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A p-adic study of the partial sums of the harmonic series. (English) Zbl 0838.11015
Let H n =1+1 2++1 n be the n-th partial sum of the harmonic series. For a given prime p, denote by J p the set of n for which p divides the numerator of H n . In 1991 A. Eswarathasan and E. Levine [Discrete Math. 91, 249-257 (1991; Zbl 0764.11018)] have determined J p for p{2,3,5,7} and made the conjecture that |J p | is finite for all p. However, they didn’t prove even that |J 11 | is finite. The author remarks that this fact appears as a problem in [R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics (Addision-Wesley, 1989; Zbl 0668.00003)], and shows that this set contains exactly 638 integers, the largest of which is a number of 31 decimal digits. He determines J p for all p<550, with three exceptions: 83, 127, 397. In this eye-opening paper the author strengthens the above conjecture on |J p |, by using the theory of branching processes. This is based on a new p-adically convergent formula for H pn -H n /p. A probabilistic model predicts that |J p |=O(p 2 (loglogp) 2+ε ) and that |J p |p 2 (loglogp) 2 for infinitely many p. Another interesting conjecture, supported by a probabilistic argument, is that the density of primes p with |J p |=3 is 1/e. This is confirmed experimentally for all p10 5 .

MSC:
11B39Fibonacci and Lucas numbers, etc.
11K99Probabilistic theory
60J80Branching processes