Logarithmic Fourier integrals for the Riemann Zeta Function

Transcription

1 Logarithmic Fourier integrals for the Riemann Zeta Function arxiv:84.489v [math.cv] 3 Apr 8 Matthias Kunik Institute for Analysis and Numerics, Otto-von-Guericke-Universität Key words: Postfach 4. D-396 Magdeburg, Germany December, 8 Zeta function, explicite formulas, Fourier analysis, symmetric Poisson-Schwarz formulas. Mathematics Subject Classification (): M6, N5, 4A38, 3D, 3D5. Preprint No. /8, Otto-von-Guericke University of Magdeburg, Faculty of Mathematics. matthias.kunik@mathematik.uni-magdeburg.de

2 Abstract We use symmetric Poisson-Schwarz formulas for analytic functions f in the half-plane Re(s) > with f(s) = f(s) in order to derivefactorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann s classical explicit formula, these representations use integrals along the critical line Re(s) = and Blaschke zeta zeroes. Introduction Due to a theorem of Balazard-Saias-Yor [4], the Riemann-hypothesis is true if and only if the integral Ω ζ := log ζ(w) dw (.) π w Re(w)=/ vanishes. These studies have their origin in Beurlings work [5], see also [3]. Using that (s )ζ(s)/s belongs to the Hardy space H (Re(s) > /) one can also show that the logarithmic integral (.) is absolutely convergent, see Burnol [6], [7]. The general result about absolute convergence of the logarithmic integral for Hardy spaces H p (Im(s) > ) is given for example in the textbooks of Koosis [] and Garnett [9]. The following factorisation formula presented in [6] is valid for Re(s) > / s ζ(s) = ζ B (s) B(s), (.) s with the function ζ B in the right half plane Re(s) > / given by ζ B (s) := exp s+w sw dw log ζ(w) π s w w, (.3) and the Blaschke product Re(w)=/ B(s) := ρ:ζ(ρ)= Re(ρ)>/ s ρ s ρ ρ ρ. (.4)

3 It follows than by putting s = into (.), (.3), (.4) that B() = ρ ρ, Ω ζ = logb(), (.5) ρ:ζ(ρ)= Re(ρ)>/ which gives the Balazard-Saias-Yor Theorem. By using the symmetry of log ζ(/+iu) we can also rewrite ζ B in the form ζ B (s) = exp ( s ) log ζ( +iu) π u +(s du, Re(s) > /. (.6) ) Thus the function ζ B results from ζ by multiplication with s and replacing s the Blaschke zeros ρ of the zeta function with Re(ρ) > by ρ, i.e. by reflecting these zeros on the critical line Re(s) = / into the left half plane Re(s) < /. Blaschke products and factorisation formulas play an important role in the theory of Hardy spaces, see for example the textbooks of Koosis [], Hoffmann [] and Garnett [9]. In the monographs [, 3] of Koosis the meaning of logarithmic integrals like (.) is highlighted by an important theoretical background including results of Beurling, Malliavin and others, but the Riemann zeta function is not considered there. In Section we derive another symmetric Poisson-Schwarz formula by using the Hadamard product decomposition of (s )ζ(s) and the Riemann-von- Mangoldt function N(t) counting the nontrivial zeta zeroes. The resulting formula is a counterpart of (.6). Then we prove a variant of the Balazard- Saias-Yor Theorem. In Section 3 we apply the Fourier transform or Mellin s inversion formula on log((s )/sζ(s)) s(s ), Re(s) >, (.7) and relate it to the Fourier transforms of log ζ( +it) πn(t) ϑ(t) arctan(t) t +, t 4 +, t R, (.8) 4 by using the Blaschke zeta zeroes and a variant of Riemann s explicite prime counting formula. If the Riemann-hypothesis is true, then (.8) can be interpreted as the trace of the real- and imaginary part of the analytical continuation of (.7) on the critical line, respectively. Conversely, some important number theoretic functions are obtained in terms of integrals along the critical line by using the two expressions in (.8).

