JACOB’S LADDERS AND THE ALMOST EXACT ASYMPTOTIC REPRESENTATION OF THE HARDY-LITTLEWOOD INTEGRAL 9 0 JANMOSER 0 2 Abstract. Inthispaperweintroduceanonlinearintegralequationsuchthat n thesystemofglobalsolutiontothisequationrepresentsaclassofaverynarrow a beam at T →∞ (an analogue to the laser beam) and this sheaf of solutions J leadstoanalmost-exactrepresentationoftheHardy-Littlewoodintegral. The 6 accuracyofourresultisessentiallybetterthantheaccuracyofrelatedresults 2 ofBalasubramanian,Heath-BrownandIvic. ] A C 1. Introduction . h Letus remindthatHardyandLittlewoodstartedtostudythe followingintegral t a in 1918: m [ T 1 2 T (1.1) ζ +it dt= Z2(t)dt, 1 2 v Z0 (cid:12) (cid:18) (cid:19)(cid:12) Z0 (cid:12) (cid:12) 3 where (cid:12) (cid:12) (cid:12) (cid:12) 7 1 1 1 t 9 Z(t)=eiϑ(t)ζ +it , ϑ(t)= tln(π)+Imln Γ +i , 2 −2 4 2 3 (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) . and they have derived the following formula (see [4], page 122, 151-156) 1 0 T 9 (1.2) Z2(t)dt T ln(T), T . 0 Z0 ∼ →∞ : v In this paper we show that except the asymptotic formula (1.2) that posses an i unbounded error there is an infinite family of other asymptotic representations of X theHardy-Littlewoodintegral(1.1). Eachmemberofthisfamilyisanalmost-exact r a representation of the given integral. The proof of this will be based on properties of a kind of functions having some canonical properties on the set of zeroes of the function ζ(1/2+it). (A) Let us remind further that in 1928 Titchmarsh has discovered a new treat- menttotheintegral(1.1)bywhichtheTitchmarsh-Kober-Atkinson(TKA)formula: ∞ N c ln(4πδ) (1.3) Z2(t)e−2δtdt= − + c δn+ δN+1 , n 2sin(δ) O Z0 n=0 X (cid:0) (cid:1) where δ 0, C is the Euler constant, c are constant depending upon N,was de- n → rived. Key words and phrases. Riemannzeta-function. 1 Jan Moser Jacob’s ladder ... The TKA formula has been published for the first time in 1951 in the funda- mental monograph by Titchmarsh (see [8], [1],[6],[7]). It was thought for about 56 years that the TKA formula is a kind of curiosity (see [5], page 139). However, in this paper we show that the TKA formula itself contains new kinds of principles. (B) Namely, in this work we introduce a new class of curves, which are the solutions to the following nonlinear integral equation: µ[x(T)] T (1.4) Z2(t)e−x(2T)tdt= Z2(t)dt, Z0 Z0 ∞ wheretheclassoffunctions µ isspecifiedas: µ C ([y , ))isamonotonically 0 { } ∈ ∞ increasing (to + ) function and it obeys µ(y) 7yln(y) . The following holds ∞ ≥ true: for any µ µ it exists just one solution to the equation (1.4): ∈{ } ϕ(T)=ϕ (T), T [T , ), T =T [ϕ], ϕ(T) as T . µ 0 0 0 ∈ ∞ →∞ →∞ Let us denote by the symbol ϕ the system of these solutions. The function ϕ(T) { } is related to the zeroes of the Riemann zeta-function on the critical line by the following way. Let t = γ be a zero of the function ζ(1/2+it) of the order n(γ), where n(γ)= (ln(γ)), (see [3], page 178). Then the points [γ,ϕ(γ)], γ >T (and 0 O only these points) are the inflection points