[Paper Review] On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals, III

Giovanni Coppola, Maurizio Laporta|arXiv (Cornell University)|Mar 1, 2010

Analytic Number Theory Research3 references13 citations

TL;DR

This paper establishes new non-trivial asymptotic formulas for the Selberg and symmetry integrals of sieve functions in short intervals using a generalized bilinear form theorem of Duke, Friedlander, and Iwaniec. It achieves improved bounds for $ J_f(N,h) $ and $ I_f(N,h) $ when the sieve function has level $ \lambda \in (1/2,1) $, yielding $ O(N^{1-\varepsilon}h^2) $ for $ \theta < 1/95 $, surpassing prior results in very short intervals.

ABSTRACT

An arithmetic function $f$ is called a sieve function of range $Q$, if it is the convolution product of the constantly $1$ function and $g$ such that $g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon&gt;0$, for $q\leq Q$, and $g(q)=0$ for $q&gt;Q$. Here we establish a new result on the autocorrelation of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain non-trivial asymptotic formulæ that are actually unreachable by the standard approach of the distribution of $f$ in the arithmetic progressions. Moreover, we apply our asymptotic formulæ to obtain new bounds for the so-called Selberg integral and symmetry integral of $f$, which are basic tools for the study of the distribution of $f$ in short intervals.

Motivation & Objective

• To derive non-trivial asymptotic formulas for the Selberg and symmetry integrals of sieve functions in short intervals.
• To overcome limitations of standard arithmetic progression distribution methods in analyzing short interval behavior.
• To extend previous results by incorporating a generalized bilinear form theorem of Duke, Friedlander, and Iwaniec for mixed autocorrelations.
• To establish improved bounds for $ J_f(N,h) $ and $ I_f(N,h) $ in the regime $ \lambda > 1/2 $, particularly in very short intervals.

Proposed method

• The authors use a generalized version of the bilinear form theorem by Duke, Friedlander, and Iwaniec on Kloosterman fractions to analyze mixed autocorrelations of sieve functions $ f_1 $ and $ f_2 $.
• They define the mixed Selberg integral $ J_{f_1,f_2}(N,h) $ and mixed symmetry integral $ I_{f_1,f_2}(N,h) $, extending the classical integrals to two functions.
• The analysis relies on the arithmetic function $ f(n) = \sum_{d|n, d \leq Q} g(d) $, where $ g(q) \ll_\varepsilon q^\varepsilon $, defining a sieve function of range $ Q = [N^\lambda] $.
• The key technical step involves bounding exponential sums via the Kloosterman sum bilinear form, leading to improved error terms in the asymptotic expansions.
• The authors introduce the notation $ \phi_1(n) \lesssim_\varepsilon \phi_2(n) $ to denote $ \phi_1(n) \ll_\varepsilon n^\varepsilon \phi_2(n) $, enabling precise error control.
• An appendix provides a Fourier-analytic proof of a formula for the first Bernoulli function on rational numbers, used in the derivation of the main results.

Experimental results

Research questions

• RQ1 Can non-trivial asymptotic formulas for the Selberg and symmetry integrals of sieve functions be derived beyond the scope of standard arithmetic progression distribution methods?
• RQ2 What improvements in error bounds can be achieved for $ J_f(N,h) $ and $ I_f(N,h) $ when the sieve function has level $ \lambda > 1/2 $?
• RQ3 How does the use of bilinear forms on Kloosterman sums enhance the analysis of short interval autocorrelations compared to classical methods?
• RQ4 To what extent can the bounds for $ J_f(N,h) $ and $ I_f(N,h) $ be improved in very short intervals, specifically when $ \theta < 1/95 $?
• RQ5 Can the mixed autocorrelation integrals $ J_{f_1,f_2}(N,h) $ and $ I_{f_1,f_2}(N,h) $ yield sharper estimates than the single-function case?

Key findings

• For sieve functions of level $ \lambda \in (1/2,1) $, the Selberg integral satisfies $ J_f(N,h) \lesssim_\varepsilon Nh + N^\delta Q^{2-\Delta}h^2 + N^{1-2\delta/3}h^2 + Qh^2 $ with $ \Delta = 1/48 $, improving on prior results with $ \Delta = 0 $.
• The symmetry integral satisfies $ I_f(N,h) \lesssim_\varepsilon Nh + N^\delta Q^{2-\Delta}h^2 + N^{1-2\delta/3}h^2 $, showing similar strength in error control.
• For $ \theta < 1/95 $, the bounds improve to $ O_{\varepsilon_0}(N^{1-\varepsilon_0}h^2) $ for both $ J_f(N,h) $ and $ I_f(N,h) $, achieving non-trivial savings in very short intervals.
• The mixed integral $ J_{f_1,f_2}(N,h) \lesssim_\varepsilon Nh + N^\delta Q_1^{53/48}Q_2^{7/8}h^2 + N^{1-2\delta/3}h^2 + Q_1h^2 $, with $ \lambda_1 \geq \lambda_2 $, demonstrates the method's robustness for asymmetric cases.
• The results show that the autocorrelation method via bilinear forms can surpass classical approaches in short interval analysis, especially when $ \theta \in (0,1/95) $.
• The appendix establishes a Fourier-analytic identity for the first Bernoulli function on rationals, crucial for handling exponential sums in the main derivation.