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I am with the following problem.

There is a closed formula or some lower bound for the $p$-adic valuation of the product $\prod_{k=1}^m(a+k\ell)$ (Pochhammer symbol)? where $a$ and $\ell$ are positive integers.

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See Chapter 5, "The $p$-adic evaluation of Pochhammer symbols", in On Dwork’s $p$-adic formal congruences theorem and hypergeometric mirror maps by E. Delaygue, T. Rivoal and J. Roques, Memoirs of the American Mathematical Society 246, 1163 (2017). [A preprint from 2013 has open access here, see page 22.

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  • $\begingroup$ DO you have this book? I don't have access to it. Thanks in advance! $\endgroup$ – Tanaku Ohara Feb 20 '17 at 13:39
  • $\begingroup$ I can't access. It ask username and password. $\endgroup$ – Tanaku Ohara Feb 20 '17 at 13:43
  • $\begingroup$ I found, but it is only for $\ell=1$ $\endgroup$ – Tanaku Ohara Feb 20 '17 at 13:46

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