Ramanujan summation of divergent series

Transcription

1 Ramanujan summation of divergent series B Candelpergher To cite this version: B Candelpergher. Ramanujan summation of divergent series. Lectures notes in mathematics, 285, 27. <hal-528v2> HAL Id: hal Submitted on 29 Mar 27 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Ramanujan summation of divergent series B. Candelpergher

3 2

4 Contents Introduction: the summation of series v Ramanujan summation. The Euler-MacLaurin summation formula Ramanujan s constant of a series A difference equation The functions ϕ f and R f The fundamental theorem The summation Definition and examples The fractional sum Relation to usual summation Properties of the Ramanujan summation Some elementary properties The unusual property of the shift Summation by parts Sums of products Summation and derivation The case of an entire function The sum of an entire function An expression of Catalan s constant Some trigonometric series Functional relations for fractional sums Dependence on a parameter Analyticity with respect to a parameter The theorem of analyticity Analytic continuation of Dirichlet series The zeta function of the Laplacian on S More zeta series A modified zeta function The sums R nk ϕ f (n) Integration with respect to a parameter i

5 ii CONTENTS 3.2. Interchanging R and I The functional equation for zeta The case of a Laplace transform Series involving zeta values Double sums Definitions and properties Some formulas for the Stieltjes constant γ A simple proof of a formula of Ramanujan The functional relation for Eisenstein function G Transformation formulas 4. A Borel summable series A formal transform Borel summation Borel summability of Euler-MacLaurin series A convergent expansion Bernoulli numbers of second kind Newton interpolation formula Evaluation of n f() The convergent transformation formula Summation of alternating series Euler summation of alternating series Properties of the summation Relation with the Ramanujan summation Generalization An algebraic view on the summation of series Introduction An algebraic formalism Properties of the general summation The linearity property The shift property The associated algebraic limit Sum of products Examples Appendix Euler-MacLaurin and Euler-Boole formulas A Taylor formula The Euler-MacLaurin formula The Euler-Boole formula Ramanujan s interpolation formula and Carlson s theorem Bibliography 67 8 Chapter VI of the second Ramanujan s Notebook 7

6 CONTENTS iii Notation * For x = a + ib where a and b are real we use the notation a = Re(x) and b = Im(x) * The derivative of a function f is denoted by f or f. * The difference operator is defined by (f) = f(x + ) f(x). * For k N and j =,..., k we use the binomial coefficient C j k = k! j!(k j)! * The harmonic numbers are H (s) n = + 2 s n s. * The Bernoulli polynomials B k (x) are defined by n B n (x) t n = text n! e t and the Bernoulli numbers B n = B n () We have B =, B = /2, B 2n+ = if n. * We define the Bernoulli numbers of second kind β n by t log( + t) = n β n n! tn = + β n+ (n + )! tn+ n * The Euler polynomials E n (x) are defined by n E n (x) z n = 2exz n! e z + and we set E n = E n (). * For x C we use the notation x f(u)du for the integral x f(u)du = f( + t(x ))(x )dt that is the integral on the segment relying to x. * Log is the logarithm function defined on C\], ] by that is for x C\], ] by Log(x) = x u du Log(x) = Log( x ) + iθ if x = x e iθ, θ ] π, π[

7 iv CONTENTS * The Stieltjes constants γ k are defined by γ k = ( n Log k (j) lim n + j j= Logk+ (n)) k + The Euler constant γ is γ. * The Catalan constant is G = + n= * The digamma function is ψ(z) = Γ (z) Γ(z) ( ) n (2n ) 2. * We define O α the space of functions g analytic in a half plane {x C Re(x) > a} with some a < and of exponential type < α: there exists β < α such that g(x) Ce β x for Re(x) > a * We say that f is of moderate growth if f is analytic in a half plane {x C Re(x) > a} with some a < and of exponential type < ε for all ε >. * For f O π the function R f is the unique solution in O π of R f (x) R f (x + ) = f(x) with 2 R f (x)dx = * For f O π the function ϕ f is the unique function in O π such that ϕ f (n) = f() f(n) for every integer n * For f O π the Ramanujan summation of f(n) is defined by f(n) = R f () If the series is convergent then + n= f(n) denotes its usual sum.

