# 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 $((x))= x-[x]- \frac{1}{2}$ if $x$ is not an integer, and let $((x))=0$ otherwise. The sum in the title, originally introduced by Hardy, is defined by $$S_2(h,k)= \sum_{a=1}^k (-1)^a \biggl(\biggl( \frac{a}{k} \biggr)\biggr) \biggl(\biggl( \frac{ah}{k} \biggr)\biggr).$$ 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 $p$, $$\mathop{{\sum}'}_{h=1}^{2p} |S_2(h,2p)|^2= \frac{1}{32} p^2+ O\Biggl(p\exp \biggl( \frac{4\log p}{\log\log p} \biggr)\Biggr).$$

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11F20 Dedekind eta function, Dedekind sums
##### Keywords:
mean value formula; asymptotic result; Hardy sum; Dedekind sum
Full Text:
##### References:
 [1] Apostol, T. M.: Introduction to analytic number theory. (1976) · Zbl 0335.10001 [2] Apostol, T. M.: Modular functions and Dirichlet series in number theory. (1976) · Zbl 0332.10017 [3] Berndt, B. C.: Analytic Eisenstein series, theta-function, and series relations in the spirit of Ramanujan. J. reine angew. Math. 303/304, 332-365 (1978) · Zbl 0384.10011 [4] Carlitz, L.: The reciprocity theorem of Dedekind sums. Pacific J. Math. 3, 513-522 (1953) · Zbl 0057.03701 [5] Conrey, J. B.; Fransen, E.; Klein, R.; Scott, C.: Mean values of Dedekind sums. J. number theory 56, 214-226 (1996) · Zbl 0851.11028 [6] Gandhi, J. M.: On sums analogous to Dedekind sums. (1975) · Zbl 0341.10010 [7] Rademacher, H.: On the transformation of $log{\eta}({\tau})$. J. indian. Math. soc. 19, 25-30 (1955) · Zbl 0064.32703 [8] Rademacher, H.: Generalization of the reciprocity formula for Dedekind sums. Duke J. Math. 21, 391-397 (1954) · Zbl 0057.03801 [9] Hardy, G. H.: On certain series of discontinuous functions connected with the modular functions. Quart. J. Math. 36, 93-123 (1905) · Zbl 35.0468.03 [10] Walum, H.: An exact formula for an average of L-series. Illinois J. Math. 26, 1-3 (1982) · Zbl 0464.10030 [11] Zhang, W.: On the mean values of Dedekind sums. J. theor. Nombres 8, 429-442 (1996) · Zbl 0871.11033 [12] Zhang, W.: A note on the mean square value of the Dedekind sums. Acta math. Hungar. 86, 275-289 (2000) · Zbl 0963.11049