Kloosterman sign changes with moduli having at most five prime factors

Abstract: On square-free moduli $q\in(X,2X]$ having at most five prime factors, we prove that each sign of the normalized Kloosterman sum $\operatorname{Kl}(1;q)$ occurs $\gg X/\log X$ times. This improves the recent unconditional result of Zhang and Zhong for moduli with at most six prime factors. Building on their analytic estimates and optimized Selberg sieve, we replace their truncated divisor penalty by a geometric half-weight. The new weight retains the $P_5$ exclusion threshold and is a positive linear combination of two standard two-parameter truncated divisor weights, so the Zhang--Zhong transference argument applies without alteration. After transference, the relevant pointwise coefficient is reduced from $5/16$ to $5/32$, which yields a positive final sieve margin.

• The paper proves that both positive and negative signs occur with frequency ≫ X/log X among square-free moduli with at most five prime factors.
• It introduces a geometric half-weight modification in the sieve majorant that reduces the error constant and enhances sign detection.
• The results refine previous bounds by eliminating one more prime factor, deepening the understanding of cancellation in Kloosterman sums.

Kloosterman Sign Changes with Moduli Having at Most Five Prime Factors

Introduction and Historical Context

This paper addresses the long-standing analytic problem of determining the distribution of signs in the normalized Kloosterman sum $\mathrm{Kl}(1;q)$ as the modulus $q$ varies. The principal hurdle has long been detection of sign changes over sparse sets of moduli. While unconditional results guaranteeing sign changes for $\mathrm{Kl}(1;q)$ in infinitely many moduli were previously available only for $q$ with significantly many prime factors, prior work had steadily lowered that threshold. In particular, the work of Fouvry and Michel established sign changes for moduli with at most 23 prime factors, and subsequent refinements by Matomäki, Xi, and most recently Zhang and Zhong reduced this to 6 prime factors.

Main Results

The paper proves that, for any sufficiently large $X$ and for each sign $\sigma = \pm 1$, there are $\gg X/\log X$ square-free integers $q \in (X,2X]$ with at most five distinct prime factors such that $\sigma\, \mathrm{Kl}(1; q) > 0$. This further narrows the support needed to guarantee sign changes and provides the sharpest unconditional distribution result to date for Kloosterman signs over almost-prime (specifically $P_5$) moduli. Compared to Zhang and Zhong's prior bound for $q$0, the current argument successfully removes an additional prime factor, matching the majorant exclusion threshold to five.

Technical Innovations

The core technical innovation is a geometric half-weight modification in the sieve majorant. Zhang and Zhong’s framework uses a truncated divisor function penalty whose support is tied to the number of prime factors; the present work replaces this with a positive linear combination of two standard two-parameter truncated divisor weights. This allows seamless integration with the Zhang–Zhong transference mechanism and preserves amenability to effective analytic evaluation.

With this modification, the paper reduces the $q$1 pointwise constant associated with the critical Selberg majorant from $q$2 to $q$3, leading to a reduction of the main error constant from $q$4 to $q$5. This new window ensures that the positive main term of the sieve outweighs the error, while strictly enforcing that only moduli with $q$6 prime factors make a positive contribution.

Analytical Structure and Quantitative Estimates

The proof framework follows a precise sequence:

• Sieve Decomposition: The admissible sieve weights and divisor support sets are fixed according to the truncated Selberg polynomial scheme, with the smooth weight $q$7 suppressing small prime divisors.
• Main Term Lower Bound: Adopts the quantitative lower bound for the sign-detecting main term from Zhang–Zhong’s framework, computed as $q$8.
• Transference and Sieve Majorant: The geometric half-weight, constructed as a convex combination of divisor functions (see equation (4.4)), ensures that the transferred weights in the sieve preserve the sign sensitivity while shaving an additional factor of $q$9 from the error constant.
• Pointwise Exclusion: The detailed combinatorial analysis (cf. Lemma 4.1) confirms the exclusion of all $\mathrm{Kl}(1;q)$0 with $\mathrm{Kl}(1;q)$1 prime factors, cementing the result for $\mathrm{Kl}(1;q)$2 moduli.
• Sieve Closure: The final sandwiching between upper and lower cutoff constants for the sieve weight guarantees a strictly positive sieve sum, localizing positivity to $\mathrm{Kl}(1;q)$3 moduli.

Numerical Strength and Robustness

The sieve parameters are chosen to maximize the positivity margin of the main term while suppressing all non-$\mathrm{Kl}(1;q)$4 contributions. With explicit constants, the final sieve sum yields a quantitatively significant main term coefficient (approximately $\mathrm{Kl}(1;q)$5 for the chosen parameter values), ensuring the result's robustness.

Implications and Future Directions

This result further elucidates the behavior and distribution of Kloosterman sums—central objects in analytic number theory. The reduction to $\mathrm{Kl}(1;q)$6 moduli sharpens the range where sign changes are provably dense, confirming that cancellation phenomena in Kloosterman sums persist deep into the almost-prime regime. The structural insight provided by the geometric penalty scheme could influence subsequent refinements, particularly the search for further reductions—potentially to $\mathrm{Kl}(1;q)$7—or generalizations to other exponential sums in trace function families.

Further, advances on Landau–Siegel zero-free regions, or improvements in the transference mechanism, could yield progress in reducing the required number of prime factors even further. The methods demonstrated here could also lead to analogous results in related settings such as Kloosterman sums at various arguments or over non-square-free moduli, and thereby inform broader sieve-theoretic phenomena.

Conclusion

The paper establishes that each sign for Kloosterman sums $\mathrm{Kl}(1;q)$8 occurs with frequency $\mathrm{Kl}(1;q)$9 among square-free moduli $q$0 with at most five distinct prime factors. The geometric modification of the sieve penalty is both analytically effective and technically efficient, achieving a significant improvement over prior bounds. This progress deepens understanding of sign distribution in exponential sums and refines the toolkit for investigating sign changes in sparse arithmetic settings.