8 Introduction: the summation of series The strange sums and n n n = n 3 = appear in Physics about the study of the Casimir effect which is the existence of an attractive force between two parallel conducting plates in the vacuum. These series are examples of divergent series in contrast to convergent series, the notion of convergence for a series was introduced by Cauchy in his Cours d Analyse in order to avoid frequent mistakes in working with series. Given a series of complex numbers n a n, Cauchy considers the sequence of the partial sums s = s = a s 2 = a + a... s n = a a n and says that the series n a n is convergent if and only if the sequence (s n ) has a finite limit when n goes to infinity. In this case the sum of the series is defined by a n = lim s n n + n= The classical Riemann series n s is convergent for every complex number s such that Re(s) > and defines the Riemann zeta function defined for Re(s) > by ζ : s + n= n s. v

9 vi INTRODUCTION: THE SUMMATION OF SERIES Non convergent series are divergent series. For Re(s) the Riemann series is a divergent series and does not give a finite value for the sums that appear in the Casimir effect. A possible strategy to assign a finite value to these sums is to perform an analytic continuation of the zeta function, this has been done by Riemann (cf. Edwards) who found an integral formula for ζ(s) which is valid not only for Re(s) > but also for s C\{}. By this method we can assign to the series nk with k > the value ζ( k), we get for example n = ζ() = 2 n = ζ( ) = 2 n 2 = ζ( 2) = n 3 = ζ( 3) = 2... For k = we have the harmonic series n = which is easily proved to be a divergent series since the partial sums s n verify s n = n 2 x dx dx x n+ n dx = Log(n + ) x But the strategy of analytic continuation of the zeta function does not work in this case since ζ has a pole at s = that is lim s ζ(s) =. Divergent series appear elsewhere in Analysis and are difficult to handle, for example by using the preceding values of nk it seems that 2 2 = ( ) = = ( ) = 2 This absurdity shows that with divergent series we cannot use the classical rules of calculation, and for a given class of series we need to define precisely some method of summation and its rules of calculation. Before and after Cauchy, some methods of summation of series have been introduced by several mathematicians such as Cesaro, Euler, Abel, Borel and

10 vii others. These methods of summation assign to a series of complex numbers n a n a number obtained by taking the limit of some means of the partial sums s n. For example the Cesaro summation assigns to a series n n a n the number C n a n = lim n + s s n n (when this limit is finite) For the Abel summation we take A a n = lim ( t) + s n+ t n t n n= (when this limit is finite) where the series n s n+t n is supposed to be convergent for every t [, [. Note that this expression can be simplified since and we have ( t) s n+ t n = s + n= n= (s n+ s n )t n = n= a n t n A a n = lim n t n= a n t n (when this limit is finite) this gives for example A ( ) n = lim n t n= ( ) n t n = lim t + t = 2 The classical methods of summation use these types of means of partial sum and can be briefly presented in the following form. Let T be a topological space of parameters and l some limit point of the compactification of T (if T = N then l = +, if T = [, [ then l = ). Let (p n (t)) n N be a family of complex sequences indexed by t T such that for all t T the series n p n(t) is convergent, then we set T n + n= a n = lim p n(t)s n t l + n= p n(t) when this limit is finite and in this case we say that the series n a n is T -summable. A theorem of Toeplitz (cf. Hardy) gives necessary and sufficient conditions on the family (p n (t)) n N to insure that in case of convergence this summation coincides with the usual Cauchy summation.

11 viii INTRODUCTION: THE SUMMATION OF SERIES These summation methods verify the linearity conditions T (a n + b n ) = n T Ca n = C n T a n + n T n b n T a n for every constant C C n and the usual translation property T a n = a + n T n a n+ This last property which seems natural is in fact very restrictive. For example the series can t be T -summable since the translation property gives an absurd relation T T = + n Thus if we need a method of summation such that the sums T n nk are well defined for any integer k then we must abandon the translation property requirement and find a way to define summation procedures other than the way of the limit of means of partial sums. This can be done by using a sort of generating function for the terms of the series. It is based on the following algebraic framework (cf. Candelpergher). Let E be a C-vector space (in general a space of functions) equiped with a linear operator D and a linear map v : E C. Given a sequence of complex numbers (a n ) n we call an element f E a generator of this sequence if We can write formally n a n = v (D n f) n a n = n v (D n f) = v ( n thus if R verifies the equation then we get D n f) = v ((I D) f) (I D)R = f () a n = v (R) n Of course, such an algebraic definition of summation needs some hypotheses, especially to assure uniqueness of the solution of equation (), this is presented in Chapter 5.

