New Properties of Fourier Series and Riemann Zeta Function

Transcription

1 New Properties of Fourier Series and Riemann Zeta Funtion Guangqing Bi a, Yuekai Bi b b Shool of Eletroni and Information Engineering, BUAA, Beijing 9, China yuekaifly@63.om arxiv:8.546v4 [math.ap] 9 De Abstrat We establish the mapping relations between analyti funtions and periodi funtions usingthe abstrat operators osh xand sinh x, inludingthe mapping relations between power series and trigonometri series, and by using suh mapping relations we obtain a general method to find the sum funtion of a trigonometri series. Aording to this method, if eah oeffiient of a power series is respetively equal to that of a trigonometri series, then if we know the sum funtion of the power series, we an obtain that of the trigonometri series, and the non-analytial points of whih are also determined at the same time, thus we obtain a general method to find the sum of the Dirihlet series of integer variables, and derive several new properties of ζn +. Keywords: Zeta funtions; Bernoulli numbers; Fourier series; Abstrat operators; Mapping. MSC Subjet Classifiation M6; 4A4; M35; 35S99 Introdution The trigonometri series, espeially the Fourier series, is of great importane in both mathematis and physis, and expanding periodi funtions into Fourier series has beome a very mature theory. During the early years of last entury, people realized the signifiane of the inverse problem, whih is how to find the sum funtion of a ertain Fourier series. So the question is: an we use the sum funtion of power series to obtain that of trigonometri series? As we know, the domain of funtions expressed by trigonometri series an be extended into the entire number axis with the existene of denumerable non-analytial points, thus funtions expressed by trigonometri series are pieewise analyti periodi funtions. If a trigonometri series onverges to an analyti sum funtion in a ertain interval, then it is quite natural that there is a mapping relation between a periodi funtion and an analyti funtion, though the two endpoints of the interval generally are its non-analytial points. Therefore, we an use the sum of power series to obtain that of orresponding trigonometri series, onverting the researh fous from trigonometri series to power series. For a long time, the first author has realized the fat that the abstrat operators osh x and sinh x an express the mapping relations between periodi funtions and analyti funtions more distintly, whih has beome a signifiant tool to obtain the sum funtion of trigonometri series. Why the abstrat operators osh x and sinh x an establish suh relations? That is beause this kind of operators is also a kind of funtions, ontaining the duality between periodi funtions and linear operators. By using suh a duality, the authors have preliminarily established theories of partial differential equations of abstrat operators in referene []-[5]. Corresponding author. yananbiguangqing@sohu.om

2 What is an abstrat operator? The operator ft, x is generally interpreted as a Taylor expansion, alled the infinite order operator. However, the need to onsider its disk of onvergene is a major onstraint of its broad appliations. Therefore, the first author has defined the operator ft, x as ft, x e ax ft,ae ax in 997. Eah operator ft, x has a set of algorithms without the need to use its Taylor expansion, and the first author has also provided a method to determine suh kind of algorithms. In this sense, the operator ft, x is known as the abstrat operator in referene []. However, the onept of abstrat operators has not yet been spread adequately. As a result, it is easy to mistake the abstrat operator for the infinite order operator by the similar symbol they are using. In fat, only several simple abstrat operators an be expanded into Taylor series under ertain onditions, suh as osh x and sinh x. However, the partiular method we use to alulate is not by using their Taylor expansions but the ertain algorithms of osh x and sinh x established by the authors in their former published papers, suh as the formulas 4, 7,, 3 and 4. Although there may be several non-analytial points in the proess of alulation, making the results at these points not tenable, yet we will not enounter the infinite series, whih benefits us a lot by avoiding the need to onsider the disk of onvergene. The abstrat operatorsosh x and sinh x haveexlusive advantagesin symbol expressions, and the summation method of Fourier series established from whih an be diretly extended into multiple Fourier series. By using this summation method of Fourier series, the abstrat operators an also have signifiant appliations in the Riemann Zeta funtion ζm of the analyti number theory. The Riemann Zeta funtion ζs defined usually by the Dirihlet series ζs ns, Rs >. In 735, Euler proved that for an arbitrary even number K >, ζk a K π K, where a K is a rational number. However, for all odd numbers K +, the arithmeti properties of ζk + are still unknown. As the summation method of trigonometri series of abstrat operators is quite suitable to find the sum of the Dirihlet series of integer variables inluding the Zeta funtion, we take an important step in studying the arithmeti properties of ζk +. Basi formulas of abstrat operators Whenatingonelementaryfuntions,theabstratoperatorsosh x andsinh x have omplete basi formulas as differential operations, now we are going to use the algorithms of abstrat operators in referene [] or [4] to establish these formulas. Firstly, aording to the definition of abstrat operators, we have osh x e bx osbhe bx, sinh x e bx sinbhe bx. Where bx b x +b x + +b n x n, bh b h +b h + +b n h n. Aording to Theorem in referene [], namely and Theorem 3: expih x fx fx+ih, osh x fx R[fx+ih], sinh x fx I[fx+ih],

