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Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $ [ABSTRACT FROM AUTHOR]

Let gcd (j , k) denote the greatest common divisor of integers k and j. Define A f (x) = ∑ k ≤ x 1 k ∑ j = 1 k f (gcd (j , k)) , for any arithmetical function f. We shall derive several asymptotic expansions of the error terms E f (x) for the partial sums of A f (k) with f = id , id 1 + a (− 1 < a < 0) , ϕ , and ψ , where ϕ and ψ are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function A f (x) , and deduce several related results. [ABSTRACT FROM AUTHOR]

Let ℤ m be the additive group of residue classes modulo m. For any positive integers m and n , let s (m , n) and c (m , n) denote the total number of subgroups and cyclic subgroups of the group ℤ m × ℤ n , respectively. Define D ̃ s (x) = ∑ m n ≤ x s (m , n) log x m n and D ̃ c (x) = ∑ m n ≤ x c (m , n) log x m n. In this paper, we study the asymptotic behavior of the functions D ̃ s (x) and D ̃ c (x). [ABSTRACT FROM AUTHOR]

Let gcd (j , k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd -sum function f (gcd (j , k)) and the function ∑ d | k , d s | j (f ∗ μ) (d) for any positive integers j and k , namely ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) and ∑ k ≤ x 1 k s (r + 1) ∑ j = 1 k s j r ∑ d | k d s | j (f ∗ μ) (d) with any fixed integer s > 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd -sum functions f (gcd (j , k)). [ABSTRACT FROM AUTHOR]

For any positive integers k and j , we consider some asymptotic formulas for weighted averages of the Cohen–Ramanujan sums s k ( s ) ( j ) = ∑ d | k , d s | j f ( d ) g ( k d ) with any fixed positive integer s ≥ 1 , and any arithmetical functions f and g . [ABSTRACT FROM AUTHOR]

Let gcd ⁡ ( k , j ) be the greatest common divisor of the integers k and j . We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ⁡ ( k , j ) ) with f = id , ϕ , ϕ s , ψ and ψ s for any fixed positive integers r and s , where ϕ , ϕ s , ψ and ψ s are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑ k ≤ x 1 k r + 1 ⋅ ∑ j = 1 k j r ϕ ( gcd ⁡ ( k , j ) ) and ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r ψ ( gcd ⁡ ( k , j ) ) . [ABSTRACT FROM AUTHOR]

We prove the fourth power moment of the Euler-Zagier type double zeta-function ζ2(s1, s2) and provide an improvement on the Ω results of Kiuchi, Tanigawa, and Zhai. We also calculate the double integral under certain conditions. [ABSTRACT FROM AUTHOR]

To clarify the lignocellulolytic process of the lower termite symbiotic protistan system, we constructed a cDNA library from an as yet uncultivated symbiotic protist community of the lower termite Reticulitermes speratus. The library was constructed by the biotinylated CAP trapper method and analyzed by one-pass sequencing. Phylogenetic analysis of actin orthologs confirmed that the resulting library reflected the intact organismal and mRNA composition of the symbiotic system. The contents of the library included abundant numbers of lignocellulolytic genes of the glycosyl hydrolase family orthologs (families 3, 5, 7, 8, 10, 11, 26, 43, 45 and 62). Our results clearly indicated that a multiple family of glycosyl hydrolase enzymes was involved in the protistan cellulose degradation system. The data also suggested that the most extensively expressed enzyme was glycosyl hydrolase family 7, a cellobiohydrolase ortholog. This family of enzymes enables the degradation of crystalline cellulose, the principal component of wood biomass. [ABSTRACT FROM AUTHOR]