12 ix It is easy to see that the Cauchy summation is a special case of this algebraic formalism: we take E as the vector space of convergent complex sequences u = (u n ) n and D : (u n ) (u n+ ) v : (u n ) u In this case the generator of a sequence of complex numbers (a n ) n is precisely this sequence f = (a n ) n since (D n f) k = a k+n. If we set R = (r n ) n equation () becomes the difference equation r n r n+ = a n The solution of this equation is defined up to an arbitrary constant, thus to get a unique solution we need to impose a condition on (r n ) n. Since we see that if we add the condition r r n = a a n lim r n = n + then we have a unique solution R of () and we get the usual convergence of the series n a n with v (R) = r = lim (a a n ) = a n n + n= With this algebraic setting we have presented (cf. Candelpergher) the summation method employed by Ramanujan in Chapter VI of his second Notebook. This is done in a way similar to the Cauchy summation but replacing the space of sequences by a certain space E of analytic functions and Df(x) = f(x + ) v (f) = f() For the Ramanujan summation the terms of the series are indexed with n as Ramanujan does and since v (D n f) = f(n) we define this summation for the series of type f(n) where f E. In this case equation () is simply the difference equation R(x) R(x + ) = f(x) (2) If we can select a solution R f of this equation then we define the Ramanujan summation by f(n) = R f () (3)

13 x INTRODUCTION: THE SUMMATION OF SERIES Existence and uniqueness of a solution of (2) can be proved by use of a general Laplace transform (cf. Candelpergher, Coppo, Delabaere). Now in Chapter we give a more simple presentation that avoids the Laplace transform and we give more explicit formulas. We start, as Ramanujan in his second Notebook, with the summation formula of Euler and MacLaurin f() f(n) = C f + n f(x)dx + f(n) + k B k k! k f(n) This formula can be viewed as an asymptotic expansion, when x goes to infinity, of the function that Ramanujan writes ϕ f : x f() f(x) which is an hypothetical interpolation function of the partial sums of the series f(n), given with the condition ϕ f () =. This expansion contains a constant C f that Ramanujan calls the constant of the series and treats like a sort of sum for the series. Since ϕ f (x + ) ϕ f (x) = f(x + ), we see that if we set R f (x) = C f + f(x) ϕ f (x) then the function R f is a solution of equation (2) and f(n) = R f () = C f We use the term Ramanujan constant for R f () since the expression Ramanujan sum is widely used for another notion in Number Theory. We give a precise definition of R f in Chapter by the use of an integral formula which is related to the Abel-Plana summation formula. For a function f sufficiently decreasing, the Euler-MacLaurin summation formula gives C f = lim n + ( n f() f(n) f(x)dx ) (4) thus if we are in a case of convergence of the series and integral then we have f(n) = n= f(n) f(x)dx (5) There is also the apparition of an integral when we compare the sums R f(n) and R f(n + ), since by (4) we have, in contrast to the usual translation property, the unusual shift property f(n + ) = f(n) f() + 2 f(x)dx (6)

14 xi When we apply this property to the function R f and use equation (2) we get 2 R f (x)dx = (7) This gives a way to define a general Ramanujan summation, independently of the Euler-MacLaurin formula, by equations (2) and (3) with R f of exponential growth < π and verifying the integral condition (7) in order to determine a unique solution R f. The shift property (6) remains valid in this general setting and is very useful, for example the Casimir series (n )k (with k a positive integer) are Ramanujan summables and we have n k = (n ) k + 2 (x ) k dx = (n ) k + k + (8) We see that the series + 2 k + 3 k + 4 k +... and k + 3 k + 4 k +... don t have the same sum! This is a consequence of the fact that the Ramanujan summation of f(n) is intimately related to the function f (in the first case we have f(x) = x k and in the second f(x) = (x ) k ). In chapter 2 we study elementary properties of the Ramanujan summation in comparison to the classical properties of the usual Cauchy summation. We prove a surprising relation between the sums ϕ f (n) and n f(x)dx We also give, in the special case of an entire function f of exponential type < π, a simple formula for the Ramanujan summation of a series f(n) in terms of a convergent series involving the Bernoulli numbers f(n) = f(x)dx 2 f() + k= k f() B k+ (k + )! In Chapter 3 we give important theorems that concern properties of sums R f(z, n) where z is a parameter, with respect to derivation, integration or summation of f in z. The simplest introduction of an external parameter in a series f(n) is to consider the series f(n)e nz. We prove that if f is a function of moderate growth then we have f(n) = lim z f(n)e nz

