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Zbl 0868.11009
Granville, Andrew; Ramaré, Olivier
Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients.
(English)
[J] Mathematika 43, No.1, 73-107 (1996). ISSN 0025-5793; ISSN 2041-7942/e

The interest in prime factors of binomial coefficients during the last ten years or so has been motivated to a great extent by a conjecture of Erdös asserting that ${{2n}\choose n}$ is not squarefree for any $n>4$. This was recently proved by {\it G. Velammal} [Hardy-Ramanujan J. 18, 23-45 (1995; Zbl 0817.11011)], and another proof is given in the present paper, among many other interesting results. As a sharpening of the Erdös conjecture, it is shown that the coefficient in question is divisible even by the square of a prime $\geq \sqrt{n/5}$ for all $n\geq 2082$. On the other hand, ${{1572}\choose {786}}$ is not divisible by the square of any odd prime (it is divisible by $2^4$), and it is the largest coefficient of this kind. \par In addition to the middle of the Pascal triangle, the authors consider it as a whole, in particular its edges. Squarefree values (other than 1) do occur near the edges, and only there. For instance, there are infinitely many integers $n$ such that ${n\choose k}$ is squarefree for all $k\leq (1/5)\log n$. On the other hand, it is shown that if ${n\choose k}$ is squarefree, then $n$ or $n-k$ is $\ll\exp(c(\log n)^{2/k} (\log\log n)^{1/3})$ for some constant $c$, and it is conjectured that this bound can be reduced to $\ll(\log n\log\log n)^2$, which would be close to being best possible. A curious statistical result indicating the scarcity of the squarefree binomial coefficients is that the average number of these in a row of the Pascal triangle is about $10.66$. \par An important tool in previous work related to the Erdös conjecture has been the exponential sums of the type $\sum_n \Lambda(n)e(x/n)$, where $n$ thus runs essentially over primes, and the same is the case also in the present paper, where explicit estimates for such sums are given and applied as a key ingredient of the argument.
[M.Jutila (Turku)]
MSC 2000:
*11B65 Binomial coefficients, etc.
11L20 Sums over primes
11L07 Estimates on exponential sums

Keywords: prime factors of binomial coefficients; Pascal triangle; squarefree; exponential sums; explicit estimates

Citations: Zbl 0817.11011

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