The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-function

Transcription

1 Mathematis Letters 6; (6): doi:.648/j.ml.66. The Geometri Interpretation of Some Mathematial Epressions Containing the Riemann ζ-funtion Yuriy N. Zayko Department of Applied Informatis, Faulty of Publi Administration, The Russian Presidential Aademy of National Eonomy and Publi Administration, Sarato ranh, Sarato, Russia address: To ite this artile: Yuriy N. Zayko. The Geometri Interpretation of Some Mathematial Epressions Containing the Riemann ζ-funtion. Mathematis Letters. Vol., No. 6, 6, pp doi:.648/j.ml.66. Reeied: Otober 6, 6; Aepted: Noember, 6; Published: Deember 7, 6 Abstrat: The artile disusses some of the mathematial results widely used in pratie whih ontain the Riemann ζ- funtion, and, at first glane, are in ontradition with ommon sense. A geometri approah is suggested, based on the onept of the urature of spae, in whih is alulated an algorithm that speifies the representation of ζ-funtion as an infinite dierging series. The analysis is based on the use of Einstein equations to alulate the metri of ured spae-time. The solution of the Einstein equations is a metri that has a singularity, like the metri in the iinity of the blak hole. The result an be interpreted in the spirit of a Turing mahine that performs the proposed algorithm for alulating the sum of a diergent series. Keywor: Riemann ζ-funtion, Einstein Equations, Metri, Metri Tensor, Energy-Momentum Tensor, Christoffel Symbols, Algorithm, Turing Mahine. Introdution Riemann ζ - funtion is an element of the mathematial apparatus, widely used in arious areas of modern siene: mathematis, physis, et. The results epressed by the ζ- funtion for humans dealing with them for the first time seem puzzling, although eperts pereie them as firmly justified. To suh results relates the formula representing ζ (-) in the form of diergent series used, for eample, in the string theory []: ς( ) = = / []. Although the theory of diergent series is suffiiently deeloped [3] neertheless, it is interesting to try to gie a different interpretation of known epressions. Typially, the sum of a diergent series is not omputed, but is searhing in another way, suh as in the ase of the known series [3] S =... = /. The results presented below an be interpreted as an attempt to find the sums of diergent series by the omputational metho. For this purpose, the method of physial analogy was used that is often useful in pratie, and whih is onfirmed in the present ase. This result is reeied from the obious relation S = - S. Physial Approah to Mathematial Problem Let us formulate a simple physial task - to alulate the distane whih is gone by a partile for a time t, moing along a straight line with onstant aeleration. Mathematially, the answer an be represented as a finite sum t t( t ) t t S( t) = n = = S, n = () S = S() = The right side of the formula () oinides with the epression for the distane traeled by a partile moing with aeleration a =, the initial speed = ½ and the initial position S =. The time t is assumed disrete and dimensionless. Let us restrit ourseles with only one alue of time t =. The answer is known and an be epressed with the help of aboe mention series: S( ) = ζ(-)=-/ [, ]. From a mathematial point of iew, it is true, beause obtained by analyti ontinuation of the Riemann ζ-funtion on the alue of the argument u = -[]:

2 43 Yuriy N. Zayko: The Geometri Interpretation of Some Mathematial Epressions Containing the Riemann ζ-funtion u ς (u) = n () n= Howeer, it is neither obious nor lear. The reason is that the alue of ζ (-) is represented by diergent series whih does not hae the sum in the usual sense, inherent in the onept of the sum of onergent series [3]. While the sum of the onergent series, whih is understood as the limit of the partial sums of the series is intuitiely lear for diergent series intuition fails to work and you hae to attribute "for the word "sum" meaning whih is different from the usual" [4]. ut een agreeing to this etension of the definition, hae to deal with the fat that different sums are asribed for one and the same diergent series [3]. Other wor, the alue of the sum depen on the "ontet". Let s try to gie a geometri interpretation of the proess of finding the "sum" of the diergent series and, thus, aoid some ontraditions that still eist in this regard. Reall that the uniformly aelerated motion of the partile requires aording to Newton's seond law, that a onstant fore operates on the partile in the diretion of its moement. This effet an be ahieed by plaing at the point = (the diretion of moement of the partile is taken as the ais OX of our frame of referene) an infinite plane oinident with the plane YOZ, haing a onstant mass density σ. The graitational potential of this plane is equal to ϕ () = πσ K, and the fore ating on a unit point mass is equal to E = ϕ = πσk and is direted along ОХ, K is the graitational onstant. The epression for the spae-time metri may be found from soling the Einstein's equations [5] 8πK R g R = T (3) ik ik 4 ik Here R is the trae of the Rii tensor R i k: R = R i i, g ik is the metri tensor; T ik - is the energy-momentum tensor; с is the speed of light in auum; indies i, k hae alues,,, 3. Let us write the epression for the interal = g dt g d g dy g dz 33 3 t,, y, z = = = = ν λ g = e,g = e,g = g = 33 The standard notations for g and g are used [5]. The solution is ery similar to the Shwarzshild solution of the problem of the finding the metri near a point mass [5]. The non-zero Christoffel symbols are the follows [5]: (4) The point means a deriatie on t, prime - on =. For where T ik = the equations (3) an be redued to the equations R ik =, whih for R and R lead to a single equation ν ν ν λ ν λ λɺɺ λɺ λɺ νɺ e = and the equation for R is redued to identity. Assuming λ = -ν, and all time deriaties equal zero, the last equation redues to the form (6) ν ( ν ) = (7) whih has a solution e ν = C C, С, are onstants. Their appropriate hoie gies the desired solution ν 4πσK e = (8) onsidering the onnetion of the metri tensor omponent g and the Newtonian potential φ[5]: g = ϕ /. Let write the final form of the interal 4πσK 4πσK = dt d dy dz Spatial metri along the diretion ОХ (d=dy=) is gien by the epression dl d = 4πσK (9) () After integration we obtain the relationship between the oordinate in the system of the remote obserer and the distane l, passed by the partile, measured in its rest frame 4πσK l = l πσk () l - is the integrating onstant. From the formula () one an obtain the relationship between the lengths of the line segments traersed by the partile, measured in different systems dl 4πσK = d / () g g g Γ = g i im mk ml kl kl l k i νɺ ν ν, e,, λɺ λɺ λ, e, ν λ Γ = Γ = Γ = λ ν Γ = Γ = Γ = (5) 3. The Solution of the Equations of Motion Let us treat the solutions of the equations of motion of the partile in question [5]

3 Mathematis Letters 6; (6): i d Γ i kl k l d d = (3) Gien the nonzero Christoffel symbols Γ = Γ = Γ = /, and the epression for the deriatie ν ν ν = e /, where = / 4πσK we reeie the system of t wo equations sign - we reeie D = () Approah the massie partile to the point aording to the formula (8) spen an infinite time from the point of iew of a distant obserer, as is seen in the Fig. : d d d d d = d = (4) The seond equation in (4) is integrated and gies d = (5) is a onstant. After substitution (5) into the first equation i n (4) it looks as follows d d = (6) Equation (6) an be integrated one that lea to an epressi on for the 4-speed (D - is a onstant) d or in another form in ariable d d = D D = 3 (7) (8) The onstants of integration an be found omparing the epression of (8) with orresponding lassial result. A lassi ase orrespon to <<. Epanding the right side of the (8) in we reeie the lassial limit of (8) d ( ) = dt D 3D = (9) It should be ompared with the lassial result following from the law of onseration of energy d ) = = ± ( ) () dt ( is the alue of х wherein the speed =, signs ± orrespond to the ases > and > or to the opposite ase, taking into aount that. That last ase we hae to hoose due to reasons whih will be lear soon. So for the Fig.. Traelling time of the partile T ersus oordinate q = /. Solid line oordinate time T = t / in the system of the distant obserer reeied from the solution of (8). Ini- tial point q = -.4 orrespon to the initial speed =-,5. 4. Algorithmi Analysis of the Result Let us analyze the resulting epression. In ase of σ the obserers in the different systems reeies different alues. Of partiular interest is the ase when = /4πσ K, and dl/d, although the alue of l stays finite. This means that a moing partile while approahing the speified point on either side, in iew of a distant obserer will neer ahiees it. This is reminisent of the behaior of a massie partile in the iinity of a blak hole, from the point of iew of the distant obserer. The alue of is an analogue of the eent horizon in the Shwarzshild problem. This makes it possible to remoe the ontradition, enlosed in the formula (). Aording to alulation of an obserer moing with the partile, he pass inreasingly larger segments of the path, while the remote obserer beliees he stan still and the resulting distane traeled by him stays finite. One an say that in the formula (), the right side is a reord of alulation algorithm whih is performing by an obserer or by a alulation deie in its rest frame. The steps of the algorithm desription look as follows: I. Take a segment of length. II. Take a segment of length and attah it to the first segment. III. Follow this proedure infinitely, proiding that length of the subsequent segment is greater the length of preeding one by. The terminology adopted in physis is used to desribe the atiity on the measurement of some quantity. So, as has often been noted [6] between the proesses of measurement and alulation is no fundamental differene, there is reason a person (omputing deie) that alulates algorithm to name him (it) an obserer.