15 xii INTRODUCTION: THE SUMMATION OF SERIES But if z > then the series f(n)e nz is convergent and by (5) we have ( + f(n) = lim f(n)e nz z + n= f(x)e zx dx ) (9) which seems to be a standard method of regularization used in Physics. A very important property of the Ramanujan summation is that the analyticity of the terms of a series implies the analyticity of the sum. A simple example is given by the zeta series n. This series is convergent only z for Re(s) > but the Ramanujan summation of this series is defined for all z C and by (5) we have for Re(z) > + n z = n z z = ζ(z) z By this property of analyticity this formula remains valid for all z and we see that the Ramanujan summation of the zeta series cancels the pole of the zeta function at z =. More precisely at z = we have by (4) n = lim ( n + n Log(n)) = γ (Euler constant) We also note that for the Casimir series (n )k (with k a positive integer) we have by (8) (n ) k = n k = ζ( k) () k + In the second section of Chapter 3 we study the possibility to integrate the sum R f(z, n) with respect to z. For example we have for < Re(s) < R u s R n + u du = u s n + u du = π sin πs n s which gives simply the functional equation for the ζ function. Finally we give a sort of Fubini theorem for double sums m f(m, n) We apply this result to the Eisenstein function G 2 (cf. Busam and Freitag) G 2 (z) = n= ( + m= m if n= ) (m + nz) 2

16 xiii Since we don t have absolute summability we cannot interchange the sums, but thanks to the Ramanujan summation we can prove simply that this function G 2 satisfies the non trivial relation G 2 ( z ) = z2 G 2 (z) 2iπz In Chapter 4 we give some relations of the Ramanujan summation to other summation formulas. First we recall the classical result (cf. Hardy) on the Borel summability of the asymptotic series B k k k! k f(n) that appears in the Euler-MacLaurin formula and we give the formula linking R f(n) with the Borel-sum of the series B k k k! k f(). In the second section of this chapter we use the Newton interpolation series (cf. Nörlund) to give a transformation formula involving a convergent series β k+ f(n) = (k + )! ( k f)() k= where the β k are the Bernoulli numbers of the second kind and is the usual difference operator. This formula is related to the classical Laplace summation formula (cf. Boole). In the third section we see that we can define the Euler summation of alternate series by using the Euler-Boole summation formula that is an analogue of the Euler-Maclaurin formula. We see that this summation is given by E ( ) n ( ) k f(n) = 2 k+ ( k f)() k= and we prove that this sum is related to the sums R R f(2n) by f(2n ) f(2n) = E ( ) n f(n) 2 2 f(2n ) and f(t)dt This can also be generalized to series of type ωn f(n) where ω is a root of unity. The unusual shift property (6) shows that the Ramanujan summation does not verify the translation property which is required (with linearity) for a summation method by Hardy in his very fine book Divergent series. But we have seen that abandon of the translation property is necessary if we want to sum series like P (n) where P is a polynomial. Thus in Chapter 5 we give a general algebraic theory of summation of series that unifies the classical summation methods and the Ramanujan summation. In this general

17 xiv INTRODUCTION: THE SUMMATION OF SERIES framework the usual translation property appears as a special case of a more general shift property. In appendix we give the classical Euler-MacLaurin and Euler-Boole formulas and a proof of Carlson s theorem. Obviously we can not claim that our version of the Ramanujan summation is exactly the summation procedure that Ramanujan had in mind, so we give an exact copy of the Chapter VI of the second Notebook in which Ramanujan introduces the constant of a series. Acknowledgements My warmest thanks to M.A. Coppo, E. Delabaere, H. Gopalkrishna Gadiyar and R. Padma for their interest and contributions to the study of the Ramanujan summation. I also wish to express my gratitude to F. Rouviere for his helpful comments and his constant encouragements.

18 Chapter Ramanujan summation In the first two sections of this chapter we recall the Euler-MacLaurin formula and use it to define what Ramanujan, in Chapter VI of his second Notebook, calls the constant of a series. But, as Hardy has observed, Ramanujan leaves some ambiguity in the definition of this constant. Thus in the third section we interpret this constant as the value of a precise solution of a difference equation. Then we can give in section.4 a rigorous definition of the Ramanujan summation and its relation to the usual summation for convergent series.. The Euler-MacLaurin summation formula Let s consider a function f C (], + [). For every positive integer n we can write n n f(k) = (k (k ))f(k) k= k= n = nf(n) k(f(k + ) f(k)) = nf(n) k= n k+ k= k [x]f (x)dx where [x] is the integral part of x. Let {x} = x [x], then n n n f(k) = nf(n) xf (x)dx + {x}f (x)dx k= and integration by parts gives n f(k) = f() + k= n f(x)dx + n {x}f (x)dx