4 A further factorisation of the Riemann zeta function with a symmetric Poisson-Schwarz integral In this section we first prove a Poisson-Schwarz representation which turns out to be a dual counterpart of equations (.) and (.6). In the next section we will combine these results with Fourier- or Mellin transform techniques in order to obtain integrals along the critical line for the representation of interesting explicite prime number formulas. The desired Poisson-Schwarz formula results from the Hadamard factorisation of ζ(s). For this purpose we need the following theorem, which provides some information about the vertical distribution of the zeros of the zeta function in the critical strip. The following result was used by Riemann in [6], it was first shown by von Mangoldt [5] and then simplified by Backlund [], see the Edwards textbook [8]. Theorem (.) For t let N(t) be the number of zeros ρ = σ + iτ of the ζ-function in the critical strip < σ < with τ t, regarding the multiplicity of the roots. For t < we put N(t) := N( t), and define the function ϑ : R R by ϑ(t) := arctan(t) t (γ +logπ)+ { } t k arctan t k +. (.) k= Then we have for t the asymptotic relations ρ:ζ(ρ)= <Re ρ< Im ρ t ρ:ζ(ρ)= <Re ρ< Im ρ >t N(t) = π ϑ(t)+o(logt), ρ = π (logt) +O(logt), Imρ = π logt t +O(/t). (.) Remark: The exact form of ϑ(t) is needed for later purposes beside its asymptotic approximation θ(t) t log t π t π +o(), t. (.3) 8 3

5 Theorem (.) We define for Re(s) > / the analytic function ζ C by ζ C (s) := exp s(s ) u(πn(u) ϑ(u) arctan(u)) π (u +(s ) )(u + ) du, (.4) 4 as well as the symmetric counterpart C(s) of the Blaschke product, ( s )( s ) ρ ρ C(s) := ρ:ζ(ρ)= Re(ρ)>/ ( s ρ+ρ Then the following representation is valid for Re(s) > /, ). (.5) s ζ(s) = ζ C (s) C(s). (.6) s Remarks: Note that the integral in (.4) is well defined by the Riemannvon Mangoldt Theorem (.). The function ζ C results from ζ by multiplication with s and projecting the Blaschke zeros ρ of the zeta function with s Re(ρ) > / on the critical line Re(s) = /. The Riemann-Hypothesis is equivalent to B(s) = C(s). Proof: We define the integration kernel K C (s,u) := π s(s ) u (u +(s ) )(u + 4 ), (.7) and for all α R the characteristic function χ α : R R by χ α (u) := {, u α, u < α. ThefollowingintegralscanbeobtainedforRe(s) > of the Poisson-Schwarz integral representations, (.8) fromelementary theory exp π χ α (u)k C (s,u)du = ( s )( +iα s iα ), α, (.9) 4

6 exp ( u arctan a) K C (s,u)du = + s a+, Re(a) >, (.) Using (.) and (.), (.) we obtain exp ϑ(u)k C (s,u)du uk C (s,u)du = s. (.) π s = s s γ e k= e s k + s. (.) k+ Note that the integral on the left hand side in (.) is absolutely convergent by the asymptotic relation (.3). By using the general product representation {( + z ) } e z k = e γz, z C, (.3) k Γ(z +) k= we can rewrite the right hand side of (.) as π s s γ e e s k+ s + s k+ π s s γ e = k= s e γ s Γ ( s + ) π s e γ(s ) = ) s Γ ( s+ ( s =Γ π ) s, Γ(s) e γ(s ) k= e s k + s k where the last step follows from the duplication formula for the Γ function, see for example the textbook of Andrews et al. []. We obtain from (.) for Re(s) > that ( exp ϑ(u)k C (s,u)du s = Γ π ) s. (.4) 5

