论文标题
关于与Borwein猜想有关的多项式系数的总和
On sums of coefficients of polynomials related to the Borwein conjectures
论文作者
论文摘要
最近，LI获得了涉及第一个Borwein猜想中多项式系数的一定部分总和的渐近公式。结果，他展示了这一总和的积极性。他的结果是基于自己和Wan发现的筛选原则。实际上，李在他的论文中指出，他的方法可以推广，以证明涉及任何主要$ p> 3 $系数的一般部分总和的渐近公式。在这项工作中，我们扩展了LI的方法，以获取几种非常通用多项式系数的部分系数的渐近公式。我们发现，在特殊情况下，$ p = 3，5 $，这些款项的迹象与三个著名的borwein猜想一致。 Zaharescu先前使用了完全不同的方法研究了类似的总和。我们还改进了Li和Zaharescu的渐近公式中的误差项。利用Borwein的最新结果，我们还获得了渐近估计值，以最大值这些系数的绝对值，用于素数$ p = 2、3、5、7、11、13 $，并且对于$ p> 15 $，我们获得了这些系数的最大绝对值，这些系数的最大绝对值对于足够大的$ n $。
Recently, Li obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan. In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein, we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.