19 2 CHAPTER. RAMANUJAN SUMMATION In order to generalize this formula we define the function b (x) = {x} 2 which is a -periodic function with b (x)dx =. We have n f(k) = f() + k= n f(x)dx + n b (x)f (x)dx + 2 thus we get the Euler-MacLaurin formula to order : n f(k) = k= n n f (x)dx f(x)dx + n 2 (f(n) + f()) + b (x)f (x)dx (.) To make another integration by parts in the last integral we introduce the function c 2 (x) = x b (t)dt which is a -periodic with c 2 (n) = for every positive integer n and then by integration by parts Thus we get n f(k) = k= n n c 2 (x) = x2 2 x 2 b (x)f (x)dx = We now replace the function c 2 by if x ], [ n c 2 (x)f (x)dx f(x)dx + n 2 (f(n) + f()) c 2 (x)f (x)dx b 2 (x) 2 with the choice of B 2 in order that = c 2 (x) + B 2 2 b 2 (x) 2 dx = which gives B 2 = /6 and b2(x) 2 = x2 2 x if x [, [. Then we get the Euler-MacLaurin formula to order 2: n f(k) = k= n f(x)dx+ n 2 (f(n)+f())+b 2 2 (f (n) f b 2 (x) ()) 2 f (x)dx

20 .2. RAMANUJAN S CONSTANT OF A SERIES 3 If we continue these integrations by parts we get a general Euler-MacLaurin formula to order m: f() f(n) = n + f(x)dx + m k=2 f() + f(n) 2 ( ) k B k [ k f(n) k f()] k! +( ) m+ n b m (x) m f(x)dx m! where b m (x) is the -periodic function b m (x) = B m (x [x]), with the Bernoulli polynomials B k (x) defined by n B n (x) t n = text n! e t and the Bernoulli numbers B n = B n () (we verify that B 2k+ = for k ). For a simple proof of this general formula see Appendix..2 Ramanujan s constant of a series Let f be a C function defined for real x >. In the beginning of Chapter VI of his Notebook 2, Ramanujan introduces the hypothetical sum f() + f(2) + f(3) + f(4) f(x) = ϕ(x), which is intended to be the solution of ϕ(x) ϕ(x ) = f(x) with ϕ() = Let s take the numbers B r defined when r = 2, 4, 6,... by (second notebook chapter V, entry 9) x e x = x 2 + k ( ) k B 2k x 2k (2k)! then Ramanujan writes the Euler-McLaurin series ϕ(x) = C + f(x)dx+ 2 f(x)+b 2 2 f (x) B 4 4 f (x)+ B 6 6 f V (x)+ B 8 8 f V II (x)+... and he says about the constant C : the algebraic constant of a series is the constant obtained by completing the remaining part in the above theorem. We can substitute this constant which is like the centre of gravity of a body instead of its divergent infinite series.

21 4 CHAPTER. RAMANUJAN SUMMATION In Ramanujan s notation the numbers B 2n are related with the usual Bernoulli numbers by B 2n = ( ) n B 2n Thus we can write the above Euler-McLaurin series in the form ϕ(x) = C + f(x)dx + 2 f(x) + B 2k (2k)! 2k f(x) k 2 The main difficulty with this formula is that this last series is not always convergent. Therefore we replace this series by a finite sum and give a precise meaning to the integral. We are thus led to write the Euler-MacLaurin summation formula in the form f() f(n) = C m (f) + where C m (f) = f() 2 m k= n n f(x)dx + f(n) 2 + b 2m+ (x) (2m + )! 2m+ f(x)dx m k= B 2k (2k)! 2k f(n) B + 2k b 2m+ (x) (2k)! 2k f() + (2m + )! 2m+ f(x)dx in this formula we assume that the function f is an infinitely differentiable function and that the integral b 2m+ (x) 2m+ f(x)dx is convergent for all m M >. Then by integration by parts we verify that the constant C m (f) does not depend on m if m M thus we set C m (f) = C(f). We use the notation C(f) = f(n) and call it the Ramanujan s constant of the series. Example If f is a constant function then f = thus R f() f(n) = 2. Thus = 2 If f(x) = x then 2 f = thus R n = 2 B2 2 = 5 2.

22 .2. RAMANUJAN S CONSTANT OF A SERIES 5 The case of convergence Assume that the integral b (x) f(x)dx is convergent, then the Euler- MacLaurin formula to order is simply with f() f(n) = C(f) + n C(f) = f() 2 + f(x)dx + f(n) 2 n b (x)f (x)dx b (x)f (x)dx Since b n (x)f (x)dx when n + we get the following expression of the Ramanujan constant as the limit f(n) = lim n + ( f() f(n) n f(x)dx f(n) ) 2 (.2) If in addition we assume that lim n + f(n) = then we have f(n) = lim n + n ( f() f(n ) k= k+ k f(x)dx ) thus we get an expression of the Ramanujan constant as a sum of a convergent series ( n+ f(n) = f(n) f(x)dx ) (.3) n= This proves the property that if the series f(n) and the integral f(x)dx are convergent then we have the relation f(n) = f(n) n f(x)dx (.4) Examples ) If f(x) = x z with Re(z) > then by (.4) + n z = + n z + x z dx = n z z thus for Re(z) > we have the relation with the classical Riemann zeta function n z = ζ(z) (.5) z

