# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A $p$-adic study of the partial sums of the harmonic series. (English) Zbl 0838.11015
Let ${H}_{n}=1+\frac{1}{2}+\cdots +\frac{1}{n}$ 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 $p\in \left\{2,3,5,7\right\}$ 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 $|{J}_{p}|=O\left({p}^{2}{\left(loglogp\right)}^{2+\epsilon }\right)$ and that $|{J}_{p}|\ge {p}^{2}{\left(loglogp\right)}^{2}$ 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 $p\le {10}^{5}$.

##### MSC:
 11B39 Fibonacci and Lucas numbers, etc. 11K99 Probabilistic theory 60J80 Branching processes