7 From (.) it also follows with a = that exp arctan(u)k C (s,u)du = s, Re(s) >. (.5) If we denote the positive imaginary parts of the nontrivial zeta zeros by < t t t 3, (.6) where the imaginary parts are listed according to the multiplicity of each root, then we can rewrite N(u) for u > as and obtain from (.9) that exp N(u) = πn(u)k C (s,u)du = ζ C (s)c(s) = n= χ tn (u), (.7) n= ρ:ζ(ρ)= Re(ρ),Im(ρ)> {( s +it n )( s it n )}. (.8) Now we use the definition of the analytic functions ζ C in (.4) and C in (.5) and calculate from (.8), (.4) and (.5) for Re(s) > the product {( )( )} s s ρ ρ Γ ( s ) π s s. ρ:ζ(ρ)= <Re ρ< Im ρ T (.9) The Theorem results from (.9) with Hadamard s product decomposition of Riemann s function ξ : C C with ξ(s) := s(s ) s π s Γ( )ζ(s) = lim ( s ). (.) T ρ Corollary (.3) With the integration kernel (.7) we have ξ(s) = C(s) exp πn(u)k C (s,u)du, Re(s) >. (.) 6

8 Remark: FortheproofofTheorem(.)wehave directlyusedtheriemann von Mangoldt Theorem (.) and the Hadamard products for ζ and ξ. In contrast, the integral (.) is divergent if we replace there ζ(w) by Γ(w/) or by Riemann s function ξ(w). In this case even the Cauchy limit lim T T T log ξ(/+it) /4+t dt is divergent due to the asymptotic behaviour log Γ(/4+ it lim ) π t t = 4 of the Gamma function. Thus it is not possible to evaluate (.) or (.6) by applying Hadamard s product representation for ζ(w) directly on log ζ(w). We have obtained Theorem (.) completely independent from the theory of Hardy spaces and the results of Balazard-Saias-Yor. Note that for all t R ζ( [ +it) = exp log ζ( ] +it) +i(πn(t) ϑ(t) πsign(t)). (.) Moreover, if the Riemann-hypothesis is true, then an analytical logarithm of s ζ(s) is defined for Re(s) > with real values for s > / and trace s log ζ( +it) +i(πn(t) ϑ(t) arctan(t)) (.3) on s = +it. Now we obtain from Theorem (.) the following counterpart of the Balazard-Saias-Yor theorem, namely Theorem (.4) We have the two inequalities π log ζ( +iu) (u + 4 ) du γ, (.4) π u(πn(u) ϑ(u) arctan(u)) (u + 4 ) du γ. (.5) In both cases equality holds if and only if the Riemann hypothesis is valid. 7

9 Proof: If we denote the left hand side of the inequalities (.4), (.5) by J and J, respectively, and put s = in the logarithmic derivatives of (.) and (.6), then we obtain γ = lim s [ s s + ζ (s) ζ(s) ] = J +Ω ζ + B () B() = J + C () C(). (.6) We will first prove that Ω ζ + B (), and that Ω B() ζ+ B () B() to the Riemann hypothesis. Put f ρ := log ρ ρ + ρ ρ = is equivalent (.7) for any ρ = σ +iτ with < σ and τ R with τ >. Then we obtain σ f ρ +f ρ = x ( x) +τ (τ 6x +6x ) (x +τ ) (( x) +τ ) dx >, (.8) such that the first part of the Theorem results from B () B() +Ω ζ = C ρ (s) := ρ:ζ(ρ)= Re(ρ)>/ f ρ. (.9) ItremainstoshowthatC ()/C() andthattheequalityc ()/C() = holds if and only if the Riemann hypothesis is valid. Put ( s )( s ) ρ ρ ( s ρ+ρ ) (.3) for any ρ = σ +iτ with < σ and τ R. Then we obtain ( ) C ρ() C ρ () + C ρ() C ρ () = (3τ σ( σ)) σ (( σ) +τ )(σ +τ )( 4 +τ ). (.3) For τ > and < σ this expression is always negative. Thus we have shown the Theorem. 8

