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:-InthispaperwestudyArithmeticfunctionsanditsdistributiveandquasidistributivepropertiesholdis 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 presentedalongwithsomeresultsbyPandmavashamma[3]BalashekharandSubrahimanyaSaitri[4].
Theorem (1.1) foranythreearithmeticfunctionsf,g,h,wherefisA-multiplicative,andAisanys-regular convolution
[1.1(1)] F(gAh)=(fg)A(fh)
HoldsifanyonlyiffiscompletelyA-multiplicative
Theorem (1.2) foranythreeA-multiplicativefunctionsf1,f2,f3,wehave
[1.2 (1)] f1A(f2 Bf3)=(f1 Af2)B(f1 Af3),whereAisanyA-regularconvolutionandBistheunitaryregular convolutionassociatedwithA.
Theorem (1.3) foranyfourcompletely-A-multiplicativefunctionsf1,f2,f3,f4 wehave
[1.3 (1)] (f1 f3)A(f1 f4)A(f2 f3)A(f3 f4)=(f1 Af2)(f3 Af4)A where
[1.3 (2)] (√ )f2 (√ ) f3 (√ ) f4(√ ),ifrisasquarew.r.to (r)= productrepresentationofA 0 , otherwise.
We also indicate the corresponding distributive properties for six completely-A-multiplicative functions,ananalogueofTheoremofSubbarao[5].
Theorem (1.4) Forcompletely-A-multiplicativefunctionf,g,h,k,u,v,wehave
[1.4 (1)] (fAg)(hAk)(uAv)=(fhuAfhuAfkuAfkvAghuAghuAgkuAgkvA ).
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
Where (n)isanA-multiplicativefunctiondefinedforanarbitraryA-primitiveptasfollows: 1, ifi=0 0, ifi=1ori=5, (pt)>4,ori>6, (pt)>6
[1.4 (2)] ( )= -w, ifi=2, (pt) 2
2abcdrs(a+b)(c+d)(r+s), ifi=3, (pt) 3 -2abcdrsw, ifi=4, (pt)>4 (abcdrs)3 , ifi=6, (pt) 6
Wheref(pt)=a;g(pt)=b;h(pt)=c;k(pt)=d;u(pt)=r;b(pt)=s
2. WemakeuseofthegeneratingpowerseriesandBellseriesdefinedbelow: ( ) ∑ ( )
iscalledthegeneratingpowerseriesorBellseriestothebasePt
IffisA-multiplicative,fisuniquelydeterminedwhenf (x)areknownforallA-primitivespt .
Example (2.1): Mobiusfunction = (associatedwithanA-divisorsystem),whereu(n)=1forallnis determinedbythegeneratingpowerseries.
[2.1 (1)] (x)=1-x.
Result (2.2): Iffiscompleted-A-multiplicative,thenf(pti)=f(pt)i ,for0 i (pt)
Wherept isanA-primitiveanditsgeneratingpowerseriestothebasept is
[2.2 (2)] f (x)=1-f(pt)x)-1 (mod )=1-f(pt)x)-1| ρ thebinomialseries(1-f(pt)x)-1 truncatedtodegree .Likewisewehave
Result (2.3): IffandgareA-multiplicativethen
[2.3 (1)] (fAg) (x)=f (x) g (x)
Where indicatesthattheproductistheformalproductoftheconcernedpowerseriesmodulo
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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
xρ+1.Wealsohave
Result (2.4): Iff1 andf2 arecompletely-A-multiplicativeandf1 (pt)=a,f2 (pt)=bthen
[2.4 (1)] (f1 f2) (x)=[1-abx]-1| ρ
Weknow ( ) ∑* + * + = ( ) ( )( )( )( )Sivaramakrishnan[6].
Henceontruncatingtothedegreexρ,wehave ( ) ∑,* +* += ( ) ( )( )( )( )
3. Weallprovetheorem(1.1)inamoregeneralform.
Definition (3.1): Aproductk=gAhforanytwoarithmeticfunctionsg,hiscalledadiscriminativeA-productifk (n)=g(1)h(n)+g(n)h(1)holdsonlywhennisA-primitive.
Inotherwords,k=gAhisadiscriminative–A-productonlyif ( ) ∑ ( ) ( ) ( )
Theorem (3.2): Letf,g,hbeanythreearithmeticfunctions,thenfiscompletely-A-multiplicativeifandonly if[1.1(1)]holdsforadiscriminative-A-productk=gAh.
Proof: Itiseasytoverifythat[1.1(1)]holdsforallarithmeticfunctionsg,hwheneverfis completely-Amultiplicative.Inparticular,itholdswhenk=gAh,adiscriminativeA-product.
Fortheconverse,weassumer=p p p ,wheretheA-primitivesp aredistinctor not.
