DISCRETE MELLIN CONVOLUTION AND ITS EXTENSIONS, PERRON FORMULA AND EXPLICIT FORMULAE

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1 DISCRETE MELLIN CONVOLUTION AND ITS EXTENSIONS, PERRON FORMULA AND EXPLICIT FORMULAE Joe Javier Garcia Moreta Graduate tudet of Phyic at the UPV/EHU (Uiverity of Baque coutry) I Solid State Phyic Addre: Addre: Practicate Ada y Grijalba 5 G P.O 6 89 Portugalete Vizcaya (Spai) joegarc@yahoo.e ABSTRACT: I thi paper we defie a ew Melli dicrete covolutio, which i related to Perro' formula. Alo we itroduce ew eplicit formulae for arithmetic fuctio which geeralize the eplicit formulae of Weil. MELLIN DISCRETE CONVOLUTION: We defie the Melli dicrete covolutio i the form c+ i a f = F( G ) = πi () c i Where ad a = G i the Dirichlet geeratig fuctio of the coefficiet a() = = F df d The proof i quite eay, firt we apply the itegral operator + f to the left of () o if the erie ivolvig a() i completely coverget, o we ca witch betwee the erie ad the itegral the, we have d a a f a t f ( t) dt u f ( u) d G F + = = = = = = () d If we apply the ivere operator of + f which i to both ide c+ i d f f πi = + the we have proved (). c i thi kid of dicrete traform i a dicrete aalogue to the Melli Covolutio theorem defied for Melli traform

2 c+ i dt (3) f g( t) = F( G ) F = df G = dg t t πi c i t> Now, if we et f = H( t ) = t t < the Coefficiet of the Dirichlet erie we recover Perro' formula [5] for c+ i a H = a = G πi ice F + = c i d = = () But oe of the bet applicatio of our Melli covolutio i related to everal a Dirichlet erie(ee [] ) i the form = G, Where G() iclude = power or quotiet of the Riema zeta fuctio for eample µ = ζ = ζ ' Λ ζ = = ζ λ = (5) ζ = ζ µ = ζ = ζ( ) ϕ = (6) ζ = The defiitio of the fuctio iide () ad () i a follow The Möbiu fuctio, µ = if the umber i quare-free (ot diviible by a quare) with a eve umber of prime factor, µ = if i ot quarefree ad if the umber i quare-free with a odd umber of prime factor. The Vo Magoldt fuctio Λ = logp, i cae i a prime or a prime power ad take the value otherwie Ω The Liouville fuctio λ = ( ) Ω i the umber of prime factor of the umber µ i if the umber i quare-free ad otherwie ϕ = p p, the meaig of p i that the product i take oly over the prime p that divide. To obtai the coefficiet of the Dirichlet erie we ca ue the Perro formula a A = G = + = c+ i A = a = G( d ) πi (7) c i If the fuctio G() iclude power ad quotiet of the Riema zeta fuctio we ca ue Cauchy theorem to obtai the eplicit formulae for eample

3 M = µ = + + ζ ' ζ '( )( ) (8) = ζ '() Ψ = Λ = + ζ() ( ) (9) = L = λ = + + ζ ζ( ) () ( / ) ζ '( ) ζ 6 ζ( ) Q = µ = ' ( ) '( ) () π ζ = ζ ζ Φ = = ( ) ( ) ϕ 6 π ζ ' = ( ) ζ '( ) () ζ Uder the aumptio that all the Riema No-trivial zero are imple. Alo we have for the Riema zeta fuctio ad it derivative ζ '( ) ( ) ζ(+ )! + π = ζ '() = log( π ) ζ () = (3) The reader will remember the relatio betwee Perro' formula ad our dicrete covolutio, uig the work of Baillie [ ] we will give differet eplicit formulae, to do o we eed to ue Cauchy' theorem o comple itegratio ad evaluate the cloed melli ivere traform by uig the reidue theorem F( G ) πi where 'C' i a cloed circuit icludig all the pole of the C Dirichlet erie G(), we ca do thi aumig all the Riema zero are imple ad that the Mellii traform F() ha o pole iide 'C', i thi cae we have the 'eplicit formulae' Λ f = F() F( ) F( ) = = () = F( ) + F( ) µ f = ζ '( ) = ζ '( ) (5) = ζ( ) F( ) λ f = F + ζ '( ) ζ (6) 3