10 Theorem (.5) For t except on a discrete singular set we define the number N B (t) of Blaschke zeta zeros ρ with Re(ρ) > and Im(ρ) t as well as the quantities N (t) := [ϑ(t)+arctan(t)], (.3) π N (t) := d π dt log t u log ζ( +iu) du, (.33) N 3 (t) := π t t B ( +iu) du. (.34) +iu) B( If we extend N B and N,N,N 3 as odd functions to the whole real axis, then we obtain for almost all t R that Remarks: N(t) N B (t) = N (t)+n (t)+n 3 (t). (.35) (a) Without using any information about the horizontal distribution of the zeta zeroes in the critical strip < Re(s) <, one can employ Theorem (.) and estimations for the logarithmic derivative of the Blaschke product on the critical line to prove the following asymptotic relations for t, N (t) = O(log t), N 3 (t)+n B (t) = O(log t). (.36) (b) If the Riemann hypothesis is valid, then we obtain from Theorem (.) with N 3 = N B the better result N (t) = O(logt) for t. Proof: For s C\{} with ζ(s) we obtain from (.) ξ (s) ξ(s) = s (γ +logπ)+ and note the functional equation n= ( n ) + ζ (s) s+n ζ(s), (.37) ξ ( s) ξ( s) = ξ (s) ξ(s). (.38) 9

11 The logarithmic derivative of (.), (.6) is given for Re(s) > with s and ζ(s) by ζ (s) ζ(s) = s s π Thus we obtain for Re(s) > log ζ( ±iu) (s iu) du+ B (s) B(s). (.39) from (.37) that ξ (s) ξ(s) = s (γ +logπ)+ π n= ( n ) s+n log ζ( ±iu) (s iu) du+ B (s) B(s). (.4) For η,t > we define the rectangular positive oriented integration path P η,t consistingonthestraightlinesegments[ +η it, +η+it], [+η+it, η+it], [ η +it, η it], [ η it, +η it]. It is centred around s =, is the boundary of the rectangle Re(s) η, Im(s) t and contains the critical strip for any η >. We decompose P η,t into two parts P η,t ± for the complex numbers s with Re(s) and Re(s) respectively, P + η,t := [ it, +η it] [ +η it, +η+it] [ +η +it, +it], P η,t := [ +it, η +it] [ η+it, η it] [ η it, it]. (.4) Let be t > given with ζ( +it). Since ζ and ξ have only isolated zeros, we can choose < η < small enough such that the compact rectangle Re(s) η, Im(s) t contains only zeros of the ζ-function on the critical line. Then we obtain due to (.) and (.38) that ξ (s) πi ξ(s) ds = N(t) N B(t). (.4) P ± η,t The direct integration of (.4) is problematic due to the presence of the pole singularities from the ζ-zeros on the critical line. Instead of this we evaluate the primitive t (N(τ) N B (τ))dτ. We have assumed ζ( ±it), and the

12 integral on the left hand side in (.4) does not depend on η for sufficiently small η >, and therefore t (N(τ) N B (τ))dτ = t π lim η + τ τ ξ ( +η+iϑ) dϑdτ (.43) +η +iϑ) ξ( by using for the short integration paths [ it, +η it] and [ +η+it, +it] of P + η,t the well known standard estimates. First we regard for each u R the double integral t π τ τ dϑ (η +i(ϑ u)) dτ = [log(η +i(t u))+log(η i(t+u)) log(η iu)]. π (.44) Its real part is an even function on u R, and its imaginary part an odd function on u. In the limit η + we obtain for u and u t that lim [log(η+i(t u))+log(η i(t+u)) log(η iu)] η + π = π log t u + i (.45) 4 [sign(t u) sign(t+u)+sign(u)]. Moreover, we have and for all α t πi t πi +η+iτ +η iτ +η+iτ dsdτ = +η iτ ds s+α dτ = π +α+η log ( + [ tarctan t π, (.46) t +α+η )] t (. (.47) +α+η)