Weprovebyinductiononm,thenecessaryresultforcompletely-A-multiplicatively,viz.,
[3.2 (1)] ( )= f( )
Ifm=1,(3.2)istrivial.
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International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
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Assume(3.2)istrueforallm<n.LetgAhbeadiscriminative
A-productandm=n.Then,sincefsatisfies[1.1(2)], ( ) ∑ ( ) ( ) ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑ ( ) ( ) ( ) ( ) ( ) +f(p . . .p )h(1)g(p . . .p ) Therefore
( ) ( ) ( )]
So,That(3.2)holdsform=n,astheproduct gAh isdiscriminative.
Inparticular,theconverseistrue,when[1.1(1)]holdsforallarithmeticfunctionsf,g,h.
Proof of the Theorem (1.2): It is enough if we that the result is true for pi t , 1 for any A-primitive pt . Note that for any A-regular convolution there exist a corresponding unique “unitary” s-regular convolutionB. Infact 1 ifi=0
(f2 Bf3)pit)= f2 (pit)+f3(pit), ifi (f1 A(f2 Bf3)(pit)= ∑ (pjt)f2 ( )+f3 ( )+f1( ) (f1 Afk)(pit)= ∑ f1(pjt)fk( ),k=2,3
Therefore (f1 A(f2 Bf3)Bf1) (pit)=(f1 Af3)(pit).
Hence f1 A(f2 Bf3)Bf1 =(f1 Af2)B(f1 Af3)
Proof of the theorem (1.3): Consideranyprimitivebasept .
Letai =f1 (pt),i =1,2,3,4, a1 a2 ;a3 a4
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
Thenforanyi,j=1,2,3,4 i jwehave, ( ) ( ) ∑[ ] and ( ) ( ) ∑
Theniff=(f1 Af2)(f3 Af4)inviewof,[2.4(1)]wehave
(x)=(1-a1 a2 a3 a4 x2)(1-a1 a3 x)-1 (1-a2 a3)-1
(1-a1 a4 x)-1 (1-a2 a4 x)-1
Further,wewillhave
(x)=1-a1 a2 a3 a4 x2 +…….+
=(1-a1 a2 a3 a4 x2)-1
Where =[ (pt)/2].
HenceTheorem(1.3)follows:
Proof of Theorem (1.4): Itfollowsonthesamelines,usinggeneratingpowerseriestobasespt modulo
Conclusion: TheproofTheorem(1.1),(1.2),(1.3)and(1.4)followswiththeuseofgeneratingpowerseriesand Bell seriesso that(3.2)holdsform=nas the productof gAh is discriminative. Inparticularthe converseistruewhen[1.1(1)]holdsforallarithmeticfunctionf,g,h. and onverifyingthatleftsideof(1.9)hasthegeneratingseriestobasept givenby
Reference
[(1-acrx)-1 (1-acsx)-1(1-adrx)-1(1-adsx)-1(1-bcrx)-1(1-bcsx)-1(1-bdrx)-1 (1-bdsx)-1]
=[1–wx2 +2abcdrs(a+b)(c+d)(r+s)x3 –abcdrswx4 –(abcdrs)3 x6] = (x).
1. Lambek,J. - Arithmeticalfunctionsanddistributivity,Amer.Math. Monthly73(1996)969-973.
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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
2. Langford,Eric - DistribuitivityoverDirichletproductandcompletely Multiplicative arithmeticfunctions,.Amer.Math.Monthly(80)(1973)411-414.
3. Padmavathamma,M.- A study of arithmetic Functions Relative to certain Convolutions in NumberTheory.Ph.dThesis,S.V.University(1990).
4. Subrahmanya Sastri., V. V. Padmavathamma, M. , Balasekher, J.- Arithmetic functions and sregularconvolutions,RanchiUniversityMath.Journal27(1996)3952.
5. Subbarao,M.V. - Arithmetic functions and distributivity, Amer. Math., Monthly 73 (1996)969-973.
6. Siyaramakrishnan - ClassicalTheoryofArithmeticfunctions,MarcelDekkerInc.,(1989)
7. VaidyanathaSwamy,R.- The Theory of multiplicative arithmetic functions, Trans. Amer. Math.Soc.33(1931)579-662.
8. Chan, heng huat (2009), Analysis Number theory for Undergraduates. Monograph in Numsber theory. World Scientific Publishing Company ISBN 978-981-427136-3.
9. Bahri,Mawardi,Ashino,Ryuiehi,Vaillancourt,Remi(2013)ConvolutionTheoremforQuaternion fourier Tranform, Properties and Application (PDF) Abstract and appliedAnalysis;2013.
10. Padvathamma.M.andSubrahmniyaV.V.–OnCertainDistributiveand Quasi – Distributive Properties of Airthmetric function, 1997, R.U.M.J..,Ranchi,Jharkhand,India.