3 we have osh x osbx oshbhosbx, osh x sinbx oshbhsinbx, sinh x osbx sinhbhsinbx, sinh x sinbx sinhbhosbx. 3 Based on 3, and by using Theorem 6 in referene []: we have sinh x u v osh x u v osh xv sinh x u sinh x v osh x u osh x v +sinh x v, osh xv osh x u+sinh x v sinh x u osh x v +sinh x v, 4 osh x tanbx osh x otbx sinh x tanbx sinh x otbx sinbx oshbh+osbx, sinbx oshbh osbx, sinhbh oshbh+osbx, sinhbh osbx oshbh. 5 For seant and oseant funtions, similarly to 5, we have osh x sebx osh x sbx sinh x sebx sinh x sbx oshbhosbx oshbh+osbx, oshbhsinbx oshbh osbx, sinhbhsinbx oshbh+osbx, sinhbhosbx osbx oshbh. 6 Aording to the following theorem: Theorem. [4] If y fbx J set of analyti funtions is the inverse funtion of bx gy, namely gfbx bx, then sinh x fbxdenoted by Y and osh x fbxdenoted by X an be determined by the following set of equations: os Y sin gx bx X gx bh Y X x R n, h R n, 7 we an derive basi formulas of the orresponding inverse funtion: 3

4 osh x lnbx ln bx +bh /, sinh x lnbx arot bx bh. 8 sinh x artanbx bh tanh +bx +bh, osh x artanbx artan bx 9 bx bh. sinh x arotbx +bx +bh oth, bh osh x arotbx +bh arotbx. bx Proof. Here we only give the detailed proof of 8 and 9. Aording to and 7, we have e X osy bx e X siny bh X osh x lnbx, Y sinh x lnbx. By solving this set of equations we have 8, and aording to 5 and 7, we have sinx oshy +osx bx sinh Y oshy +osx bh X osh x artanbx, Y sinh x artanbx. By solving this set of equations we have +bx +bh + sin X +sinh Y oshy +osx oshy oshy +osx. Aordingtotheseondexpressionofthissetofequations,wehaveoshY+osX sinhy/bh, and by substituting it into the above expression, we have +bx +bh bhoshy sinh Y or tanhy bh +bx +bh. Thus the first expression of 9 is proved, similarly we an prove the seond one. For hyperboli and inverse hyperboli funtions, we an also derive the orresponding basi formulas. For instane, orrespondingly to 3, we have osh x oshbx osbhoshbx, osh x sinhbx osbhsinhbx, sinh x oshbx sinbhsinhbx, sinh x sinhbx sinbhoshbx. 4

5 Irrational funtions an be onsidered as the inverse funtions of rational funtions. For instane, bx / is the inverse funtion of bx, thus similarly we have sinh x bx / osh x bx / bx +bh bx, bx +bh +bx. To alulate onisely, the algorithms of produts and omposite funtions in referene [] are listed as follows, while the algorithm of quotients has already been given in 4: sinh x vu osh x v sinh x u+sinh x v osh x u, osh x vu osh x v osh x u sinh x v sinh x u. osh x fgx os Y sinh x fgx sin fx, X fx. Y X 3 4 Where x R n, h R n, X osh x gx, Y sinh x gx. 3 Summation method of trigonometri series Theorem. Let St J be the sum funtion of the power series n a nt n, fx L [a,b] be the sum funtion of the orresponding osine series, and gx L [a,b] be that of the orresponding sine series, namely St fx gx a n t n, n n n a n os nπx, t R, t r, < r < +. a n sin nπx, x R, a < x < b, then we have the following mapping relations: fx os Se z z, gx sin Se z z, z R. 5 And the endpoints a and b of the interval a < x < b are non-analytial points singularities of Fourier series, whih an be uniquely determined by the detailed omputation of the right-hand side of 5. 5

