Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

UNITARY ANALOGUES OF A GENERALIZED NUMBER-THEORETIC SUM

  • Traiwat Intarawong (Algebra and Applications Research Unit Division of Computational Science Faculty of Science Prince of Songkla University) ;
  • Boonrod Yuttanan (Algebra and Applications Research Unit Division of Computational Science Faculty of Science Prince of Songkla University)
  • Received : 2022.05.20
  • Accepted : 2022.08.17
  • Published : 2023.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate the sums of the elements in the finite set $\{x^k:1{\leq}x{\leq}{\frac{n}{m}},\;gcd_u(x,n)=1\}$, where k, m and n are positive integers and gcdu(x, n) is the unitary greatest common divisor of x and n. Moreover, for some cases of k and m, we can give the explicit formulae for the sums involving some well-known arithmetic functions.

Keywords

References

  1. J. D. Baum, A number-theoretic sum, Math. Mag. 55 (1982), no. 2, 111-113. https://doi.org/10.2307/2690056 
  2. D. M. Burton, Elementary Number Theory, McGraw-Hill, New York, 2011. 
  3. E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66-80. https://doi.org/10.1007/BF01180473 
  4. N. R. Kanasri, P. Pornsurat, and Y. Tongron, Generalizations of number-theoretic sums, Commun. Korean Math. Soc. 34 (2019), no. 4, 1105-1115. https://doi.org/10.4134/CKMS.c180414 
  5. N. R. Kanasri and Y. Tongron, A simple proof of generalizations of number-theoretic sums, Songklanakarin J. Sci. Technol. 44 (2022), no. 2, 450-458. https://doi.org/10.14456/sjst-psu.2022.62 
  6. K. Nageswara Rao, On the unitary analogues of certain totients, Monatsh. Math. 70 (1966), 149-154. https://doi.org/10.1007/BF01297269 
  7. I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, fifth edition, John Wiley & Sons, Inc., New York, 1991. 
  8. R. Sivaramakrishnan, Classical theory of arithmetic functions, Monographs and Textbooks in Pure and Applied Mathematics, 126, Marcel Dekker, Inc., New York, 1989.