13 Thus we obtain by direct calculation from the last two equations and (.) t lim η + πi = t +η+iτ +η iτ N (τ)dτ. [ s (γ +logπ)+ ( n ) ] dsdτ s+n n= (.48) Next we use(.44), (.45) and obtain from Fubini s theorem and the Lebesgue dominated convergence theorem that t lim η + πi +η+iτ +η iτ π log ζ( ±iu) (s du dsdτ = iu) t N (τ)dτ. (.49) Finally we recall that for η > sufficiently small the rectangle bounded by P ± η,t does not contain Blaschke ζ-zeros, such that t lim η + πi +η+iτ +η iτ B (s) t B(s) dsdτ = N 3 (τ)dτ. (.5) From (.4), (.43) and (.48), (.49), (.5) we conclude (.35). 3 Logarithmic Fourier integrals and their relation to the distribution of prime numbers In this section we derive Fourier-Mellin transforms from the boundary integral formulas obtained in Section which give interesting relations to the distribution of prime numbers. AsafirstbasicbuildingblockweneedarepresentationtheoremfortheMellin transforms of certain functions involving the exponential integral, which is generally useful for the study of Hadamard s product decomposition of entire functions with appropriate growth conditions. One of the various representations of the exponential integral is Ei(z) := γ +logz +Ei (z) (3.)

14 with Euler s constant γ = lim n Ei (z) := k= ( n k= z k z k k! = ) k logn and the entire function e t t dt = e uz u du. (3.) IncontrasttotheentirefunctionEi, thefunctionseiandlogareonlydefined on the cut plane C := C\(,]. The representation theorem formulated below was proved in [4], which also contains its application to Riemann s explicit formula for the number of primes less than a given limit x > and related formulas. Theorem (3.) Let s be any complex number with Re(s) >. (a) The expression (γ + log(logx))/x s+ is Lebesgue integrable on the interval (, ), and there holds the relation s = exp γ +log(logx) s dx. x s+ (b) The expression Ei(logx)/x s+ is Lebesgue integrable on the interval (, ), and with Li(x) := Ei(logx) there holds the relation s = exp Li(x) s dx. xs+ (c) Assume that ρ C\[, ) and Re(ρ) < Re(s). Then s ρ = exp γ +log(logx)+log( ρ)+ei (ρlogx) s dx. x s+ If we note that for u > and Re(s) > x sin(ulogx) x s+ dx = 3 u u +(s ) (3.3)

15 and define f C : (, ) R by f C (x) := x π πn(u) ϑ(u) arctan(u) u + 4 sin(ulogx)du, (3.4) then we obtain from (.) and (.4) with Fubini s theorem f C (x) ζ C (s) = exp s(s ) dx. (3.5) xs+ Next we define for α / C the functions ϕ α,φ α : (, ) C by ϕ α (x) := γ +log(logx)+log( α)+ei (αlogx), Φ α (x) := x x ϕ α (y) y dy. (3.6) FromTheorem(3.)aboveweobtainbypartialintegrationthatforRe(s) > and Re(s) > Re(α) > s α = exp Φ α (x) s(s ) dx. (3.7) xs+ The integral expression for Φ α in (3.6) can be solved explicitely. For any α C with Im(α) and for all x > we obtain that Φ α (x) = x ϕ α (x) ϕ α (x)+x [log( α) log( (α ))]. (3.8) Equation (3.8) can be checked easily by forming the derivative of Φ α (x)/x with respect to x and by regarding that limφ α (x) =. We have not expressed Φ α (x) in terms of the exponential integral, because ϕ α (x) in (3.6) x has a better asymptotic behaviour than Ei(αlogx) for Im(α) and x >, ϕ α (x) = Ei(αlogx) iπsign(im(α)) = xα xα e y αlogx +xα (αlogx y) dy, e y (αlogx y) dy x Re(α) Im(α) log x. (3.9) 4

16 Aproofofthefollowing theoremcanbefoundinthetextbook[8] ofedwards. Theorem (3.) For any integer number n we define the von Mangoldt function { lnp, for n = p Λ(n) := m, m, p prime, otherwise, and for x the functions ψ(x) := n x Λ(n), π (x) := <n x Λ(n) lnn = n= π( n x) n where π(x) is the number of primes x. Then we obtain for Re(s) > ζ(s) = exp( n= ζ (s) ζ(s) = Λ(n)n s = s n= Λ(n) lnn n s ) = exp(s π (x) dx) (3.) xs+ ψ(x) dx. (3.) xs+, From Theorem (3.)(a,b) and (3.) we obtain with the entire function Ei defined in (3.) for Re(s) > that s π (x) Ei (logx) ζ(s) = exp s dx. (3.) s x s+ If we apply partial integration on the last integral, we can also rewrite (3.) in the form s f (x) ζ(s) = exp s(s ) dx, (3.3) s xs+ where the function f : (, ) R is given by f (x) = x = x x π (y) Ei (logy) dy y ( ) Λ(n) nlogn +Ei ( logx) n x 5 (π (x) Ei (logx)). (3.4)