6 Proof. By substituting Se z n a ne nz into 5, we an prove Theorem. The sum funtion of infinite power series an be an elementary funtion, whih in most ases an be expressed as the definite integral of an elementary funtion, thus in the appliation of Theorem, the following theorem an be partiularly useful: Theorem 3. Let Sx J be an arbitrary analyti funtion integrable in the interval [,], then we have e z os z Sξdξ π Se z de z sin [Se z e z ] z dx. 6 e z sin Se z de z z π os z [Se z e z ] dx. 7 Proof. 6 and 7 are operator formulas. Aording to the analyti ontinuous fundamental theorem in referene [], we only need to prove this set of formulas when Sx x n, n N, this is obvious. Theorem 4. Let Sx J be an arbitrary analyti funtion integrable in the interval [,], if t Stdt is the sum funtion of the power series a nt n, let fx L [a,b] be the sum funtion of the orresponding osine series, and gx L [a,b] be that of the orresponding sine series, namely t St dt fx gx a n t n, a n os nπx, t R, t r, < r < +, a n sin nπx, x R, a < x < b. Then we have the following mapping relations: fx gx π Sξdξ π os sin [Se z e z ] z dx, [Se z e z ] dx. z 8 And the endpoints a and b of the interval a < x < b are non-analytial points singularities of Fourier series, whih an be uniquely determined by the detailed omputation of the right-hand side of 8. Proof. Combining Theorem with Theorem 3 will lead us to the proof. Apparently, suh summation method of Fourier series an be extended into other trigonometri series. Theorem 5. Let St J be the sum funtion of power series n a nt n, fx be the sum funtion of the orresponding n a nosnxos n x, gx be the sum 6

7 funtion of the orresponding n a nsinnxos n x, namely St a n t n, t R, t r, < r < +, fx gx n a n osnxos n x, n a n sinnxos n x, x R, a < x < b, n then we have the following mapping relations: fx os xρ Sρ ρ os ρosx gx sin xρ Sρ ρ sin ρosx Y X Y X SX, SX. 9 Where X ρosx os x, Y ρsinx osxsinx. And the endpoints a and b of the interval a < x < b are non-analytial points singularities of trigonometri series, whih an be uniquely determined by the detailed omputation of the right-hand side of 9. Proof. Aording to the Definition 5 in referene[4], the following two expressions are obvious: fx os xρ ρ gx sin xρ ρ Sρ, ρosx Sρ, ρosx where os xρ ρ and sin xρ ρ are the abstrat operators taking ρ ρ as the operator element. By using the Theorem 3 in referene [4], namely Let ρ R n, θ R n, X ρ osθ,...,ρ n osθ n, Y ρ sinθ,...,ρ n sinθ n, then for an arbitrary analyti funtion fρ, we have where os θρ fρ osy X fx, ρ sin θρ fρ siny X fx, ρ θρ ρ θ ρ + +θ n ρ n, ρ ρ n we an obtain 9 immediately. Corollary. If the power series expansion of an analyti funtion is unique, then aording to the mapping relations between analyti funtions and periodi funtions, the Fourier series expansion of a periodi funtion is unique as well. It is easy to prove the uniqueness of power series expansions of analyti funtions, thus Corollary an atually derive the uniqueness of Fourier series expansions of periodi funtions easier. 7

8 Theorem an be diretly extended into the multiple Fourier series while the form of expressions remains essentially onstant, namely Theorem 6. Let St JD be the sum funtion of power series n a nt n, fx L Ωbethesumfuntionoftheorrespondingosineseries,andgx L Ω be that of the orresponding sine series, namely St a n t n, t D R m. fx gx n n n a n os nπx, a n sin nπx, n Nm, x Ω R m, then we have the following mapping relations: fx os Se z z, gx sin Se z z, z R m. And the non-analytial points singularities on the border Ω an be uniquely determined by the detailed omputation of the right-hand side of. Here the signs appearing in these formulas should be interpreted as the following universal abbreviation, namely nx n x +n x + +n m x m, a n a n,n,...,n m, e z e z,e z,...,e zm, x z x +x + +x m, z z z m n n n n m And the meanings of other t,x,t n are the same with the universal signs. Example. Trigonometri series: the sum funtion of /nsinnxosn x is gx π/ x, and its non-analytial points are x and x π, namely. π x n sinnxosn x, < x < π. Proof. By using Theorem 5, and the algorithms and basi formulas in this paper, we have St n tn ln t, t R, t <, gt gx sin xρ Sρ ρ arot X Y n sinnxosn x, x R, a < x < b. ρosx sin Y SX sin Y ln X X X arot os x sinxosx arot sin x sinxosx. 8

