Recent Progress on the Large Sieve with Sparse Sets of Moduli and Its Application to Square Moduli

• Main Authors: Chew, Xiao-Xian, 周孝賢
• Other Authors: Yang, Yi-Fan
• Format: Others
• Language: en_US
• Published: 2015
• Online Access: http://ndltd.ncl.edu.tw/handle/60760829604233832540

碩士 === 國立交通大學 === 應用數學系所 === 104 === The large sieve inequality for square moduli has the following form: \[ \sum_{q=1}^Q\sum_{\substack{a~\mathrm{mod}~q^2\\\mathrm{gcd}(a,q)=1}} \left|\sum_{n=M+1}^{M+N} a_ne\left( \frac{a}{q^2}n\right) \right|^2\ll\Delta \sum_{n=M+1}^{M+N} \left| a_n\right|^2. \] From the classical large sieve inequality, we can deduce two natural $\Delta$s, namely $\Delta=Q^4+N$ and $\Delta=Q(Q^2+N)$. L. Zhao \cite{Zhao 2004} gives a $\Delta$, namely $\Delta=Q^3+(N\sqrt{Q}+\sqrt{N}Q^2)N^\varepsilon$ in (\ref{Zhao's bound}), it is sharper than the former two $\Delta$s in the range $N^{2/7+\varepsilon}\ll Q\ll N^{1/2-\varepsilon}$. Extending a method of D. Wolke \cite{Wolke 1971/2}, S. Baier \cite{Baier 2006} establishes a general large sieve inequality (see Theorem \ref{thm 2:Baier} below), for the case when $\mathcal{S}$ is a sparse set of moduli which is in a certain sense well-distributed in arithmetic progressions. As an application, he then employs Theorem \ref{thm 2:Baier} with $\mathcal{S}$ consists of squares. In this case, he obtains Theorem \ref{thm 3:Baier} with a $\Delta=(\log\log10NQ)^3(Q^3+N+N^{1/2+\varepsilon}Q^2)$, it is sharper than the two natural $\Delta$s and Zhao's bound (\ref{Zhao's bound}) within the range $N^{1/4+\varepsilon}\ll Q\ll N^{1/3-\varepsilon}$.