17 Now we are able to prove the following result, Theorem (3.3) For α C with Im(α) we define Φ α : (, ) C by Φ α (x) := xϕ α (x) ϕ α (x). (3.5) Then we obtain for all x > for the function f in (3.4) (a) f (x) = x π ρ:ζ(ρ)= Re(ρ)> πn(u) ϑ(u) arctan(u) u + 4 [ Φρ (x)+ Φ ] ρ (x) Φ (+ρ ρ)(x), sin(ulogx)du (3.6) (b) f (x) = x π ρ:ζ(ρ)= Re(ρ)> log ζ( +iu) u + cos(ulogx)du 4 [ Φρ (x) Φ ρ (x)]. (3.7) In equations (a) and (b), the integrals as well as the sums (or series) with respect to the Blaschke zeroes all converge in the absolute sense. Proof: We define the functions f,f,f,f : (, ) R by f (x) := x π f (x) := ρ:ζ(ρ)= Re(ρ)> f (x) := x π f (x) := ρ:ζ(ρ)= πn(u) ϑ(u) arctan(u) u + sin(ulogx)du, 4 [ Φρ (x)+ Φ ] ρ (x) Φ (+ρ ρ)(x), log ζ( +iu) u + cos(ulogx)du, 4 [ Φρ (x) Φ ρ (x)]. (3.8) Re(ρ)> 6

18 The absolute convergence of the f -integral was already mentioned for the formulation of the Balazard-Saias-Yor Theorem, and the absolute convergence of the f -integral results from the first equation in (.). For the absolute convergence of the sums f,f we notice the third equation in (.) and for x >, Im(α) the asymptotic law (3.9), which implies that Φ α (x) = α(α ) x α x +R(α,x), R(α,x) logx Re(α) Im(α) log x. (3.9) We also obtain that the convergence of the expressions in (3.8) is uniform on each compact interval < x x x, such that the functions f,f,f,f are continuous. Using the following asymptotic behaviour for x, π (x) Li(x) = O(e c logx ), c > constant, (3.) we conclude from (3.) in the limit s for real s > that f (x) lim x x = π (t) Ei (logt) t dt =. (3.) In accordance with this we also obtain from (3.9) that f mn (x) lim x x On the other hand we have from the definition of f =, m,n {,}. (3.) lim f (x) =, (3.3) x and the symmetry of the zeta zeroes with respect to complex conjugation and (.), (3.6), (3.8) imply that lim f (x) =, limf (x) =, x x lim f (x) = Ω ζ, limf (x) = Ω ζ. x x (3.4) From Theorem (.), (3.3) and (3.7), (3.8) we obtain for Re(s) > with a constant K C that f (x) (f (x)+f (x)+k x) = exp s(s ) dx. (3.5) x s+ 7

19 We conclude for x > from (3.) to (3.4) with Mellin s inversion formula f (x) = f (x)+f (x). (3.6) Thus we have shown the first part of Theorem (3.3). In order to prove the second part, we first recall that the function ζ B in (.3) is analytic for Re(s) > with B()ζ B (s) = s ζ(s) B(s) s B() Since we have the limit = exp Ω ζ + π ( s ) B() lim s ζ B (s) = B() B() the real valued function Q : (/, ) R with log ζ( +iu) u +(s ) du. (3.7) =, (3.8) Q(s) := log(b()ζ B(s)) s(s ) has an analytical continuation Q in Re(s) > with Q (s) = Ω ζ s + s πs(s ) = Ω ζ s π (s ) = Ω ζ +f (x) x s+ dx. [ log ζ( +iu) (u +(s ) ) log log ζ( +iu) (u + 4 )(u +(s ) ) du ζ( +iu) ] (u + ) 4 du (3.9) (3.3) For the last equation in (3.3) we have used Fubini s theorem with xcos(ulogx) x s+ dx = because all integrals converge in the absolute sense. s u +(s ), Re(s) >, (3.3) 8