9 When sinx, namely x and x π thus a, b π, we have gx arot sinx osx arottanx arototπ x π x. Example. Fourier series: the sum funtion of [ n /3n 3n+ ]os3nωt is ft 3π/9osωt /, and its non-analytial point is ωt π/3, namely 3 9 πosωt n 3n 3n+ os3nωt, π 3 < ωt < π 3. 3 Proof. By using Theorem, and the algorithms and basi formulas in this paper, let Sx ft n x 3n 3n 3n+, x R, x <, n 3n 3n+ os3nωt, t R, a < ωt < b. As it is diffiult to obtain the sum funtion Sx diretly, we an use the operators d d dx x and dx x to transform the power series into the series familiar to us, then we an obtain Sx, and then ft, namely d d dx xdx xsx n x 3n x +x 3. To desribe onisely, the following results will be given diretly: Sx x x xdx x +x 3 dx ln x x+ x x 6 x ln+x+ 6x artan x + π 4x 3 6. ft os ωt Se z z 6 os ωt z [sinhzln e z e z + ] 3 os ωt [sinhzln+e z ] z + os ωt [ oshz artan ez + π 3 z 3 6] 6 sinωtarotosωt osωt osωt sinωt + 3 sinωtarot os ωt/ sinωt/ osωt/ + 3osωt 3 osωtartan + π osωt 6 3 osωt. 9

10 When osωt, and osωt/, namely ωt π/3 and ωt π, we have ft 6 sinωtarototωt+ ωt sinωtarotot osωtartan 3+ π 6 3 osωt 3 9 πosωt. There are four non-analytial points in the interval [ π,π]: ωt π, π/3,π/3,π, thus a π/3, b π/3. Therefore, Example is proved. 4 The Zeta funtion of odd variables Definition. Let S t be a funtion analyti in the neighborhood of t and S t then S m t is defined as S m t a n t n, t dt t m t t < r, < r < +, S t dt t t n a n nm. 4 Apparently S m is the sum funtion of the Dirihlet series taking m as the variable. Lemma. Aording to 4, S m t satisfies the following reurrene relation: t S m t dt t S mt. 5 Lemma. The sum funtion S m has the following reurrene property: os S m e z z S m π sin S m e z z dx. 6 Proof. Taking Sx S m x/x in 6, it is proved by using Lemma. Similarly, Lemma 3. S m t has the following reurrene property: sin S m e z z π os S m e z z dx. 7 Theorem 7. The sum funtion S m has the following property: os S m e z z ksm k k! + r π r dx r sin z S m r e z dx. 8

11 Proof. We an use the mathematial indution to prove it. Aording to Theorem 3, it is obviously tenable when r in 8. Now we indutively hypothesize that it is tenable when r K, namely os z S m e z ksm k K k! + K π K dx K sin z S m K e z dx. Using Lemma 3 and respetively, then the above expression turns into os S m e z z ksm k K k! + K π K dx K os z S m K e z dx K ksm k + K π K Sm K xk k! K! + K+ π K+ dx sin S m K e z z dx K+ K ksm k + K+ π k! K+ dx K+ sin S m K e z z dx. Thus it is tenable when r K +, and then Theorem 7 is proved. Similarly we an prove the following theorems: Theorem 8. The sum funtion S m has the following property: sin S m e z z k k! k + r π r dx k Sm k r os z S m r e z dx. 9

12 Theorem 9. The sum funtion S m has the following property: sin S m e z z r k k! k + r π r dx r k Sm k sin S m r e z z dx. Theorem. The sum funtion S m has the following property: os S m e z z ksm k k! + r π r dx r os z S m r e z dx. Lemma 4. In the interval,, we have: sin ln e z z π πx, < x <. sin ln+e z z πx, x <. os artane z z π, x < / Proof. By using the algorithms and basi formulas in this paper, we have sin ln e z z sin Y lnx X arot X Y arot osπx/ sinπx/ sin πx/ arot sinπx/osπx/. When sinπx/ or x, x, the above expression turns into sin ln e z z arottan πx π πx. Similarly we have sin z arot +osπx/ sinπx/ ln+e z sin Y X lnx arot X Y os πx/ arot sinπx/osπx/.

13 When osπx/ or x, the above expression turns into sin ln+e z z arotot πx πx. os z artane z os Y X artan X X +Y artan artan X osπx/ os πx/+sin πx/. When osπx/ or x /, the above expression turns into os artane z z artan π 4. Theorem. In the interval,, we have the following Fourier series expressions: nπx os nr k! + r π kζr k r! r r! r. 33 nπx sin nr+ r k k ζr + k k! k π + r r r+. r! r+! 34 Proof. Aording to Definition, if S t ln t, then S m t t n n m, S m ζm. In Theorem 7, let m r, and by using Theorem, we have nπx os nr ksr k k! + r π r dx r 3 sin z ln e z dx.

