Representations of Integers as Sums of Squares
Representations of Integers as Sums of Squares
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DOI：
10.1006/jnth.2001.2765
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发表时间：
2002-08
影响因子：
0.7
通讯作者：
K. Ono
中科院分区：
文献类型：
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作者：
K. Ono
确定r（s;n）的精确公式的一般问题是数论中的经典问题。人们可以参考E. Grosswald [G]对这个主题进行了详尽的描述（截至20世纪80年代初），并附有参考文献。级数Θ（z）是一个模形式，因此有抽象公式r（s;n）作为模形式的傅里叶系数。具体地说，众所周知，Θ（z）= E s（z）+cs（z），其中E s（z）是具有显式系数的爱森斯坦级数，cs（z）是尖点形式。利用这一事实，可以推导出r（s;n）的渐近信息。兰金证明了[R]中对任意s > 8，cs（z）是非平凡的.因此，计算r（s;n）的非平凡公式的问题仍然存在，因为尖点形式的系数虽然很小，但很少有简单的描述。在一个令人吃惊的转变中，米尔恩[M1]宣布了r（4s;n）和r（4s+4s;n）对每个s的公式。他的公式是综合各种方法和观察所得的
The general problem of determining exact formulas for r(s;n) is classical in number theory. One may consult the popular book by E. Grosswald [G] for a thorough account (as of the early 1980’s) of the subject complete with references. The series Θ(z) is a modular form, and so there are abstract formulas for r(s;n) as the Fourier coefficients of modular forms. Specifically, it is well known that Θ(z) = E∗ s (z)+cs(z), where E ∗ s (z) is an Eisenstein series with explicit coefficients and cs(z) is a cusp form. Using this fact, one may deduce asymptotic information for r(s;n). Rankin proved [R] that cs(z) is non-trivial for every s > 8. Therefore, the problem of computing non-trivial formulas for r(s;n) remains since the coefficients of cusp forms, although small, rarely have simple descriptions. In a startling turnabout, Milne [M1] announced formulas for r(4s;n) and r(4s+4s;n) for every s. His formulas were obtained by combining a variety of methods and observations