# 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)
On sparsely totient numbers. (English) Zbl 0538.10006
Let $\phi$ denote Euler’s totient function. A positive integer n is said to be sparsely totient if $\phi \left(m\right)>\phi \left(n\right)$ for all $m>n$; this is analogous to Ramanujan’s definition of a highly composite number. The authors give constructions for sparsely totient numbers, and they deduce that the ratio of successive ones tends to 1. They also prove several results about the prime factorizations of sparsely totient numbers n (for example, n is divisible by all primes up to $\left(\sqrt{2}-1-ϵ\right)logn$ for $n>{n}_{0}\left(ϵ\right)\right)$, and, with the aid of a plausible but unproved gap hypothesis, they deduce conditional improvements of these results that are asymptotically sharp ($\sqrt{2}-1$ is replaced by 1 in the above example).

##### MSC:
 11A25 Arithmetic functions, etc.