20 Next we obtain from (3.7) and (.4) for Re(s) > that B(s) B() = exp s(s ) {Ω ζ (x ) ρ:ζ(ρ)= Re(ρ)> and with (3.3), (.), (3.9), (3.3) for Re(s) > s f (x) ζ(s) = exp s(s ) dx s xs+ (Φ ρ (x) Φ ρ (x))} dx = (ζ B (s)b()) B(s) B() Ω ζ +f (x) = exp s(s ) dx B(s) x s+ B(). x s+, (3.3) (3.33) Note the relation (3.8) between Φ α (x) and Φ α = x ϕ α (x) ϕ α (x). Then we combine (3.3) and (3.33) and use (3.)-(3.4) to conclude the second part of the theorem from Mellin s inversion formula. Finally we mention without going into details that the mathematical technique developed here can be used as well in order to derive an integral form for other explicite formulas related to the distribution of prime numbers, for example for the following result Theorem (3.4) For x >, α,r C and u R we define Θ(x,α) := K(x,r,u) := { x α +Ei α (αlogx)logx, α, logx, α =, { π π,r (x) := n x [ x /+iu r (/+iu r) logx log x π, /+iu r Λ(n) n r logn, ψ r(x) := n x ], r /+iu, r = /+iu, Λ(n) n r. (3.34) 9

21 Then we have x π,r (y) y dy = π,r (x)logx ψ r (x) = Θ(x, r) Θ(x, r) + (K(x,r,u)+K(x,r, u))log ζ( +iu) du ρ:ζ(ρ)= Re(ρ)> {Θ(x,ρ r) Θ(x, ρ r)}. (3.35) On the right hand side in (3.35) the integral as well as the sum (or series) with respect to the Blaschke zeroes converge in the absolute sense. References [] G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press (). [] R. Backlund, Sur les zéros de la fonction ζ(s) de Riemann, C.R. Acad. Sci. Paris 58, (94). [3] M. Balazard, E. Saias, The Nyman-Beurling equivalent form for the Riemann hypothesis, Expositiones Mathematicae 8, 3-38 (). [4] M. Balazard, E. Saias, M. Yor, Notes sur la fonction ζ de Riemann,, Advances in Math. 43, (999). [5] A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. 4, 3-34 (955). [6] J.-F. Burnol, A note on Nyman s equivalent formulation of the Riemann hypothesis, Algebraic methods in statistics and probability (Notre Dame, IN, ), Contemp. Math., 87, Amer. Math. Soc., Providence, RI, 3 6 (). [7] J.-F. Burnol, An adelic causality problem related to abelian L- functions, J. Number Theory 87, ().

22 [8] H. M. Edwards, Riemann s Zeta Function, Dover Publications, New York (). [9] J. B. Garnett, Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 36. Springer, New York (7). [] K. Hoffman, Banach spaces of analytic functions, Dover Publication, (988). [] P. Koosis, Introduction to H p -spaces, second edition, Cambridge University Press (998). [] P. Koosis, The logarithmic integral. I. Corrected reprint of the 988 original. Cambridge Studies in Advanced Mathematics,. Cambridge University Press (998). [3] P. Koosis, The logarithmic integral. II. Cambridge Studies in Advanced Mathematics,. Cambridge University Press (99). [4] M. Kunik, On the formulas of π(x) and ψ(x) of Riemann and von-mangoldt, Preprint Nr. 9/5, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (5). [5] H. von Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion ξ(t). Math. Ann 6, -9 (95). [6] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (859), in Gesammelte Werke, Teubner, Leipzig (89), wieder aufgelegt in Dover Books, New York (953).