4 45 Yuriy N. Zayko: The Geometri Interpretation of Some Mathematial Epressions Containing the Riemann ζ-funtion IV. Measure the length of the resulting segment. The final measurement is performed by the remote obserer or deie. From the standpoint of general relatiity their results differ as desribed aboe, whih eplains the ontradition of the formula (). The use of suh "physial" approah implies that the numbers in the formula () are regarded as oordinate alues in some different frames of referenes (oordinate maps) applied to the real numbers ais. 5. Calulation the Sum of the Diergent Series As was mentioned earlier the sums of the diergent series in pratie are not alulated but is found from the onsiderations of another kind [5]. Neertheless we ll try to alulate the sum S( ) () with the help of formulas deried aboe. To this end, using the formula (8) we try to take into aount the initial alues for the aeleration a = a()=, the eloity = () = ½ and the position S = S() =. It appears that the ondition for the eloity annot be satisfied for real as seen from the Fig. 6. Analysis of the Equation (6) The desribed method an be used to find the sums of other diergent series related to the ζ-funtion. Consider the formula for the triial zeros of the ζ-funtion k ζ ( k) = n = (3) where k is a positie integer []. To apply the method desribed aboe, one should onsider the motion of the alulating deie along the OX-ais with the alternating aeleration. This, obiously, requires finding other solutions to the equation (6). Put ν = λ = α () β (t). Equation (6) then redues to the system of two equations (m is a parameter of the ariables separation) ( ) ( ) n= α α α me = βɺɺ β ɺ = Their solutions look as follows β me π = ± e erfi m mα C C /m π t = t ie erf m m mβ C C /m m (4) (5) where C, are onstants, and erf и erfi probability integrals of a real and imaginary argument [7]. Fig.. Veloity of the partile (8) in the system of the distant obserer ersus oordinate; V( q) = ( q) q, q = /. Thus we slightly hange our task and instead searhing of the sum S( ) () we will find an another sum S( ), where t t t S( t) = ( n ) = S S = () It is obiously that S( )=S( ) ζ()=s( )-,5 []. The initial alues for the S are just the same as for the S eept the alue for the initial speed whih looks as follows: = - ½, and easily an be satisfied what gies the alue for the q =-,4. The ondition S = means that the initial point should be plaed at the point q = -,4. Then the distane whih has been traelled by a massie partile before its stop at point q = / = - is equal to - (-,4) -,5 = -,8 from the point of iew of the distant obserer. This oinides with the eat alue for ζ(-)=-,833(3) up to two deimal plaes, whih is due to the finite preision of alulations ( -3 ). 7. Conlusions erf() = π erfi() = π u e du u e du (6) The result obtained an be interpreted in the spirit of the theory of relatiity of A. Einstein if we treat the alulation as a physial proess implemented by material bodies and displayed on the ais of real numbers, whih seres as a spae of one dimension. The transformation () make it possible to eplain the results of the alulations with the Riemann ζ-funtion mentioned aboe and eliminate their ontradition with ommon sense. Riemann ζ- funtion for more than a entury, is the fous of mathematiians all oer the world in onnetion with attempts to proe the famous Riemann hypothesis [8] onerning its non-triial zeroes. Numerous attempts both numerial [9] and theoretial [,, ] were made in order to proe or rejet the Riemann hypothesis. The present work ould shed new light to this problem. The used aboe physial objets an be interpreted in the spirit of a Turing mahine that performs the aboe proposed

5 Mathematis Letters 6; (6): algorithm, assuming the read-write head has a mass and moing with the ariable speed along the resting tape on whih the result will be written and whih is identified as the system of distant obserer. An oeriew of the different ersions of the relatiisti Turing mahine is gien in the paper [3]. From the physial point of iew, the result reeied aboe an be onsidered as yet another onfirmation of the general theory of relatiity, along with known eperimental onfirmations [4]. Aknowledgments The author epresses his thanks to prof. Tuyen Truong for his interest in this work and disussions. Referenes [] Zwiebah. A First Course in String Theory, -nd Ed., MIT.- 9. [] Janke E., Emde F, Lösh F., Tafeln Höherer Funktionen,.G. Teubner Verlagsgeselshaft, Stuttgart, 96. [3] Hardy G. H. Diergent series.-oford, 949. [4] Eiler L. Differential Calulation.- Aademy Pub., St. Petersburg [6] Mawell s Demon. Entropy, Classial and Quantum Information, Computing. Ed. by Leff H.S., and Re A.F., IoP Publishing, 3. [7] Prudniko A. P., ryhko Yu. A., and Marihe O. I., Integrals and Series, ols. 3; Gordon and reah, New York, 986, 986, 989. [8] Derbyshire J., Prime obsession. ernhard Riemann and the Greatest Unsoled Problem in Mathematis, Joseph Henry Press, Washington, D.C., 3. [9] ZetaGrid Homepage: zeta_grid.html [] Pei-Chu Hu and ao Qin Li. A Connetion between the Riemann Hypothesis and Uniqueness of the Riemann zeta funtion, arxi:6.583 [math.nt] 5 Ot 6. [] MPhedran R.C., Zeros of Lattie Sums: 3. Redution of the Generalised Riemann Hypothesis to Speifi Geometries, arxi:6.793 [math-ph] Ot 6. [] May M. P., On the Loation of the Non-Triial Zeros of the RH ia Etended Analyti Continuation, arxi: [math.gm] 6 Sep 6. [3] Andr eka H., N emeti I., N emeti P. General relatiisti hyperomputing and foundation of mathematis, Natural Computing, 9, V. 8, 3, pp [4] Weinberg S., Graitation and Cosmology. Priniples and Appliations of the General Theory of Relatiity, MTI, John Villey & Sons, NY, 97. [5] Landau L. D., Lifshitz E. M., The Classial Theory of Fiel. Vol. (4th ed.). utterworth-heinemann, 975.