14 Where S r k ζr k, by using Lemma 4 we have 33. Let m r + in Theorem 9, using Theorem, and onsidering S r+ k ζr+ k and Lemma 4, we have 34. Corollary. In Theorem, when r, the Fourier series expression is tenable at the endpoints of the interval. Proof. Observing the summation formulas of trigonometri series given by Theorem, if we use the integral operator dx to integrate both sides of 33, we an obtain 34. It means that, for any point x in the interval of onvergene of summation formulas of trigonometri series, if x is within the interval or at the endpoints of it, then the termwise integration of the trigonometri series by the integral operator dx onverges uniformly to the integration of the sum funtion. Atually, in the theory of Fourier series, aording to the integrability, if there is a ertain summation formula of trigonometri series in the open interval a < x < b, then the termwise integration of the trigonometri series by the integral operator x dx, x [a,b] onverges uniformly to the integration of the sum funtion. In other a words, for a summation formula of trigonometri series in a < x < b, if we use the integral operator dx, x [a,b] to integrate both sides of the equation, then the a new summation formula of trigonometri series obtained is definitely tenable in the losed interval a x b. Thus Inferene 3 is proved. Theorem. The Riemann Zeta funtion ζn satisfies the following reurrene formula: k π k k +! ζr k π r r r. 35 r +! Proof Let x in the seond expression of Theorem, then it an be proved. Theorem 3. The Riemann Zeta funtion ζn satisfies the following reursion formula: k πk k +! ζr k π r r r. 36 4r+! Proof. Let x in the seond expression of Theorem, then it an be proved. Theorem 4. In the interval,, we have the following Fourier series expressions related to ζn+: nπx os nr+ kζr + k k! + r π r dx ln r sin πx dx. 37 Proof. Taking m r +, S t ln t in Theorem, onsidering it is in the interval,, and by using the algorithms and basi formulas in this paper, we have os z ln e z 4

15 os Y X ln os πx lnx lnx +Y +sin πx ln sin πx and S r k ζr+ k, thus it is proved by using Theorem. Lemma 5. Let m be an arbitrary natural number, then for lnx we have the following integral formula: dx m, lnxdx xm m! lnx H m. 38 Where H m are the harmoni numbers. Theorem 5. Let r N be an arbitrarynaturalnumber, and Bk be the Bernoulli numbers, Bk satisfies the following reurrene formula: r+ k Bk+ k +, 39 then the Riemann Zeta funtion ζn + an be reursively determined by the following reurrene formula, namely ζr+ 4r+ 4r+ + r k π kζr+ k k! k + r r+ π r 4r+ + r H r ln π r! + r r π r 4r+ + r k k B k r+k! π k, 4 or, equivalently, ζr+ 4r+ 4r+ + r k π kζr+ k k! k + r r+ π r 4r+ + r H r ln π r! + r r+ π r 4r+ + r k k! 4 k r+k!k ζk. 4 Proof. Aording to Theorem 4 and Lemma 5, in the interval, if r, then in the losed interval [,] we have nπx os nr+ k! kζr r + k+ r! rln 5

16 ln πx + r π r dx r kζr + k k! + r r! + r π r H r ln πx r dx ln r kζr + k sin πx πx k! + r r H r ln πx r! + r π r dx k r kζr + k k! + r r H r ln πx r! + r k k B k r+k! r+k. πx sin +ln πx dx dx k Bk k k! k dx Let x /, onsidering os k π sinkπ, os kπ oskπ k we have r r+ηr+ π kζr + k k! + r π r H r ln r! π + r k k Bk π r+k. r+k! Where ηr + is the Dirihlet Eta funtion, defined usually by the Dirihlet series: Apparently ηs n n s, Rs >. 4 ηr+ r r ζr+, r N. 43 Substituting 43 into the above expression, then it is proved. 6

17 Example 3. In Theorem 5, taking r, we have ζ3 6π 35 4π 35 ln π + π 35 k B k πk k4 k k +!. 44 In general, of whih a useful generalization is Theorem 6. We have the following relation between the Riemann Zeta funtions ζr+ and ζr, r N: nπx os nr+ kζr+ k k! + r r H r ln πx 45 r! + r r k!ζk x k, x. k r+k! k In 999, by using another method in referene [6], H. M. Srivastava has obtained the following results n N: ζn+ n π n [ Hn ln π 4n+ + n n! n k ζk + + n k! + πk k k k! n+k! ] ζk 4 k. 46 ζn+ n π n [ Hn ln 3 π 3 n n ++ n n! ] n k ζk + k! ζk + n k! + 3 πk n+k! 6 k. k k 47 Obviously 4 and 46 are idential. By using Theorem 6 and properties of the Hurwitz Zeta funtion, we also obtain the 47. The Hurwitz s generalized Zeta funtion ζs,a and the Lerh transendent Φz,s,a are usually defined by and so that ζs,a Φz,s,a n n+a s, Rs > ; a,,, 48 n z n n+a s, z ; a,,,, 49 ζs, ζs, ζs, ζs and ζs,a+φ,s,a a s ζs,, 5 are known to be meromorphi that is, analyti everywhere in the omplex s-plane exept for a simple pole at s with residue. 7