with the horizontal tangent. In more details, it holds true the following system of equations: (1.5) ϕ′(γ)=ϕ′′(γ)= =ϕ(2n)(γ)=0, ϕ(2n+1)(γ)=0, ··· 6 where n=n(γ). With respectto this propertyanelement ϕ ϕ is to be namedas the Jacob’s ∈{ } ladder leading to [+ ,+ ] (the rungs of the Jacob’s ladder are the segments of ∞ ∞ the curve ϕ lying in the neighborhoods of the points [γ,ϕ(γ)], γ >T [ϕ]). Finally, 0 also the composition of the functions G[ϕ(T)] is to be named the Jacob’s ladder if ∞ the following conditions are fulfilled: G C ([y ,+ )), G grows to + and G 0 ∈ ∞ ∞ has a positive derivative everywhere. Let us mention that the mapping (the operator) Hˆ : µ ϕ { }→{ } can be named the Z2-mapping of the functions of the class µ . { } (C) Jacob’s ladder implies the following results: (a) AnalmostexactasymptoticformulafortheHardy-Littlewoodintegral. Let us mention that our new formula makes more exact also the leading term in (1.2): T ϕ(T) ϕ(T) Z2(t)dt ln , T + , ϕ ϕ , ∼ 2 2 → ∞ ∀ ∈{ } Z0 (cid:18) (cid:19) i.e. theleadingterminthisformulaisalsotheJacob’sladder,andtherefore we can say that the leading term has a very fine structure. (b) The system ϕ has the property of an ”infinitely close approach” of any { } two Jacob’s ladders at T + . → ∞ Page 2 of 10 Jan Moser Jacob’s ladder ... (c) Our new formula for the Hardy-Littlewood integral is stable with respect to the choice of the elements from some subset ϕ ⋆ in ϕ . { } { } 2. Results The following theorem holds true: Theorem 1. The TKA formula implies: (A) T (2.1) Z2(t)dt=F[ϕ(T)]+r[ϕ(T)], T T [ϕ], 0 ≥ Z0 where (2.2) y y y ln(ϕ(T)) ln(T) F[y]= ln +(c ln(2π)) +c , r[ϕ(T)]= = . 0 2 2 − 2 O ϕ(T) O T (cid:16) (cid:17) (cid:26) (cid:27) (cid:18) (cid:19) (B) For all ϕ (T),ϕ (T) ϕ we have 1 2 ∈{ } 1 (2.3) ϕ (T) ϕ (T)= , T max T [ϕ ],T [ϕ ] . 1 2 0 1 0 2 − O T ≥ { } (cid:18) (cid:19) (C)Iftheset µ(y ) isboundedthenforallϕ(T)in ϕ andforanyfixedϕ (T) 0 0 { } { } ∈ ϕ : { } A A (2.4) ϕ (T) <ϕ(T)<ϕ (T)+ , T T =sup µ(y ) . 0 0 0 0 − T T ≥ { } µ Letusmentionthattheconstantsinthe -symbolsdonotdependuponthechoice O of ϕ(T). Let us remind the Balasubramanian’s formula (see [2]): T (2.5) Z2(t)dt=T ln(T)+(2c 1 ln(2π))T + T1/3+ǫ , − − O Z0 (cid:16) (cid:17) and Ω-theorem of Good (see [3]): T (2.6) Z2(t)dt T ln(T) (2c 1 ln(2π))T =Ω T1/4 . − − − − Z0 (cid:16) (cid:17) Remark 2. Combining the formulae (2.1), (2.2) and (2.5) one obtains that: Formula(2.5)possessesquitelargeuncertaintysincethedeviationfromthe • value of (1.1) is given by R(T)= (T1/3+ǫ), and (see (2.6)) since O lim R(T) =+ , T→∞| | ∞ this cannot be removed. Following (2.2) we have • lim r[ϕ(T)]=0, T→∞ and this means that formula (2.1) seems to be almost exact. Remark 3. WehavefoundanewfactthattheleadingtermintheHardy-Littlewood integral is a ladder (see (2.1)), i.e. it has a fine structure. There is no analogue of this in the formulae (1.2) or (2.5). Remark 4. We say explicitly that: Page 3 of 10 Jan Moser Jacob’s ladder ... Formula (2.3) contains a new effect, namely, any two Jacob’s ladders ap- • proach each other at T . →∞ (2.4) implies that the representations (2.1) and (2.2) of the integral of • Hardy-Littlewood (1.1) are stable under the choice of ϕ(T) ϕ in the ∈ { } case of a bounded set µ(y ) . 