On sums of coefficients of polynomials related to the Borwein conjectures

Ankush Goswami; Venkata Raghu Tej Pantangi

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arXiv · 2004.08954

On sums of coefficients of polynomials related to the Borwein conjectures

Ankush Goswami·Venkata Raghu Tej Pantangi

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Abstract

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 $n$.

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Ankush Goswami, Venkata Raghu Tej Pantangi. 2020-04-19. On sums of coefficients of polynomials related to the Borwein conjectures. https://arxiv.org/abs/2004.08954

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