18 There is also the multipliation theorem see also referene [7] m s ζs of whih a useful generalization is m k ζs, k, m N, 5 m so that m ζs,a+ k m ms ζs,ma, m N, 5 ζs,/3+ζs,/3 3 s ζs and ζs,/3+ζs,4/3 3 s ζs, ζs, 53 are known to be meromorphi. In Theorem 6, taking x /3, we an apply the identities 5 and 53 in order to prove the following series representations for ζn + : Theorem 7. Let r N be an arbitrarynaturalnumber, and Bk be the Bernoulli numbers, then the Riemann Zeta funtion ζn + an be reursively determined by the following reurrene formula, namely ζr+ r+ 3 r 3 r r ++ r k π kζr + k k! 3 k + r r+ π r 3 r r ++ r H r ln π r! 3 + r 4π r 3 r r ++ r k k! ζk r+k! 6 k, 54 or, equivalently, ζr+ r+ 3 r 3 r r ++ r k π kζr + k k! 3 k + r r+ π r 3 r r ++ r H r ln π r! 3 + r π r 3 r r ++ r k B k r+k!k π 3 k. 55 Obviously 54 and 47 are idential. Proof. Taking x /3 in Theorem 6, we obtain n r+ os nπ 3 π kζr + k k! 3 + r π r H r ln π r! r π r k k!ζk 3 k. r+k! 3 k 56 8

19 Where n r+ os nπ 3 os π nπ os 3n r+ 3 π 3 3nπ os 3n r+ 3 + os 3n+ r+ 3nπ 3 + π 3 Φ,r+,/3 3 r+ ηr+ 3 r+ Φ,r+,4/3 3 r+ r ζr +,/3 ζr +,/3 3 r+ r ζr + 3 r+ r ζr +,/3 ζr +,4/3 3 r+ ζr +,/3+ζr +,/3 r+ 3 r+ r ζr + 3 r+ + ζr+,/3+ζr+,4/3 3 r+ 3r+ ζr+ r+ 3 r+ r ζr + 3 r+ + 3r+ [ζr+ ] ζr+ 3 r+ + r+ 3 r r+ 3 r ζr +. So the 54 is proved. Connon [8], Srivastava and Tsumura [9] reported for Rs > nπ os ns 3 6 s 3 s s +ζs, 57 nπ os ns 3 3 s ζs, 58 nπ sin 3 ns 3 { 3 s ζs+3 s ζs, 3 }, 59 nπ os ns s s ζs, 6 Taking x /3 in 34, we an apply the 59 in order to prove the following Corollary: Corollary 3. For Hurwitz Zeta funtion and Riemann Zeta funtions, we have following identities r N: 3 [ζr+, ] 3r+ + ζr+ 3 r kπk+ k +! 3r k ζr k+ r r π r+ 6r+. r+! 6 9

20 Theorem 8. For r N, in the interval [,] we have the following Fourier series expressions: n nπx os nr k! kηr k+ r r! 6 r. n nπx sin nr+ r k k ηr+ k k! k + r r+. r+! 63 Proof. Aording to Definition, if S t ln+t, then S m t n tn n m, S m ηm. In Theorem 7, let m r, then S r k ηr k. By using Theorem and Lemma 4, thus the 6 is proved. In Theorem 9, let m r +, then S r+ k ηr + k. By using Theorem and Lemma 4, thus the 63 is proved. Theorem 9. The Dirihlet Eta funtion ηn satisfies the following reurrene formula: k π k k +! ηr k r π r r+!. 64 Proof. Let x in the seond expression of Theorem 8, then it an be proved. 5 Sum-a K π K of the Dirihlet series We have already known many Dirihlet series of integer variables have a sum similar to the one given by ζn n Bn π n, 65 n! therefore, we introdue the definition of the sum-a K π K : Definition. The sum-a K π K of the Dirihlet series of integer variables are numbers like a K π K, where K is a natural number, and a K is a rational number. Of ourse we expet the Riemann Zeta funtion ζr+ has the sum-a K π K, but to our disappointment, the alulation formula of ζr + ontains the infinite power series of π, though it onverges very fast. Then we would like to ask: Whether or not ζr+ has a sum-a K π K?

