大篩其稀疏集合之元素為模數及其在平方模數之應用的新進展

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

周孝賢(Xiao-Xian Chew)

指導教授 ： 楊一帆

國立交通大學/理學院/應用數學系暨研究所/碩士(2015年)

https://doi.org/10.6842/NCTU.2015.00925

摘要

模數為平方數時的大篩不等式有以下形式：
[
sum_{q=1}^Qsum_{substack{a~mathrm{mod}~q^2\mathrm{gcd}(a,q)=1}} left|sum_{n= M+1}^{M+N} a_neleft( frac{a}{q^2}n
ight)
ight|^2llDelta sum_{n=M+1}^{M+ N} left| a_n
ight|^2
]。由古典大篩不等式，我們可得兩個自然的$Delta$，即$Delta=Q^4+N$和$Delta=Q(Q^2+N)$。趙良軼 cite{Zhao 2004} 給出一個$Delta$，即$Delta=Q^3+(Nsqrt{Q }+sqrt{N}Q^2)N^varepsilon$，它在$N^{2/7+varepsilon}ll Qll N^{1/2-varepsilon}$比以上兩個自然的$Delta$準確。

延用D. Wolke cite{Wolke 1971/2}的某些方法，對於某個稀疏集合$mathcal{S}$其元素構成等差數列，S. Baier得到了一般形式的大篩不等式，即定理
ef {thm 2:Baier}。之後，令定理
ef{thm 2:Baier}的集合$mathcal{S}$其元素為平方數，他推得定理
ef{thm 3:Baier}，這時$Delta=(loglog 10NQ)^3(Q^3+N+N^{1/2+varepsilon}Q^2)$，它在$N^{1/4+varepsilon}ll Qll N^{1/ 3-varepsilon}$比以上兩個自然的和趙良軼的$Delta$準確。

關鍵字

大篩

並列摘要

The large sieve inequality for square moduli has the following form:
[
sum_{q=1}^Qsum_{substack{a~mathrm{mod}~q^2\mathrm{gcd}(a,q)=1}} left|sum_{n=M+1}^{M+N} a_neleft( frac{a}{q^2}n
ight)
ight|^2llDelta sum_{n=M+1}^{M+N} left| a_n
ight|^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 $Delta=Q^3+(Nsqrt{Q}+sqrt{N}Q^2)N^varepsilon$ in (
ef{Zhao's bound}), it is sharper than the former two $Delta$s in the range $N^{2/7+varepsilon}ll Qll N^{1/2-varepsilon}$.

Extending a method of D. Wolke cite{Wolke 1971/2}, S. Baier cite{Baier 2006} establishes a general large sieve inequality (see Theorem
ef{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
ef{thm 2:Baier} with $mathcal{S}$ consists of squares. In this case, he obtains Theorem
ef{thm 3:Baier} with a $Delta=(loglog10NQ)^3(Q^3+N+N^{1/2+varepsilon}Q^2)$, it is sharper than the two natural $Delta$s and Zhao's bound (
ef{Zhao's bound}) within the range $N^{1/4+varepsilon}ll Qll N^{1/3-varepsilon}$.

並列關鍵字

Large Sieve

參考文獻

[1] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, Heidelberg, 1976.

[3] E. Bombieri and H. Davenport, Some inequalities involving trigonometrical polynomials, Annali Scuola Normale Superiore - Pisa 23 (1969), 223-241.

[7] H. Davenport, Multiplicative number theory, Third Edition, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, etc., 2000.

[8] H. Davenport and H. Halberstam, The values of a trigonometric polynomial at well spaced points, Mathematika 13 (1966), 91-96.

[9] H. Davenport and H. Halberstam, Corrigendum and addendum, Mathematika 14 (1967), 229-232.