2y))(n2−1)/2n, Δ1/(3· 2n−2) (log(2y))(n3−1)/(3· 2n−1) } Biggr], where Δ= 1/q+1/y+q/Kyn. The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fejér kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.
Related
- Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
- A Bombieri-Vinogradov theorem for exponential sums over products of k primes
- Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
- Polynomial configurations and pointwise averages along Piatetski--Shapiro primes
- Polynomial bounds for the stability of associated primes of monomial ideals
BibTeX
@misc{mirzoabdughafurov2026mean,
title = {{Mean Estimates for Short Polynomial Exponential Sums over Primes}},
author = {Karimjon Ibrohimjonovich Mirzoabdughafurov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08886},
eprint = {2610.08886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08886},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mean Estimates for Short Polynomial Exponential Sums over Primes
AU - Karimjon Ibrohimjonovich Mirzoabdughafurov
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08886
UR - https://arxiv.org/abs/2610.08886
AB - Let n≥2 be a fixed integer, and let K,x,y be positive integers satisfying 2≤ K≤ y<x. Consider a polynomial f(u)=αu^n+α_n−1u^n−1+⋯+α_1u+α_0 with real coefficients. We study the quantity V_n(K,x,y) = ∑_k=1^K | ∑_x−y<p≤ xe(kf(p)) |, e(t)=e^2πit, where the inner sum is over primes. For α=a/q+θ/q^2, a∈ℤ, q∈ℕ, (a,q)=1, |θ|≤1, we establish the bound V_n(K,x,y) &≪ KyBiggl[ 1/√(Klog(2y))+ min{ Δ^1/2^n (log(2y))^(n^2−1)/2^n, Δ^1/(3· 2^n−2) (log(2y))^(n^3−1)/(3· 2^n−1) } Biggr], where Δ= 1/q+1/y+q/Ky^n. The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fejér kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.
ER -