21 Theorem. If the Dirihlet series of integer variables denoted by fm has a sum-a K π K, then the orresponding osine series definitely onverges to the polynomial funtion in a ertain interval. Proof. Denoting the sum of the Dirihlet series of integer variables as fm, namely a n fm nm, m N. 66 Espeially, we have fm { ζm, an, ηm, a n n. If the orresponding osine series of fm has an analyti sum funtion in the interval a < x < b, we an expand this analyti funtion into a power series in the neighborhood of x, denoted by Pπx/, namely a n nπx os P nm, a < x < b. Differentiating both sides of the equation to obtain derivatives of all orders less than m, and let x, we derive that the first several terms of oeffiients of Pπx/ are related to fm k, k,,...,[m/], namely [m/] P fm k k! k +R. If x is in the interval a,b, then differentiating at x we have [m/] k k fm k πk k! +R π. 67 We would like to ask: Is R π in 67 unique? Is there another power series R π of π making 67 tenable? As we know, if the analyti funtion gx is analyti in a neighborhood of x, then in a ertain interval a x b of x, we have the unique power series expansion: gx a +a x+a x +, a x b. In other words, at any point x x in the interval a x b, the power series of the funtion gx is unique. Aording to suh a uniqueness of power series expansions, the power series R π of π in 67 is definitely unique. Therefore, if R π is an infinite power series of π, then it definitely annot be a polynomial of π, vie versa. If the Dirihlet series fm has a sum-a K π K, aording to 67, R π definitely has the form of the sum-a K π K. But unless Rπx/ is a polynomial, R π annot be the sum-a K π K, therefore, Rπx/ is definitely a polynomial and annot be an infinite power series at the same time, thus Pπx/ is definitely a polynomial. Then Theorem is proved. Theorem. Let r, and B n be the Bernoulli numbers, then in the interval [,], we have the following Fourier series expansion related to ηn+:

22 n nπx os nr+ r k! + r+ kηr+ k n B n nr+n! 68 r+n. Proof. Aording to Definition, if S t ln+t, then n tn S m t n m, S m ηm. In Theorem, let m r +, using Theorem, and onsidering S r k ηr+ k we have n nπx os nr+ k! + r π r dx kηr + k r os z ln+e z dx. In the above expression, by using the algorithms and basi formulas in this paper we have os ln+e z z lnx +Y ln +os πx +sin πx ln os πx n n B n πx n, ln x <. nn! Substituting this result into the above expression, and onsidering η ln, then when r, the formula is tenable at the endpoint x, thus we have 68 immediately. Theorem. The Riemann Zeta funtion ζn + of odd variables does not have a sum-a K π K. Proof. Aording to Theorem, the neessary ondition for the Dirihlet Eta funtion ηn+havingthesum-a K π K isthattheorrespondingosineseriestakesa polynomial as its sum funtion in a ertain interval. Aording to Theorem and the uniqueness of power series expansions, the orresponding osine series of ηn+ has an analyti sum funtion only in the interval [,], but suh a sum funtion annot be a polynomial, thus the Dirihlet Eta funtion ηn+ annot have a sum-a K π K. In addition, ζn+ n n ηn+, n N. 69

23 Therefore, the Riemann Zeta funtion ζn + of odd variables does not have a sum-a K π K as well. In preisely the same manner, we an prove the following results for the Dirihlet Beta funtion βn+: Corollary 5. For r N, in the interval [ /,/] we have the following Fourier series expressions: n n+πx os n+ r+ n k! kβr+ k+ r π 4 r! 7 r. Where the Dirihlet Beta funtion is defined as n βs n+ s, Rs >. 7 n By using the 7 we an easily obtain π kβr + k r k! π π r, r N. 7 4 r! For any positive integer k: βk + k E k π k+ 4 k+, 73 k! where E n represent the Euler numbers. By using the 7 and 73 we an easily obtain r E r k, r N. 74 k Aording to Definition, if S t ln +t t, then S m t t n n m, S m λm. In Theorem 7, let m r, and by using Theorem, onsidering S r k λr k and z +ez sin ln z e π 4, < x <, thus we obtain Corollary 6. For r N, in the interval [,] we have n πx os n r k! kλr k+ r π 4 r! 75 r. 3

24 Where λs are the Dirihlet Lambda funtion defined by λs n+ s, Rs >. 76 Letting x /, we obtain n π kλr r π r, k r N. 77 k! r! Letting x /4 in 75, we obtain [n ] n r π kλr k k+ r π 78 r. k! 4 r! 4 For example, letting r, in 78, and using 77 we obtain [n ] n [n ] n 4 π, 6 79 π Theorem 3 Let St n a nt n, t R, t r, < r < +, if a n os nπx os Se z z, n n a n sin nπx sin z are tenable in a < x < b, x R, then x [, b a, n n a n os nπx a n os nπx os nπx sin nπx osh x x osh x x Se z os sin Se z z, 8 Se z z are tenable in a+x < x < b x. Aordingly, for any definite value x a,b, x in 8 takes values in the following interval: { x a, a < x a+b/, < x < b x, b > x a+b/. Proof Clearly a n os nπx n os nπx os 4 πx os z z Se z