23 6 CHAPTER. RAMANUJAN SUMMATION 2) If f(x) = x n = then by (.2) lim ( n n + k Log(n) 2n ) = k= lim ( n n + k Log(n)) k= thus = γ where γ is the Euler constant (.6) n 3) If f(x) = Log(x) then by (.2) Log(n) = lim ( n Log(k) (nlog(n) n + + n + 2 Log(n)) k= Using the Stirling formula we have this gives lim n + k= n Log(k) (nlog(n) n + 2 Log(n)) = Log( 2π) Log(n) = Log( 2π) (.7) 4) If f(x) = xlog(x) then we have f(x) = Log(x) + and 2 f(x) = x. Thus by the preceding Euler-MacLaurin formula with m = we have nlog(n) = this gives nlog(n) = thus lim n + lim n + ( n klog(k) k= ( n k= n xlog(x)dx nlog(n) 2 Log(n) + ) 2 klog(k) Log(n)( n2 2 + n ) + n2 ) 4 3 nlog(n) = Log(A) 3 (.8) where A is the Glaisher-Kinkelin constant (cf. Srivastava and Choi p.39). This constant is related to the zeta function by the following relation that we prove later on nlog(n) = ζ ( ) 4

24 .2. RAMANUJAN S CONSTANT OF A SERIES 7 Remark Note that with the Euler-MacLaurin formula we have for all a > : f() f(n) = a f() f(x)dx + 2 m B 2k k= (2k)! 2k f() + b 2m+(x) (2m+)! 2m+ f(x)dx + n f(n) f(x)dx + a 2 + m B 2k k= (2k)! 2k f(n) b 2m+(x) n (2m+)! 2m+ f(x)dx f(n) is re- This is a summation formula where the constant C(f) = R placed by a C a (f) = f(x)dx + f(n) It seems that Ramanujan leaves the possibility that the choice of a depends on the series considered. In the special case where the series f(n) and the integral f(x)dx are convergent then if we take a = + we get C (f) = and with the relation (.4) we get C (f) = f(x)dx + n= f(n) f(n) This explains an assertion of Ramanujan in Notebook 2 Chapter 6 p.62: If f () + f (2) f (x) be a convergent series then its constant is the sum of the series. But if for example f(x) = x, then it is not possible to take a = + since C (f) is not defined but if we take for example a = then C (f) = x dx + n = B 2 2 To get simple properties of the Ramanujan summation we fix the parameter a in the integral, and we make the choice a = in order to have n = γ

25 8 CHAPTER. RAMANUJAN SUMMATION Conclusion With the use of Euler-Maclaurin formula we have the definition of the constant of a series by n f(n) = lim (f()+...+f(n) [ n + this needs convergence of the integral f(x)dx+ f(n) 2 + m b 2m+ (x) 2m+ f(x)dx k= B 2k (2k)! 2k f(n)] (.9) This last hypothesis is not always satisfied, for example if we look at a series like en. Thus we need to avoid the systematic use of Euler- McLaurin formula and define in a more algebraic way the Ramanujan summation..3 A difference equation.3. The functions ϕ f and R f In his Notebook Ramanujan uses the function ϕ formally defined by ϕ(x) = f() f(x) It seems he has in mind a sort of unique interpolation function ϕ f of the partial sums f() + f(2) f(n) of the series f(n) associated to f. This interpolation function must verify ϕ f (x) ϕ f (x ) = f(x) (.) and Ramanujan sets the additional condition ϕ f () =. With this condition the relation (.) gives for every n integer ϕ f (n) = f() + f(2) f(n) Note that if the series f(n) is convergent we have lim ϕ f (n) = f(n) n + n= Now in general the Euler-Mclaurin summation formula gives an expansion of the function ϕ f which we can write ϕ f (n) = C(f) + f(n) R f (n)

