[Paper Review] Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

Igor E. Shparlinski, Yixiu Xiao|arXiv (Cornell University)|Jan 15, 2026

Analytic Number Theory Research0 citations

TL;DR

The paper derives nontrivial upper bounds for shifted bilinear sums involving Salié sums modulo a large prime and uses them to study the distribution of modular square roots of shifted primes, yielding asymptotics and discrepancy bounds for primes up to P.

ABSTRACT

We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b ot \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b ot \equiv 0 \bmod q$ exhibits some new difficulties.

Motivation & Objective

• Motivate and study upper bounds for Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime q.
• Develop bounds that enable analysis of the distribution of solutions to x^2 ≡ a p + b (mod q) for primes p ≤ P.
• Translate bounds on bilinear sums into results about the distribution of modular square roots of shifted primes.
• Explore both sums over integers and primes, including smooth/weighted variants and hyperbolic domain restrictions.

Proposed method

• Define Salié sums S(t; q) and relate them to quadratic congruences via S(t;q) = ε_q q^{1/2} sum_{x^2 ≡ t (mod q)} e_q(2x).
• Bound Type-I and Type-II shifted bilinear sums W_{a,b,λ} and V_{a,b,λ} using exponential-sum techniques, Poisson summation, and Weil bounds.
• Apply Vaughan’s identity to sums over primes to derive S_{a,b,λ}(P) bounds across dyadic ranges of P.
• Use amplification, Fourier/Poisson analysis, and exponential-sum bounds to control bilinear forms and smoothed variants (V, U, T sums).
• Derive corollaries on distribution discrepancies N_{a,b}(H,P) via Erdős–Turán type inequalities.
• Obtain specialized bounds for smooth/weighted sums and hyperbolic domains to handle shifted-prime square roots.

Experimental results

Research questions

• RQ1 What are nontrivial upper bounds for shifted bilinear sums of Salié sums modulo a large prime q?
• RQ2 How do these bounds translate into the distribution of modular square roots for shifted primes, i.e., x^2 ≡ a p + b (mod q) with p primes?
• RQ3 What are the optimal ranges for M, N, P, and related parameters where the bilinear bounds remain nontrivial?
• RQ4 Can smoothing, hyperbolic domain restrictions, and prime-sum techniques yield asymptotics or discrepancy bounds for R_{a,b}(P) (the set of square roots for shifted primes) in intervals?
• RQ5 Do the results extend to Type-I sums with shifts and to smoothed/spectral variants beyond the diagonal case?

Key findings

• Theorem 2.1: U_{a,b,λ}(m,N) ≪ q^{1/2} log q for gcd(amλ,q)=1.
• Theorem 2.2: V_{a,b,λ}(α; M, N) ≪ sqrt(||α||_1 ||α||_2) M^{1/12} N^{7/12} q^{1/4+o(1)} under M ≤ q, MN ≤ q^{3/2}, M ≤ N^2.
• Theorem 2.3: A smoothed variant V_{a,b,λ}(α, φ; M, N) with bounds depending on M, N, q, and smoothness parameters; applicable when MN ≪ q.
• Theorem 2.5: W_{a,b,λ}(α,β; M, N) ≪ ||α||_2 ||β||_∞ (M^{1/2} N^{1/2} + M^{1/2} N q^{-1/4} + N q^{1/4} (log q)^{1/2}).
• Theorem 2.10 and Theorem 2.11: Bounds for sums over primes S_{a,b,λ}(P) in various P-ranges, including P^{13/18} q^{1/4}, P^{5/6} q^{1/8}, P^{2/3} q^{1/3}, P^{5/6} q^{1/12}, and P q^{-1/4}, with an improved bound for q^{3/4} ≤ P ≤ q^{3/2} giving P^{7/9+o(1)} q^{1/6}.
• Corollary 2.13: Erdős–Turán-type discrepancy bound for N_{a,b}(H,P) in terms of the same P-intervals, showing nontrivial bounds for P in q^{3/4+ε}..