0 { } Remark 5. Following the second part of Remark 3 the representations (2.1) and (2.2) of the Hardy-Littlewoodintegral(1.1) is microscopically unique in sense that any two Jacob’s ladders ϕ ,ϕ , T T (see (2.4)) cannot be distinguished at 1 2 0 ≥ T . →∞ InthefifthpartofthisworkweestablishtherelationbetweentheJacob’sladder and the prime-counting function π(T): 1 ϕ(T) π(T) T , T . ∼ 1 c − 2 →∞ − (cid:26) (cid:27) 3. Existence of the Jacob’s ladder The following lemma holds true: Lemma1. Letµ(y) µ befixed. Thenthereexistsanuniquesolutionϕ(T), T ∈{ } ≥ T [ϕ] to the integral equation (1.4) that obeys the property (1.5). 0 Proof. By the Bonnet’s mean-value theorem in the case y >0 and µ>0 we have µ M (3.1) Z2(t)e−y2tdt= Z2(t)dt, M >0, Z0 Z0 where e−y2t, t [0,µ] is decreasing and equals 1 at t=0. ∈ First of allwe will show that the formula (3.1) maps to any fixed µ>0 just one M >0, since the case M =M is impossible because it would mean that 1 2 6 M2 (3.2) Z2(t)dt=0. ZM1 Let µ(y) µ . Then the following formula ∈{ } µ(y) M(y) (3.3) Z2(t)e−y2tdt= Z2(t)dt Z0 Z0 defines a function M(y)=M (y), y y . Letthe symbol M denote the class of µ 0 ≥ { } the images of the elements µ(y) µ . ∈{ } The function M(y) is positive and increases to + . In fact, let ∞ µ(y) (3.4) Φ(y)= Z2(t)e−y2tdt. Z0 Then (3.5) Φ′(y)= 2 µ(y)tZ2(t)e−y2tdt+Z2[µ(y)]e−y2µ(y)dµ(y) >0 y2 dy Z0 Page 4 of 10 Jan Moser Jacob’s ladder ... (the first term is obviously positive and the second one is non-negative). Thus, we see that the function Φ(y), y [y ,+ ) is increasing. Subsequently, for any 0 ∈ ∞ y,∆y >0,y y one has 0 ≥ M(y+∆y) 0<Φ(y+∆y) Φ(y)= Z2(t)dt M(y+∆y)>M(y). − ⇒ ZM(y) The function M(y), y y is continuous. In fact, for any fixed yˆ [y ,+ ) we 0 0 ≥ ∈ ∞ have M (yˆ)=limsupM(y), M (yˆ)=liminfM(y). 1 2 y→yˆ+ y→yˆ+ And with respect to the continuity of the left-hand side of eq. (3.3) we obtain (see (3.2)): M2(yˆ) Z2(t)dt=0 M (yˆ)=M (yˆ)=M(yˆ+0). 1 2 ZM1(yˆ) The existence ofM(yˆ 0)andthe identities: M(yˆ)=M(yˆ 0)=M(yˆ+0)can − − be shown by analogy. The function M(y), y y obeys the following properties: 0 ≥ (a) It has a continuous derivative at any point y such that M(y)=γ, where γ 6 is a zero of the function ζ(1/2+it). (b) It has a derivative equal to + at any point y such that M(y) = γ, γ ∞ mentioned above. Thesepropertiesofthe M functioncanbeprovedasfollows. By(3.3)and(3.4)we have (3.6) Φ(y+∆y) Φ(y) M(y+∆y) M(y) − = − Z2 M(y)+θ [M(y+∆y) M(y)] , ∆y ∆y { · − } ′ whereθ (0,1). UsingthefactthatΦ(y)>0(see(3.5))wecandeducefrom(3.6) ∈ that: (i) for any values of y suchthat Z2[M(y)]>0 there exists a continuousderiv- ative, and (ii) for any values of y such that Z2[M(y)]=0 we have M(y+∆y) M(y) (3.7) lim − =+ . ∆y→0 ∆y ∞ Since our function T = M(y), y y is continuous and increasing (to + ) 0 ≥ ∞ thereexistsuniqueinversefunctionthatisalsocontinuousandincreasing(to+ ): ∞ (3.8) y =ϕ(T)=ϕ (T), T T [ϕ]=M (y ). M 0 µ 0 ≥ As a consequence of eqs. (3.5) and (3.6) we have dϕ(T) Z2(T) ′ ′ (3.9) = , Φ =Φ [ϕ(T)]>0, T T [ϕ], dT Φ′[ϕ(T)] y ≥ 0 ′ ∞ (ϕ(γ) = 0, see (3.7)). Eq. (3.9) implies that ϕ(T) C ([T [ϕ],+ )) and also 0 ∈ ∞ that the property (1.5) holds true. Inserting (3.8) into (3.3) one obtains that X(T)=ϕ (T), T T [ϕ] is a solution to the integral equation (1.4). (cid:3) µ 0 ≥ Page 5 of 10 Jan Moser Jacob’s ladder ... 4. Consequences from the TKA formula The following Lemma holds true: Lemma 2. Let M(y) M be arbitrary, then ∈{ } M(y) (4.1) Z2(t)dt=F(y)+r(y), Z0 where y y y ln(y) (4.2) F(y)= ln +E +c , r(y)= , E =c ln(2π), 0 2 2 2 O y − (cid:16) (cid:17) (cid:18) (cid:19) and the constant within the -symbol is an absolute constant. O Proof. We start with the formula (1.3) and N =1: ∞ c ln(4πδ) (4.3) Z2(t)e−2δtdt= − +c +c δ+ (δ2). 0 1 2sin(δ) O Z0 As long as Z(t) <At1/4, t t , we have 0 | | ≥ 1 1 f(t,δ)=t1/2e−δt f ,δ = , ≤ 2δ √2eδ (cid:18) (cid:19) ∞ e−δU A2 Z2(t)e−2δtdt<B , B = , U t . δ3/2 √2e ≥ 0 ZU The value U =µ(1/δ) is to be chosen by the following rule: 1 7 1 1 B Bδ−3/2e−δU δ2 µ ln > ln . ≤ ⇒ δ ≥ δ δ δ δ7/2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Now, (4.3) implies: µ(1/δ) c ln(4πδ) 1 7 1 (4.4) Z2(t)e−2δtdt= − +c +c δ+ δ2 , µ ln , 0 1 2sin(δ) O δ ≥ δ δ Z0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) (seetheintroduction,part(B)-theconditionforµ(y)),andfortheremainderterm we have: ∞ (4.5) Z2(t)e−2δtdt= δ2 . − O Zµ(1/δ) (cid:0) (cid:1) Let δ (0,δ ] with δ being sufficiently small, then 0 0 ∈ 1 1 δ2 = 1+ + δ4 , sin(δ) δ 6 O (cid:26) (cid:27) (cid:0) (cid:1) and c ln(4πδ) 1 1 D 1 − = ln + + δln , 2sin(δ) 2δ δ 2δ O δ (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) and (see (4.4)) µ(1/δ) 1 1 D 1 (4.6) Z2(t)e−2δtdt= ln + +c + δln , D =c ln(4π). 0 2δ δ 2δ O δ − Z0 (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) Putting δ = 1/y, y = 1/δ into eq. (4.6) and using eq. (3.3) we obtain the 0 0 formulae (4.1) and (4.2), respectively. Since the constants in the eqs. (4.3), (4.5) are absolute, the constant entering the -symbol in (4.2) is absolute, too. (cid:3) O Page 6 of 10 Jan Moser Jacob’s ladder ... 5. Proof of the theorem Putting T =y/2 into eq. (2.5) and comparing with the formula (4.1) we obtain y <M(y), y . 2 →∞ Furthermore, putting into eq. (2.5) y A T = 1+ , A>1 c, 2 ln y ! − 2 and comparing with eq. (4.1) we have (cid:0) (cid:1) y A M(y)< 1+ , y . 2 ln y ! →∞ 2 Subsequently (see (3.8)), (cid:0) (cid:1) y A y ϕ(T) (5.1) 0<M(y) < 0<2T ϕ(T)<B , − 2 2 ln y ⇒ − ln[ϕ(T)] 2 i.e. the following equation holds t(cid:0)ru(cid:1)e (5.2) 1.9T <ϕ(T)<2T. Insertingy =ϕ(T) ϕ into eq. (4.1) (see (3.8)) we obtainthe formula(2.1) and ∈{ } the estimate (2.2), (see (5.2)). ′ The relation (2.3) follows from F (y) = 1/2ln(y/2)+E +1 and from the eq. (2.1) written for ϕ and ϕ , respectively, with help of (5.2). 