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 sum analogous to the Dedekind sum and its mean value formula. (English) Zbl 0997.11076

Let ((x))=x-[x]-1 2 if x is not an integer, and let ((x))=0 otherwise. The sum in the title, originally introduced by Hardy, is defined by

S 2 (h,k)= a=1 k (-1) a a kah k·

For even k it can be expressed in terms of the classical Dedekind sum S(h,k) by the relation

S 2 (h,k)=2S(h,k/2)-S(h,k)·

The author uses some of his earlier work on S(h,k) to deduce corresponding results for S 2 (h,k). A corollary of the main theorem gives, for an odd prime p,

' h=1 2p |S 2 (h,2p)| 2 =1 32p 2 +Op exp 4logp loglogp·

MSC:
11N37Asymptotic results on arithmetic functions
11F20Dedekind eta function, Dedekind sums