Fractional parts and their relations to the values of the Riemann zeta function

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1 Arab. J. Math. (08) 7: 8 Arabia Joural of Mathematic Ibrahim M. Alabdulmohi Fractioal part ad their relatio to the value of the Riema zeta fuctio Received: 4 Jauary 07 / Accepted: 0 Augut 07 / Publihed olie: 6 September 07 The Author() 07. Thi article i a ope acce publicatio Abtract A well-ow reult, due to Dirichlet ad later geeralized by de la Vallée Poui, expree a relatiohip betwee the um of fractioal part ad the Euler Macheroi cotat. I thi paper, we prove a aymptotic relatiohip betwee the ummatio of the product of fractioal part with power of iteger o the oe had, ad the value of the Riema zeta fuctio, o the other had. Dirichlet claical reult fall a a particular cae of thi more geeral theorem. Mathematic Subject Claificatio M06 Bacgroud I 849, Dirichlet etablihed a relatiohip betwee the Euler Macheroi cotat γ ad the average of fractioal part. More pecifically, writig [x for the itegral (floor) part of the umber x R ad {x x [x for it fractioal part, Dirichlet [3,5 proved that { ( γ + O ). (.) Thi urpriig coectio betwee γ ad the average of fractioal part wa, i tur, ued by Dirichlet to prove that the umber of divior of a iteger i of the order log. The techique itroduced by Dirichlet to prove thee reult i ofte called the hyperbola method, which i a coutig argumet to the umber of lattice poit that lie beeath a curve [3,6. I. M. Alabdulmohi (B) Computer, Electrical ad Mathematical Sciece ad Egieerig Diviio, Kig Abdullah Uiverity of Sciece ad Techology (KAUST), Thuwal , Saudi Arabia ibrahim.alabdulmohi@aut.edu.a 3

2 Arab. J. Math. (08) 7: 8 The error term i (.) i ow to be peimitic. Fidig the optimal expoet θ>0 uch that { ( γ) O( θ+ɛ ), for ay ɛ>0i ow a the Dirichlet divior problem, which remai uolved to thi date. A well-ow reult of Hardy i that θ 4, which i cojectured to be the true awer to thi problem [3. I 898, de la Vallée Poui geeralized (.). He howed that for ay iteger w N, w w { w + ( γ + O ). (.) A oted by de la Vallée Poui, thi reult i quite remarable becaue the limitig average of the fractioal part remai uchaged regardle of the arithmetic progreio that oe wihe to ue [3. More recetly, Pillichhammer obtaied a differet geeralizatio of Dirichlet reult. He howed that for ay β>, β { β ( γ /β ) β ( ) + O β+, (.3) where γ /β i a family of cotat whoe firt term i γ γ [5. I thi paper, we loo ito a differet lie of geeralizig (.). Specifically, we addre the quetio of derivig the aymptotic expreio to ummatio of the form f () {, (.4) for poitive real umber > 0. Thi i the ummatio of the product of fractioal part ad power of iteger. Iteretigly, we will how that the aymptotic behavior of thi ummatio i coected to the value of the Riema zeta fuctio ζ(), ad we will recover Dirichlet reult i (.) a a particular cae. More pecifically, we prove that for ay real umber > 0, + { ζ( + ) ( + + O ). (.5) We coclude thi ectio with two claical theorem that we will rely o i our proof: Theorem. (Abel ummatio formula) Let a be a equece of complex umber ad φ(x) be a fuctio of cla C. The, a φ() A()φ() A(t)φ (t)dt, (.6) where A(x) [x a x. Theorem. (Euler Maclauri ummatio formula) We have φ() A + φ(t)dt + r B r r! φ(r) () + O(φ () ()), (.7) for ome cotat A, where B, B 6, B 3 0,...are the Beroulli umber. Thee reult ca be foud i may place, uch a [,4. 3

3 Arab. J. Math. (08) 7: 8 3 Notatio We will ue the followig otatio: [x deote the itegral (floor) part of x ad {x x [x deote the fractioal part. N deote the et of poitive iteger, ofte called the atural umber; Z + i the et of o-egative iteger; R i the et of real umber; C i the et of complex umber. R() deote the real part of C. 3 The fractioal traform 3. Overview The ey iight we will employ to derive the aymptotic expaio of the fuctio f () i (.4) ithatwe ca olve thi problem idirectly by awerig a differet quetio. Specifically, we will be itereted i the followig fuctio: (). (3.) + More geerally, whe () φ(), (3.) + we will call () the fractioal traform of φ(). () allow u to awer our origial quetio becaue () + { ) ( ( ), where we have ued the fact that {/ { 0. By expadig the right-had ide uig the biomial theorem, we obtai a method of olvig our origial quetio. 3. Prelimiary We preet a few ueful lemma related to the fractioal traform defied above. Before we do thi, we itroduce the followig ymbol: { [ () N : +, N. (3.3) I other word, () i the et of poitive iteger N that are le tha, ad for which the iterval [/( + ), / cotai, at leat, oe iteger. For itace, (5) becaue the iterval [5/3, 5/ cotai the iteger two, wherea 3 / (5) becaue the iterval [5/4, 5/3 lie trictly betwee ad. Lemma 3. { { + ( + ) where S deote the ize (cardiality) of the et S. Proof We have: { { ( + + ( + ) ) [ ([ + [ [ + [ +, N, (3.4) ) + [ ( + ) +, N. 3

