Recently, Li (Int J Number Theory, 2020) 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 (Sci China Math, 2010). 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 (Ramanujan J, 2006) 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 (JNT 1993), 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.

最近，李（Int J Number Theory, 2020）得到了第一博尔温猜想多项式系数的某个部分和的渐近公式。结果，他显示了这笔款项的积极性。他的结果是基于他和万发现的筛分原理（Sci China Math，2010）。事实上，李在他的论文中指出，他的方法可以推广到证明涉及任何素数\(p>3\)的系数的一般部分和的渐近公式。在这项工作中，我们扩展了李的方法以获得一个非常一般多项式的系数的几个部分和的渐近公式。我们发现在特殊情况下\(p=3, 5\)，这些和的符号与三个著名的 Borwein 猜想一致。Zaharescu (Ramanujan J, 2006) 早些时候使用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。使用 Borwein (JNT 1993) 的最新结果，我们还获得了对于素数\(p=2, 3, 5, 7, 11, 13\)和\( p>15\)，对于足够大的n，我们获得了这些系数的最大绝对值的下界。