Sums of Logarithmic Averages of gcd-Sum Functions

@article{Kiuchi2018SumsOL,
  title={Sums of Logarithmic Averages of gcd-Sum Functions},
  author={Isao Kiuchi and Sumaia Saad Eddin},
  journal={Results in Mathematics},
  year={2018},
  volume={75},
  pages={1-22}
}
Let $$ \gcd (k,j) $$ gcd ( k , j ) be the greatest common divisor of the integers k and j . For any arithmetic function f , we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\begin{aligned} \sum _{k\le x}\frac{1}{k} \sum _{j=1}^{k}f(\gcd (k,j)) \log j. \end{aligned}$$ ∑ k ≤ x 1 k ∑ j = 1 k f ( gcd ( k , j ) ) log j . More precisely, we give asymptotic formulas for various multiplicative functions such as $$f=\mathrm… Expand

References

SHOWING 1-10 OF 18 REFERENCES
On sums of weighted averages of $\gcd$-sum functions
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 functionExpand
Sums of averages of gcd-sum functions
Abstract 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 ∑Expand
Handbook of Number Theory I
Preface. Basic Symbols. Basic Notations. I. Euler's phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analoguesExpand
Exponential sums and lattice points III
The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. InExpand
Sums of averages of generalized Ramanujan sums
For any positive integers k and j, we consider some asymptotic formulas for weighted averages of the Cohen–Ramanujan sums sk(s)(j)=∑d|k,ds|jf(d)g(kd) with any fixed positive integer s≥1, and anyExpand
Averages of Ramanujan sums: note on two papers by E. Alkan
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of theExpand
Introduction to analytic number theory
This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introductionExpand
Étude moyenne di plus grand commun diviseur de deux nombres
1 . (u, v) d6signant le plus grand commun diviseur de u et v, et F(u, v) 4tant une fonction quelconque de (u, v), fAchons d' ISomme des va' leurs que prend F(u, v), lorsque it et v varient s4par4meitExpand
  • J. Ramanujan Math. Soc.
  • 2016
...
1
2
...