26 .3. A DIFFERENCE EQUATION 9 where the function R f is defined by R f (n) = f(n) 2 m and C(f) = f() 2 k= B + 2k (2k)! 2k f(n)+ n m k= We observe that C(f) = R f (). Thus we get b 2m+ (t) (2m + )! 2m+ f(t)dt n B + 2k b 2m+ (x) (2k)! 2k f() + (2m + )! 2m+ f(x)dx f(t)dt (.) f(n) = R f () (.2).3.2 The fundamental theorem To avoid the systematic use of the Euler-MacLaurin summation formula we now find another way to define the function R f. The difference equation ϕ f (n + ) ϕ f (n) = f(n + ) and the relation ϕ f (n) = C(f) + f(n) R f (n) gives for R f equation R f (n) R f (n + ) = f(n) the difference By (.2) it seems natural to define the Ramanujan summation of the series f(n) by R f(n) = R() where the function R satisfies the difference equation R(x) R(x + ) = f(x) Clearly this equation is not sufficient to determine the function R, so we need additional conditions on R. Let us try to find these conditions. First we see, by the definition (.) of R f, that if f and its derivatives are sufficiently decreasing at + then we have lim R f (n) = n + f(x)dx

27 CHAPTER. RAMANUJAN SUMMATION But we can t impose this sort of condition on the function R because it involves the integral f(x)dx which, in the general case, is divergent. Thus we now translate it into another form. Suppose we have a smooth function R solution of the difference equation R(x) R(x + ) = f(x) for all x > Now if we integrate the two sides of this equation between k and k + for all integer k, and formally take the infinite sum over k, we obtain f(x)dx = 2 R(x)dx lim R(x) x + We see that for the function R we have the equivalence lim R(x) = f(x)dx if and only if x + 2 R(x)dx = Thus we can try to define the function R f by the difference equation R f (x) R f (x + ) = f(x) with the condition 2 R f (x)dx = Unfortunately this does not define a unique function R f because we can add to R f any combination of periodic functions x e 2iπkx. To avoid this problem we add the hypothesis that R f is analytic in the half plane {x C Re(x) > } and of exponential type < 2π: Definition A function g analytic for Re(x) > a is of exponential type < α (α > ) if there exists β < α such that g(x) Ce β x for Re(x) > a We define O α the space of functions g analytic in a half plane {x C Re(x) > a} with some a < and of exponential type < α in this half plane. We say that f is of moderate growth if f is analytic in this half plane and of exponential type < ε for all ε >. With this definition we have the following lemma : Lemma (uniqueness lemma) Let R O 2π, be a solution of then R =. R(x) R(x + ) = with 2 R(x)dx =

28 .3. A DIFFERENCE EQUATION Proof By the condition R(x) R(x + ) =, we see that R can be extended to an entire function. And we can write R(x) = R (e 2iπx ) where R is the analytic function in C\{} given by R (z) = R( 2iπ Log(z)) (where Log is defined by Log(re iθ ) = ln(r) + iθ with θ < 2π). The Laurent expansion of R gives R(x) = n Z c n e 2iπnx the coefficients c n are c n = 2πr n 2π R (re it )e int dt = 2πr n 2π where r >. The condition that R is of exponential type < 2π gives c n r n Ce α 2π ln(r) with α 2π <. R( t 2π + 2iπ ln(r))e int dt If we take r we get c n = for n < and if we take r + then we get c n = for n >. Finally the condition 2 R(x)dx = then gives c =. Theorem If f O α with α 2π there exists a unique function R f O α such that R f (x) R f (x + ) = f(x) with 2 R f (x)dx =. This function is R f (x) = x f(t)dt + f(x) 2 + i f(x + it) f(x it) e 2πt dt (.3) Proof a) Uniqueness is given by the preceding lemma. b) The function R f defined by (.3) is clearly in O α. c) Let us prove that R f (x) R f (x + ) = f(x). By analyticity it is sufficient to prove this for real x. Consider the integral f(z) cot(π(z x))dz 2i with γ the path γ

29 2 CHAPTER. RAMANUJAN SUMMATION iy x x+ -iy By the residue theorem we have f(z) cot(π(z x))dz = f(x) 2i γ To evaluate the different contributions of the integral we use the formulas: and 2i cot(π(z x)) = 2 e 2iπ(z x) 2i cot(π(z x)) = 2 + e 2iπ(z x) when Im(z) > when Im(z) <. Let us examine the different contributions of the integral: * the semicircular path at x and x + gives when ε * the horizontal lines give 2 f(x) f(x + ) 2 ( x+ 2 ) f(t + iy)dt + 2 x x+ x f(t iy)dt and two additional terms which vanish when y + (by the hypothesis that f of exponential type < 2π). * the vertical lines give and 2 i y ε y ε f(x + it) f(x it) e 2πt dt i f(x+it)idt 2 y ε f(x it)idt 2 y ε y ε f(x + + it) f(x + it) e 2πt dt f(x++it)idt+ 2 y ε f(x+ it)idt

