International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 11 Issue: 07 | July 2024
www.irjet.net
p-ISSN: 2395-0072
Dirichet And Unitary Convolutions Of Arithmetic Functions. Dr. Ram Kumar Sinha Associate Professor in Mathematics, E:mail ID – rksinha.rcit@gmail.com. Ramchandra Chandravansi Institute of Technology, Bishrampur, Palamu, Jharkhand- 822132 ----------------------------------------------------------------------------***-------------------------------------------------------------------------Abstract:- In this paper we study Arithmetic functions and its distributive and quasi distributive properties hold is general set-up of s-regular and A-regular convolutions. They hold in the case of Dirichet and Unitary Convolutions. We study the theorems of Pandmavashamma [3], Vidyanatha Swamy [7], Langford [2], Lambek [1]. (1) Introduction:- A system study of Vasu’s S-regular and A-regular i.e. strong regular convolutions has been presented along with some results by Pandmavashamma [3] Balashekhar and Subrahimanya Saitri [4]. Theorem (1.1) convolution
for any three arithmetic functions f, g, h, where f is A- multiplicative, and A is any s-regular
[1.1(1)]
F(g A h) = (f g) A (f h)
Holds if any only if f is completely A-multiplicative Theorem (1.2)
for any three A-multiplicative functions f1, f2, f3, we have
[1.2 (1)] f1A (f2 B f3) = (f1 A f2) B (f1 A f3), where A is any A- regular convolution and B is the unitary regular convolution associated with A. Theorem (1.3) [1.3 (1)]
for any four completely-A-multiplicative functions f1, f2, f3, f4 we have
(f1 f3) A (f1 f4) A (f2 f3) A (f3 f4) = (f1 A f2) (f3 A f4) A where
[1.3 (2)] (r) =
(√ ) f2 (√ ) f3 (√ ) f4(√ ), if r is a square w. r. to product representation of A 0 , otherwise.
We also indicate the corresponding distributive properties for six completely-A-multiplicative functions, an analogue of Theorem of Subbarao [5]. Theorem (1.4) [1.4 (1)]
© 2024, IRJET
For completely-A-multiplicative function f, g, h, k, u, v, we have
(f A g) (h A k) (u A v)=(f hu A f hu A f ku A f kv A ghu A ghu A gku A gkv A ).
|
Impact Factor value: 8.226
|
ISO 9001:2008 Certified Journal
|
Page 925