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Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. (English) Zbl 0868.11009

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 $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ is not squarefree for any $n>4$. This was recently proved by 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 $\ge \sqrt{n/5}$ for all $n\ge 2082$. On the other hand, $\left(\genfrac{}{}{0pt}{}{1572}{786}\right)$ 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.

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 $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is squarefree for all $k\le \left(1/5\right)logn$. On the other hand, it is shown that if $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is squarefree, then $n$ or $n-k$ is $\ll exp\left(c{\left(logn\right)}^{2/k}{\left(loglogn\right)}^{1/3}\right)$ for some constant $c$, and it is conjectured that this bound can be reduced to $\ll {\left(lognloglogn\right)}^{2}$, 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 $10·66$.

An important tool in previous work related to the Erdös conjecture has been the exponential sums of the type ${\sum }_{n}{\Lambda }\left(n\right)e\left(x/n\right)$, 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.

Reviewer: M.Jutila (Turku)

##### MSC:
 11B65 Binomial coefficients, etc. 11L20 Sums over primes 11L07 Estimates on exponential sums