4 6 ζ( ) F( ) F( ) ζ( ) = + + π ζ '( ) ζ '( ) ϕ f F() = = (7) ζ F 6 F( ) ζ( ) π ζ '( ) ζ '( ) µ f F() = + + = = (8) If the Melli traform ha pole iide the cloed circuit 'C' C F( G ), the thi pole will cotribute with a remaider term due to the Reidue theorem [] i thi cae we have the etra term r = Re F( G ) with k { } = k F( k) df k = = (9) thi i what happe i Perro formula, due to the tep fuctio H( ) i thi cae it Melli traform ha a pole at = ice F = thi i why i formulae (8-) there i a cotat term. A a curiou fial eample of our Melli dicrete covolutio, if we ue the Dirichlet geeratig fuctio G = ζ( k) ad the floor fuctio a a tet fuctio o [ ] d ζ =, the our Melli dicrete covolutio become the + idetity for the k-th order um of the divior fuctio c+ i k k = π c i σ d = = ( k) i ζ ζ () We have previouly ivetigated thi kid of eplicit formula [3] but itead of the Melli traform we ued the Fourier traform ad Fourier covolutio theorem for tet fuctio g() ad h() related by a dualfourier traform, o the itegral hc = dg( e ) ic eit ad i fiite for every real umber (poitive i or egative) c, ad g( α) = dh( e ) α π or g( α) = dhco( α) π depedig o if the tet fuctio are eve or ot h = h( ). For the cae of the Liouville fuctio, there i o cotributio due to the otrivial Riema zeroe -,-,-6,... ice the Dirichlet geeratig fuctio for ζ thi cae ( ) i Holomorphic o the regio of the comple plae Re( ) < ζ

5 I our previou work [3] we have tablihed imilar formulae to (-8) but i term oly of the imagiary part of the Riema zero ( γ ) ' ζ '( ) µ h + g (log ) = + dg ( e ) ζ () = γ = = ( ) h( γ) ζ ( ) λ ζ g (log ) = dg ( ) ζ / + () ' 3 ϕ h( γ ) = + ζ ( ) + = γ = γ 6 ζ( ) + (3) g(log ) dg( e ) dg( e ) π ζ ' ζ '( ) γ h 3 6 µ ζ + g(log ) dg( e ) = + ζ dg( e ) + = π ' γ ζ ( ) = ζ '( ) () Ad fially the eplcit formula for the divior fuctio σ which i the um of divior of '' σ () = = 8, give by γ h σ 5 ζ(3) (log ) = + + ζ ζ + ζ γ ζ '( ) 3 7 g g e d g e d g e 5 = π Where the um iide (-5) are over the imagiary part of the zero of the Riema zeta fuctio o the critical lie, ad = + iγ. Equatio (-8) are equivalet to the equatio (-5) but i oe had we ue the Melli traform ad i the other had we ue the Fourier traform i g( α) = dh( e ) α π, the ue of the Fourier traform i i aalogy to the Riema-Weil eplicit formula for the Vo Magoldt fuctio Λ Γ' ir i h() g(log ) = dr h logπ h( γk) π + + (6) Γ = k= I the formula (6) the um i over the poitive imagiary part of the Riema zero. For the cae of the eplicit formulae which ivolve the tet fuctio g() the Laplace Bilateral traform of thi fuctio defied by (5) 5

6 { } t L g( t) = dtg( t) e (, ) (7) mut be fiite, or at leat regularizable REFERENCES: [] Apotol Tom Itroductio to Aalytic Number theory ED: Spriguer-Verlag, (976) [] Baillie R. Eperimet with zeta zero ad Perro formula Ariv: 3.66 [3] Garcia J.J "O the Evaluatio of Certai Arithmetical Fuctio of Number Theory adtheir Sum ad a Geeralizatio of Riema-Weil Formula [] H. W. Gould ad Temba Shohiwa "A catalog of iteretig Dirichlet erie" avaliable at [5] Perro O.," Zur Theorie der Dirichletche Reihe", J. reie agew. Math. 3, 98, pp [6] Titchmarh, E. C. The Theory of Fuctio, d ed. Oford, Eglad: Oford Uiverity Pre, 96. 6