1 2 Since µ(y)>M(y), y y (see (3.3)) and in the case of boundedness of the set 0 ≥ µ(y ) the choice of values (see (2.4)): 0 { } T =sup µ(y ) µ(y )>M (y ), µ(y) µ , 0 0 0 µ 0 { }≥ ∀ ∈{ } µ is regular, i.e. the interval [T ,+ ) is the common domain of the functions 0 ∞ ϕ (T), µ(T) µ . µ ∈{ } 6. Relation between the Jacob’s ladders and prime-counting function π(T) Comparing the formulae (2.1) and (2.5) we obtain ϕ ω ω(T)=(1 c)T + T1/3+ǫ , ω(t)=tln(t)+(c ln(2π))T. 2 − − O − (cid:16) (cid:17) (cid:16) (cid:17) Let us consider the power series expansion in the variable ϕ/2 T of the previous − formula. We obtain the following nonlinear equation ∞ xk T ϕ (6.1) x(ln(T) a) =1 c+ T−2/3+ǫ , x= − 2, − − k(k 1) − O T Xk=2 − (cid:16) (cid:17) where a=ln(2π) 1 c. Because of (see (5.1) and (5.2)) − − 1 x= , O ln(T) (cid:18) (cid:19) Page 7 of 10 Jan Moser Jacob’s ladder ... we obtain from (6.1): 1 c 1 ϕ(T) T 1 x= − + T = 1+ , ln(T) a O ln3(T) ⇒ − 2 ln(T) O ln(T) − (cid:18) (cid:19) (cid:26) (cid:18) (cid:19)(cid:27) and furthermore, by using the Selberg-Erdo¨s theorem, we obtain the formula: 1 ϕ(T) (6.2) π(T) T , T , ϕ ϕ . ∼ 1 c − 2 →∞ ∀ ∈{ } − (cid:26) (cid:27) Remark 6. As aconsequenceoftheabovewrittenwehavethatthe Jacob’sladders are connected (along to the zeroes of the function ζ(1/2+it)) also to the prime- counting function π(T), see (6.2). More interesting information can be deduced from the nonlinear equation (6.1). Namely, inserting (6.3) Te−a =τ, e−aϕ(eaτ)=ψ(τ) into (6.1) we obtain: ∞ xk τ ψ(τ) (6.4) xln(τ) =1 c+ τ−2/3+ǫ , x= − 2 . − k(k 1) − O τ Xk=2 − (cid:16) (cid:17) The following statement holds true: if in the equation: A A A A 1 3 n n+1 (6.5) x= + + + + +... , ln(τ) ln3(τ) ··· lnn(τ) lnn+1(τ) the coefficients A ,A ,...,A are already known, then the coefficient A is de- 1 3 n n+1 termined by (6.4). We obtain: 1 1 A =1 c, A = (1 c)2, A = (1 c)3, 1 3 4 − 2 − 6 − 1 1 A = (1 c)3+ (1 c)4, ... . 5 2 − 12 − Changing the variables in (6.5) into the initial ones (see (6.3)) we obtain the following asymptotic formula (6.6) 1 ϕ(T) A A A B B 1 3 1 2 3 T + + + + +... , T , T − 2 ∼ ln(T) a (ln(T) a)3 ···∼ ln(T) ln2(T) ln3(T) →∞ (cid:26) (cid:27) − − where B =aA , B =a2A +A , .... 2 1 3 1 3 Remark 7. Let us remark that the asymptotic formula (6.6) is an analogue to the following asymptotic formula 1 T dt 1 1 2! + + +... , T ln(t) ∼ ln(T) ln2(T) ln3(T) Z2 for the Gauss logarithmic integral. Page 8 of 10 Jan Moser Jacob’s ladder ... 7. Fundamental properties of Z2-transformation Let us mention explicitly that the key idea of the proof of the theorem was to introduce the new integral transformation: Z2-transformation. By using the appropriate terminology from optics (see example: Landau& Lif- shitz, Field theory, GIFML, Moscow 1962,page 167): the elements µ(y) µ are called the rays and the set µ itself is called • ∈{ } { } the beam, thebeamscrossingeachotherinagivenpointarecalledhomocentricbeams. • The fundamental property of the Z2-transformation lies in the following: iftheset µ(y ) isboundedthenthebeam µ (and,atthesametime,also 0 • { } { } anyotherhomocentricbeam)istransformedintothehomocentricbeam ϕ { } oftheJacob’sladders,withrespecttothepoint[+ ,+ ](see(2.3),(2.4)), ∞ ∞ thetransformedbeam ϕ isverynarrowinsenseof(2.4),i.e. anZ2-optical • { } system generates an analogue of a laser beam. Let us consider for example the homocentric (with respect to the point [y ,y2]) 0 0 sheaf of rays: (7.1) u(y;ρ,n)=y2[1+ρ(y y )n], ρ [0,1], n N. 0 − ∈ ∈ Following the equation u (y+∆;0,n)=u (y;1,n) we obtain that 0 1 y(y y )n 0 ∆= − , at y , 1+(y y )2 →∞ →∞ 0 − i.e. (7.1) is a diverging beapm. Anyway, the Z2-mapping transforms (7.1) into an analogue of a laser beam ϕ . u { } 8. On intervals that cannot be reached by estimates of Heath-Brown and Ivic We will show the accuracy of our formula (2.1) in comparison with known esti- mates of Heath-Brown and Ivic (see [5], (7.20) page 178, and (7.62) page 191) T+G 1 (8.1) Z2(t)dt= Gln2(T) , G=T1/3−ǫ0, ǫ = . 0 O 108 ZT (cid:0) (cid:1) First of all, one can easily obtain the tangent law from both (2.1) and (8.1): T+U ϕ(T) 1 (8.2) Z2(t)dt=Uln e−a tan[α(T,U)]+ ZT (cid:18) 2 (cid:19) O(cid:18)T1/3+2ǫ0(cid:19) for 0 < U < T1/3+ǫ0, where α = α(U,T) is the angle of the chord of the curve y = 1ϕ(T) crossing the points [T,1ϕ(T)] and [T +U,1ϕ(T +U)]. Further, from 2 2 2 (2.1) and (2.5) we have 1 (8.3) tan(α )=tan[α(T,U )]=1+ , U =T1/3+2ǫ. 0 0 0 O ln(T) (cid:18) (cid:19) And finally, considering the set of all chords of the curve y = 1ϕ(T) which are 2 paralleltoourfundamentalchordjoiningpoints[T,1ϕ(T)]and[T+U,1ϕ(T+U)], 2 2 Page 9 of 10 Jan Moser Jacob’s ladder ... we obtain the continuum of formulae: b 1 Z2(t)dt=(b a)ln(T)+ (b a)+ , Za − O − O(cid:18)T1/3+2ǫ0(cid:19) (8.4) 0<b a<1,(a,b) T,T +T1/3+ǫ0 . − ⊂ (cid:16) (cid:17) Remark 8. It is quite evident that the interval (0,1) cannot be reached in known theories leading to estimates of Heath-Brown and Ivic. References [1] F. V. Atkinson, ‘The mean value of the zeta-function on critical line’, Quart. J. Math. 10 (1939) 122–128.. [2] R. Balasubramanian, ’An improvement on a theorem of Titchmarsh on the mean square of |ζ(1/2+it)|’,Proc. London. Math. Soc.336(1978)540–575. [3] A. Good, ‘Ein Ω-Resultat fu¨r quadratische Mittel der Riemannschen Zetafunktion auf der kritischeLinie’,Invent. Math. 41(1977) 233–251. [4] G. H.HardyandJ. E.Littlewood, ‘Contributiontothe theoryofthe Riemannzeta-function andthetheorypfthedistributionofPrimes’,Acta. Math. 41(1918) 119–195. [5] A.Ivic,‘TheRiemannzeta-function’,AWilley-IntersciencePublication,NewYork,1985. [6] H. Kober, ‘Eine Mittelwertformeln der Riemannschen Zetafunktion’, Composition Math. 3 (1935) 174–189. [7] E. C. Titchmarsh, ‘The mean-value of the zeta-function on the critical line’, Proc. London. Math. Soc.(2)27(1928) 137–150. [8] E.C.Titchmarsh,‘ThetheoryoftheRiemannzeta-function’,ClarendonPress,Oxford,1951. Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni- versity,MlynskaDolina M105,84248Bratislava,SLOVAKIA E-mail address: [email protected] Page 10 of 10