A problem of D. H. Lehmer in short intervals. II

Abstract: A problem of D. H. Lehmer suggests to study the number of integers, each of which has different parity from its multiplicative inverse modulo $q$. For large prime $q$, we obtain an asymptotic formula for the number of such integers up to $N$, where $N$ is a bit smaller than $q<sup>{1/2}$. This beats the barrier $q<sup>{1/2}$ in the prime modulus case. An estimate for the second moment of the error term on average over $q$ is also established. The main inputs are estimates for several bilinear forms with Kloosterman fractions.

• The paper establishes an asymptotic formula for L(N, q), breaking the q^(1/2) barrier in short intervals for prime moduli.
• It leverages refined estimates on incomplete Kloosterman sums and advanced bilinear form analysis to control error terms.
• The work has implications for cryptographic applications and advances analytic techniques in modular arithmetic.

Asymptotics for Lehmer's Problem in Short Intervals for Prime Moduli

Introduction

The paper "A problem of D. H. Lehmer in short intervals. II" (2607.06575) investigates the distribution of integers in the set

$$\mathcal{L}(q) = \{1 \leq a < q:~ (a, q) = 1,~2 \nmid (a + \overline{a})\}$$

where $a$ and $\overline{a}$ are multiplicative inverses modulo an odd modulus $q$. The focus is on estimating the cardinality of $\mathcal{L}(q)$ in short intervals $[1, N]$ for $N$ significantly smaller than $q$, particularly in the case where $q$ is a large prime.

This problem, introduced by D. H. Lehmer and recorded in Guy’s compendium, quantifies the number of residues coprime to $q$ whose multiplicative inverse has different parity. Previous work had established asymptotics for $a$0 for general moduli $a$1 and for intervals $a$2 larger than $a$3, with the barrier $a$4 being a significant challenge for prime moduli. The present work breaks this barrier for primes, leveraging deep estimates for incomplete Kloosterman sums and refined bilinear form analysis.

Main Results

The central result is an asymptotic evaluation of

$a$5

establishing the following: for any $a$6, for all $a$7,

$a$8

The implied constant depends only on $a$9.

The result is obtained under the assumption that $\overline{a}$0 is an odd prime, thus resolving a key extremal case not covered by earlier work, which relied strongly on the factor structure of composite $\overline{a}$1.

Additionally, the paper proves an average-case result for the second moment of the error term, establishing for $\overline{a}$2, and arbitrary coefficients $\overline{a}$3, that

$\overline{a}$4

where the difference $\overline{a}$5 measures the deviation from the expected main term in weighted counts over $\overline{a}$6. This demonstrates nontrivial control of the error term over almost all moduli $\overline{a}$7 in a given dyadic range.

Technical Approach

A core component is the transformation of the original combinatorial problem into exponential sums using orthogonality and parity considerations. This yields expressions involving sums of the type

$\overline{a}$8

where $\overline{a}$9 denotes relevant intervals and $q$0.

Central to the analysis are several advanced estimates on bilinear forms and incomplete Kloosterman sums:

• Bourgain–Garaev bound: For very short Kloosterman sums, the maximum over $q$1 of $q$2 is bounded by $q$3 for $q$4, essential for controlling exponential sums over short intervals.
• Bilinear forms with Kloosterman fractions: The estimates due to Duke–Friedlander–Iwaniec [DFI97] and their refinement by Bettin–Chandee [BC18] enable power saving in the second moment analysis and are interpolated using the duality principle.
• "Shift by $q$5" method: This approach, inspired by Vinogradov and further used by Friedlander–Iwaniec et al., is a crucial combinatorial device allowing one to introduce additional averaging and thus amplify cancellation in sums over inverses.

The proofs carefully balance the contributions from "short" and "long" ranges in the dyadic decomposition of the dual parameter and optimize parameters in the shift and bilinear sum construction to maximize savings.

Numerical Strength and Claims

The main theorem outperforms prior art by obtaining nontrivial asymptotics for $q$6 in a range slightly below $q$7. Specifically, the regime

$q$8

surpasses the $q$9 restriction that had been the best known for prime $\mathcal{L}(q)$0. The error term

$\mathcal{L}(q)$1

shows a logarithmic power saving relative to the main term, enabled by the accurate control of incomplete Kloosterman sums' fluctuations.

The average-case second moment bound demonstrates that, except possibly for a sparse set of moduli $\mathcal{L}(q)$2, the expected main term holds for nearly all $\mathcal{L}(q)$3 in a short interval. The decay rate is dictated by the exponent $\mathcal{L}(q)$4, which is a refinement over trivial square-root cancellation and leverages the strongest bilinear form estimates available.

Theoretical and Practical Implications

From a theoretical perspective, the results progress towards understanding correlations between arithmetic functions in short intervals and the fine-scale structure of residue classes modulo large primes. The techniques further exemplify the potency of modern analytic methods—specifically, the utilization of spectral methods and exponential sum estimates derived from deep algebro-geometric or automorphic considerations (e.g., via Weil's bound and amplification machinery).

The practical relevance appears, for instance, in cryptographic applications where understanding the structure of invertibility and parity in residue classes is critical, or in sieve-theoretic questions where parity phenomena are exploited.

The approach set forth in this work may inspire further refinements. In particular, improvements to bounds on short exponential sums (potentially via new automorphic or algebro-geometric input), or leveraging additional averaging (over both $\mathcal{L}(q)$5 and $\mathcal{L}(q)$6), could further push the admissible range below $\mathcal{L}(q)$7. The methods also clarify the limitations of existing analytic techniques, notably for a single modulus, where bilinear sum methodology appears to reach natural barriers.

Conclusion

This paper advances the study of Lehmer's parity problem for modular inverses in short intervals, particularly for prime moduli, by developing and applying refined estimates for incomplete Kloosterman sums and bilinear forms involving inverses. By transcending the $\mathcal{L}(q)$8 barrier, it opens new directions for the analysis of arithmetic statistics in thin sets and strengthens the toolkit for studying additive and multiplicative structures in finite fields and residue rings. The established second moment estimates further suggest the effectiveness of average-case analyses in understanding fine distributions of arithmetic functions modulo $\mathcal{L}(q)$9.