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

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 $q$. We obtain an asymptotic formula for the number of such integers in very short intervals as long as $q$ is squarefree and has good factorizations, using arithmetic exponent pairs to estimate incomplete Kloosterman sums. This improves an early result of Z. Zheng in the above special cases. We also prove a prime-variable version with the aid of an estimate for incomplete Kloosterman sums in prime variables, due to Bourgain.

• The paper establishes a refined asymptotic formula for L(N, q) that significantly improves error estimates for Lehmer's parity problem in short intervals.
• It leverages advanced arithmetic exponent pairs and bounds for incomplete Kloosterman sums to obtain nontrivial results for squarefree, well-factored moduli.
• The methods extend to prime variables and reveal new avenues for exploring additive and multiplicative structures in residue classes.

Asymptotic Analysis of a Problem of D. H. Lehmer in Short Intervals

Background and Problem Statement

This paper addresses a question first posed by D. H. Lehmer, as recorded in Guy's book (Problem F12), concerning the enumeration of integers $a$ coprime to a modulus $q$ such that $a$ and its multiplicative inverse $\bar{a}$ modulo $q$ have different parity. Formally, define the set

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

and its cardinality $L(q) = |\mathcal{L}(q)|$. Traditional results, starting with W. Zhang, established an asymptotic formula

$$L(q) = \frac{1}{2}\varphi(q) + O\left(q^{1/2}\tau(q)^2\log^2 q\right)$$

where $\varphi$ is Euler's totient function and $\tau$ the divisor function. Generalizations to short intervals $q$0 were achieved by Z. Zheng, who proved analogous results with nontrivial error terms only for $q$1. The current work targets substantial improvement for $q$2 with restricted prime factorizations, leveraging recent advances in the estimation of incomplete Kloosterman sums.

Main Results

The paper establishes a refined asymptotic formula for $q$3—the count of integers $q$4 in $q$5, coprime to $q$6, with $q$7 odd—when $q$8 is squarefree and composed solely of small prime factors ($q$9 for small $a$0). Using bounds for arithmetic exponent pairs, the principal result (Theorem 1) states:

$a$1

where

$a$2

for any integer $a$3 and any $a$4. By taking $a$5 large and $a$6 sufficiently small, nontrivial estimates for all $a$7 are proved for $a$8 with sufficiently good factorizations.

A critical corollary follows: for such $a$9, there is always some $\bar{a}$0 with $\bar{a}$1 and $\bar{a}$2 odd. The structure of the arguments also permits these results on any interval $\bar{a}$3 (not just starting at $\bar{a}$4).

Methodology and Analytic Techniques

The central analytic device is the reduction of the original counting problem to the estimation of exponential sums of the form

$\bar{a}$5

where $\bar{a}$6 is a short interval and $\bar{a}$7 denotes the modular inverse. The paper builds upon the recent advances by Wu and Xi, deploying so-called arithmetic exponent pairs to achieve stronger bounds for incomplete Kloosterman sums when $\bar{a}$8 is squarefree and well-factored [WX21]. Notably, the error term is decisively reduced below the classical Pólya–Vinogradov threshold ($\bar{a}$9), which was the bottleneck in earlier work.

The paper also provides a comprehensive transformation of the counting function into weighted exponential sums, carefully managing the contributions from characters, and exploits fine analytic properties of incomplete Kloosterman sums as established in [WX21]. For the prime-modulus case, the methodology does not yield improvements, delineating a critical difference between the composite, well-factored $q$0 and the irreducible, prime $q$1 scenarios.

Extensions to Prime Variables

The findings extend to the distribution of such Lehmer-type integers among primes. By utilizing Bourgain's estimate for incomplete Kloosterman sums in prime variables [Bo05], the paper proves that for prime $q$2 and $q$3, the count of primes $q$4 with $q$5 odd satisfies

$q$6

with $q$7 the number of primes up to $q$8 and $q$9 depending on $$\mathcal{L}(q) = \{ 1 \leq a < q : (a, q) = 1, \; 2 \nmid (a + \bar{a}) \}$$0. This connects the combinatorial structure of the Lehmer problem to deep analytic questions about prime distributions in exponential sums.

Implications and Future Directions

The results represent a significant improvement in the granularity with which additive and multiplicative structures in residue classes can be analyzed, showing that for highly factorizable, squarefree moduli, arithmetic exponent pairs enable nontrivial distribution estimates far into "short interval" regimes. This not only pushes the limits of current exponential sum techniques but also refines the understanding of "parity with inverse" phenomena for multiplicative group elements.

The methodology indicates that further refinements in the theory of arithmetic exponent pairs, Kloosterman sum estimates for additional classes of moduli, or analogous breakthroughs in the analytic theory of character sums could increase the scope and depth of such results. Also, as outlined by the failure for $$\mathcal{L}(q) = \{ 1 \leq a < q : (a, q) = 1, \; 2 \nmid (a + \bar{a}) \}$$1 prime, prime moduli continue to pose significant obstacles, likely requiring new conceptual advances. Detailed exploration of analogous problems for other symmetry or parity constraints—and their distributions across sets with arithmetic structure—remains a fruitful area of ongoing analytic number theory research.

Conclusion

This work advances the analysis of Lehmer's problem on parity of residues and their modular inverses, particularly in short intervals and for moduli with highly composite, squarefree structure. By integrating the latest bounds for incomplete Kloosterman sums from arithmetic exponent pair theory, the error terms are pushed below long-standing barriers, and the results approach optimal ranges in $$\mathcal{L}(q) = \{ 1 \leq a < q : (a, q) = 1, \; 2 \nmid (a + \bar{a}) \}$$2 for almost all practical composite $$\mathcal{L}(q) = \{ 1 \leq a < q : (a, q) = 1, \; 2 \nmid (a + \bar{a}) \}$$3. This highlights the intimate interplay between combinatorial residue class problems and analytic exponential sum estimates, and opens new avenues for the study of parity phenomena in arithmetic settings (2607.07710).