ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN SQUARE VALUE FORMULA

The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.


Introduction. For a positive integer k and an arbitrary integer h, the classical Dedekind sum S(h, k) is defined by
where , if x is not an integer; 0, if x is an integer. (1. 2) The various properties of S(h, k) were investigated by many authors.For example, Carlitz [1] obtained a reciprocity theorem of S(h, k).Conrey et al. [2] studied the mean value distribution of S(h, k), and first proved the following important asymptotic formula: + O k 9/5 + k 2m−1+1/(m+1) log 3 k , (1.3) where h denotes the summation over all h such that (k, h) = 1, and (1.4) The author [4] improved the error term of (1.3) for m = 1.In October, 2000, Todd Cochrane (personal communication) introduced a sum analogous to Dedekind sum as follows: where a defined by equation aa ≡ 1 mod k.Then he suggested to study the arithmetical properties and mean value distribution properties of C(h, k).About first problem, we have not made any progress at present.But for the second problem, we use the estimates for character sums and the mean value theorem of Dirichlet L-functions to prove the following main conclusion.
Theorem 1.1.Let k be any integer with k > 2. Then we have the asymptotic formula where exp(y) = e y , φ(k) is Euler function, p α k denotes the product over all prime divisors of k with p α |k and p α+1 k.
It seems that our methods are useless for mean value 2. Some lemmas.To complete the proof of Theorem 1.1, we need the following lemmas.
Lemma 2.2.Let k be any integer with k > 2. Then we have the identity where µ(d) is Möbius function.(2.3)

Proof. From the definition of S(h, k) and C(h, k), we have
Since (ab, k) = 1, so if h round through a complete residue system modulo k, then bh also round through a complete residue system modulo k.Therefore, note that the identities we have (2.5) This proves Lemma 2.2.
Lemma 2.3.Let u and v be integers with (u, v) = d ≥ 2, χ 0 u the principal character mod u, and χ 0 v the principal character mod v. Then we have the asymptotic formula where p|n denotes the product over all prime divisors of n, (u, v)denotes the greatest common divisor of u and v, and m = max(u, v).
where A(y, χ) = N<n≤y χ(n)r (n).Note that the partition identities (2.8) Applying Cauchy inequality and the estimates for character sums (2.9) and note that the identities where ω(u) denotes the number of all different prime divisors of u.We have (2.11) Thus from (2.11) and Cauchy inequality we get (2.12) Note that for (ab, d) = 1, from the orthogonality relation for character sums modulo d we have where τ(n) is the divisor function and r (n (2.15) Taking parameter N = d 3 and note the identity ln ln m . (2.17) This proves Lemma 2.3.
Lemma 2.4.Let p be a prime, and let α, β be nonnegative integers with β ≥ α.Then we have the identity Proof.Note that d = (d 1 ,d 2 ), we have Now combining (2.19) and (2.20) we have This proves Lemma 2.4.

Proof of the theorem.
In this section, we complete the proof of Theorem 1.1.Let k be an integer with k ≥ 3. Then applying Lemmas 2.1 and 2.2 we have For each χ 1 mod u, it is clear that there exists one and only one k 1 |u with a unique primitive character χ 1 k 1 mod k 1 such that χ 1 = χ 1 k 1 χ 0 u , here χ 0 u denotes the principal character mod u.Similarly, we also have Note that u|k and v|k, from the orthogonality of characters we have Let So from (3.1), (3.2), and Lemma 2.3 we have Since φ(n) and µ(n) are multiplicative functions, so from the multiplicative properties of these functions, (3.3) and Lemma 2.4 and note that the identities (for any multiplicative functions f (u) and g(v)) we have

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: .18) where d = (d 1 ,d 2 ) denotes the greatest common divisors of d 1 and d 2 .