4 4 Arab. J. Math. (08) 7: 8 Lemma 3. If ad (), the { { + ( + ). Proof The iterval [/( + ), / ca cotai, at mot, a uique iteger ice: + ( + ) Thi fact ad Lemma 3. both imply the tatemet of the lemma. ( + ) <. 4Maireult We begi with the followig lemma: Lemma 4. For ay real umber > 0, [ ɛ where the cotat i O( ) deped o. Proof Firt, let u coider the followig fuctio: Sice we obtai w { w O( ɛ ), (4.) + g (w) { w w [{ { + { g (w) w + w + { { ( ) g (w) + w, w + w { Coequetly, we coclude that w g (w) w. Uig Theorem., [ ɛ + [ ɛ + [ ɛ g ([ ɛ ) ( ) [ ɛ ( ) [ ɛ + Here, we ued the fact that g(w) w. Similarly, [ ɛ [{ [ ɛ [ ɛ { ( + + Therefore, the tatemet of the lemma follow. Now, we are ready to prove our firt mai reult. 3 [ ɛ g (t) t dt t dt g (t) t dt ( + ) [ ɛ. ) [ ɛ.

5 Arab. J. Math. (08) 7: 8 5 Theorem 4. For ay >, Proof We plit the um ito two part: () ( + + ζ() + O ). [ + + [ +. (4.) + The firt term i O( ) a proved i the previou lemma, which i O( ) whe >. Next, we examie the ecod term. We have by Lemma 3., [ + + [ + [ + [ + [ + ( + ) ( + ) ( + ) [ + [ [ [ ( [ +, N ) + [ ( + ) + O( ). { Uig the fact that for R() >0(ee[), we coclude that ζ() w w + w {t dt, w t+ w + ζ() + O(w ). Alteratively, the error term O(w ) i the above expreio ca be derived from Theorem.. Hece, [ [ [ [ + ζ() + O ( { ) ( + O + ζ() + O ( ) ( ) + ζ() + O ( )) ( ) + ζ() + O ( ). 3

6 6 Arab. J. Math. (08) 7: 8 Aareult, [ ( ) + ζ() + O( ) (ζ() ) O( ). Fially, we loo ito the remaiig term. Uig Theorem.,wehaveforay >, [ + [ + [ + [ [ + [ [ + + ( + ) ( 3 + ) + (+) + O( ). Here, we ued the fact that for ay u, we have by Theorem., u C(u) + u+ u + u + [ u C(u) + u+ u + + u + m m ( ) u B u+ + O( um ) ( ) u B u+ + O( um ), for ome cotat C(u) that i idepedet of. Puttig everythig together, we coclude that for ay real umber >, which i the tatemet of the theorem. [{ { [ ( ) + + ζ() + O, Theorem4. i illutrated i Fig.. Clearly, thi theorem geeralize Dirichlet reult, a promied earlier, becaue { lim ζ() γ ad the fact that { ( ) { ( ). + Now, we are ready to derive the aymptotic expreio of the fuctio f () give i (.4). Theorem 4.3 For ay real umber > 0, 3 + { ( ζ( + ) + + O ).

7 Arab. J. Math. (08) 7: x x Fig. A compario betwee the value of () mared i blue ad the aymptotic expreio derived i Theorem 4. mared i red. Thex-axi i while the y-axi i (). Theleft, middle, adright figure correpod to, ad 3, repectively Proof Let f () be a defied i Eq. (.4). The, writig by Theorem 4., [ ζ( + ) + + O( + ) + + { ) ( + ( ) + { + ( ) + () + + ( ) + () + f + () + + ( ) ( + ) f () + () + f + (). However, Therefore, Hece for > 0, 0 f () { O( + ) + ( ) + () + f + () O( ). { [ + ζ( + ) f () + + O( + ), + which implie the tatemet of the theorem. 3

8 8 Arab. J. Math. (08) 7: 8 5Cocluio I thi paper, we geeralized Dirichlet claical reult o the coectio betwee the Euler Macheroi cotat ad the average of fractioal part. Our theorem reveal that the fractioal part are, i geeral, coected to the value of the Riema zeta fuctio ζ(). Hece, ζ() with > ca be expreed a a limitig average of the product of fractioal part with power of poitive iteger. Ope Acce Thi article i ditributed uder the term of the Creative Commo Attributio 4.0 Iteratioal Licee ( creativecommo.org/licee/by/4.0/), which permit uretricted ue, ditributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author() ad the ource, provide a li to the Creative Commo licee, ad idicate if chage were made. Referece. Cvijovic, D.; Srivatava, Hari M.: Limit repreetatio of Riema zeta fuctio. Amer. Math. Moth. 9(4), (0). Hardy, G.H.: Diverget Serie. Oxford Uiverity Pre, New Yor (949) 3. Lagaria, J.: Euler cotat: Euler wor ad moder developmet. Bull. Am. Math. Soc. 50(4), (03) 4. Lampret, V.: The Euler Maclauri ad Taylor formula: twi, elemetary derivatio. Math. Mag. 74(), 09 (00) 5. Pillichhammer, F.: Euler cotat ad average of fractioal part. Am. Math. Moth. 7(), (00) 6. Stopple, J.: A primer of aalytic umber theory: from Pythagora to Riema. Cambridge Uiverity Pre, Cambridge (003) Publiher Note Spriger Nature remai eutral with regard to juridictioal claim i publihed map ad ititutioal affiliatio. 3