25 os πx x Se z z + os πx+x Se z z osh x os Se z x z. a n os nπx sin nπx os sin Se z z z n sin πx x Se z z + sin πx+x Se z z osh x sin Se z x z. These two equations are both tenable in the ommon interval a+x < x < b x of a < x x < b and a < x+x < b. For 75, using Theorem 3 we an easily obtain: For any definite value x [,/, we have n r os n πx os n πx πx λr k osh x k! x + r π r! 4 osh πx r x x k 8 are tenable in x x x. On the ontrary, for any definite value x,, x in 8 takes value in the following interval: { x, < x /, x x, > x /. When r, the equation above an be expressed as: n os n πx os n πx λ π x 4. As the equation is tenable in x x x, x annot be and unless x. When x /4 [,/, we get [n/] n πx os n λ π x 4 is tenable in /4 x 3/4, where λ π /8. If we take not x /4 but x /4,/], then in x /4 we have [n/] n os n πx λ π π

26 Combining these two equations into one, we have [n ] n πx os n { π /6, x /4, λ π x /4, /4 x 3/4. In 8, if we take not r but r, then similarly we have [n/] n πx os n 4 5π π4 x 8 π4 x 6 + π4 x 3 4 3, 4 x [n/] n 4 os n πx π4 536 π4 3 x, x /4. If the following Dirihlet series is denoted by Dr, 83 Dr [n/] n r, r N, 84 then by using 75 and Theorem, we know that Dr has a sum-a K π K. The 79 gives D4 π 4 /536, D π /6. The 8 indiates that the orresponding osine series of D4 onverge to polynomials. By observing 83 we an find that oeffiients of the polynomial funtions on the right side of the equation are definitely related to D4 and D, and an be expressed as: [n/] n 4 os n πx D4! D πx. The result is the same as that pointed out in the proof of Theorem. Generally, we an easily prove: Corollary 7. For r N, in the interval [,/4] we have [n/] n r os n πx k! kdr k, x /4. 85 Letting x /4 in 85, we get the reurrene formula of Dr as π kdr k λr. 86 k! 4 Clearly λr satisfies ζr λr + ζr/ r, then by solving this equation for λr and substituting it into 86, we have 6

27 π kdr k r ζr. 87 k! 4 r+ Using 87 or 86, we an reursively obtain the sum-a K π K of Dr. For instane, let r 3 in 87, using the sum-a K π K of D, D4 and ζ6, we an easily obtain the sum-a K π K of D6: D6 [n/] n 6 36π The result is the same as that obtained by 78. In preisely the same manner, we an prove the following results: Corollary 8. If the following Dirihlet series is denoted by Dr+, namely Dr + [n/] n+ r+, r N, 89 n then for r N, in the interval [,/4] we have [n/] n+ r+ os n+πx n r kdr+ k, x /4. k! 9 Where D π/4, and r N Dr+ k π k+λr r π r+. k+ 9 k +! 4 r! 4 Letting x /4 in 9, we get the reurrene formula of Dr + as r βr+ k k! π 4 kdr+ k, r N. 9 For example, letting r,,3 in 9, and using 7 we obtain D3 3π3 57π5 37π7, D5, D The results is the same as that obtained by 9. Similarly, differentiating 9 at x /4 we get r λr k π k+dr k, r N. 94 k +! 4 Aknowledgments: We are grateful to Wenpeng Zhang for some helpful remarks. The authors also would like to take this opportunity to thank Armen Bagdasaryan for his generous help. 7

28 Referenes [] G.Q. Bi, Appliations of abstrat operator in partial differential equationi, Pure and Applied Mathematis. 3997, 7-4 [] G.Q. Bi, Appliations of abstrat operator in partial differential equationii, Chinese Quarterly Journal of Mathemati. 4999, 8-87 [3] G.Q. Bi, Operator methods in high order partial differential equation, Chinese Quarterly Journal of Mathematis. 6, 88- [4] G.Q. Bi and Y.K. Bi, A new operator theory similar to pseudo-differential operators, arxiv:8.388 [5] G.Q. Bi and Y.K. Bi, Stationary solutions of the Klein-Gordon equation in a potential field, arxiv:8.44 [6] H. M. Srivastava, Some rapidly onverging series for ζn +, Proeedings of The Amerian Mathematial Soiety. 7999, [7] Zeta funtion [8] Donal F. Connon, Some infinite series involving the Riemann Zeta funtion, arxiv:5.73 [9] H.M. Srivastava and J. Choi, Series assoiated with the Zeta and related funtions. Kluwer Aademi Publishers, Dordreht, the Netherlands, 8