30 .3. A DIFFERENCE EQUATION 3 If we add this term with the contributions of the horizontal lines we obtain the sum of the integrals of f on the paths iy ιε ιε x x+ -iy By the Cauchy theorem this sum is 2 x+ x f(t + iε)dt + 2 x+ x f(t iε)dt which gives the contribution x+ x f(t)dt when ε. Finally when ε and y + we get f(x) = 2 f(x) f(x + ) 2 f(x + it) f(x it) +i e 2πt dt i + x+ x f(x + + it) f(x + it) e 2πt f(t)dt This is f(x) = R f (x) R f (x + ) with R f given by (.3). d) It remains to prove that 2 R f (x)dx =. By Fubini s theorem 2 f(x + it) f(x it) e 2πt dt dx = We have for < x < f(x + it) f(x it) e 2πt dx dt (f(x + it) f(x it))dx = F (2 + it) F (2 it) (F ( + it) F ( it))

31 4 CHAPTER. RAMANUJAN SUMMATION where F (x) = x f(t)dt. Thus 2 f(x + it) f(x it) e 2πt dt dx = F (2 + it) F (2 it) e 2πt dt F ( + it) F ( it) e 2πt dt By the preceding result (applied with F in place of f) we have With x = we get i 2 This gives 2 R f (x)dx = F (x) = 2 F (x) F (x + ) 2 F (x + it) F (x it) +i e 2πt dt i + x+ x F (x + + it) F (x + it) e 2πt F (t)dt f(x + it) f(x it) F () + F (2) e 2πt dt dx = F (x)dx+ 2 2 f(x)dx ( F () + F (2) ) F (t)dt F (t)dt =.4 The summation.4. Definition and examples Consider f O 2π, then by Theorem we can define the Ramanujan summation of the series f(n) by f(n) = R f () where R f is the unique solution in O 2π of R f (x) R f (x + ) = with 2 R f (x)dx =.

32 .4. THE SUMMATION 5 We immediately note that for some f O 2π this definition can give surprising results. Let us consider for example the function f : x sin(πx), since sin(πx) sin(π(x + )) = sin(πx) 2 2 we get thus R f (x) = sin(πx) 2 2 sin(πx) dx = sin(πx) π sin(πn) = π But sin(πn) = g(n) with the function g =, and we have trivially R g = which gives = This example shows that for f O 2π the sum R f(n) depends not only on the values f(n) for integers n but also on the interpolation function f we have chosen. To avoid this phenomenon we restrict the Ramanujan summation to functions f in O π, since with this condition we can apply Carlson s theorem (see Appendix) which says that such a function is uniquely determined by its values f(n) for all integers n. Note that in this case the function R f given by theorem is also in O π. Definition 2 If f O π, then there exist s a unique solution R f O π of R f (x) R f (x + ) = f(x) with 2 R f (x)dx = We call this function R f the fractional remainder of f and we set f(n) = R f () By (.3) we have by the integral formula f(n) = f() 2 + i f( + it) f( it) e 2πt dt (.4)

33 6 CHAPTER. RAMANUJAN SUMMATION We call this procedure the Ramanujan summation of the series f(n) and R f(n) the Ramanujan constant of the series. Some properties are immediate consequences of this definition: Linearity If a and b are complex numbers and f and g are in O π, then we verify immediately that R af+bg = ar f + br g thus the Ramanujan summation has the property of linearity af(n) + bg(n) = a f(n) + b g(n) Reality Consider g O π such that g(x) R if x R. Then for all t > we have by the reflection principle g( it) = g( + it) thus i(g( + it) g( it)) R and by the integral formula (.4) we have g(n) R Real and imaginary parts Take f O π defined in the half plane {x C Re(x) > a} with a <. We define for x ]a, + [ the functions f r : x Re(f(x)) and f i : x Im(f(x)) and assume that the functions f r and f i have analytic continuations on the half plane {x C Re(x) > a} that are in O π. Then we can define the sums R Re(f(n)) and R Im(f(n)) by By linearity we have f(n) = Re(f(n)) = Im(f(n)) = f r (n) f i (n) (f r (n)) + i f i (n)) = f r (n) + i f i (n)

34 .4. THE SUMMATION 7 Since f r (x) and f i (x) are real for x R then by the reality property we get Re( f(n)) = Re(f(n)) Im( f(n)) = Im(f(n)) For example if f(z) = z+i then f r : z z z 2 + and f i : z analytic functions in O π. Thus z 2 + are Re( Im( R n + i ) = R n + i ) = n n 2 + n 2 + Remark Note that generally we can t write R f(n) = R f(n) since the function f is not analytic (if f is non constant). Examples ) Take f(x) = e zx with z C then for Re(x) > we have f(x) = e Re(zx) = e Re(z x x ) x Thus if we set x x = eiθ then f O π for z U π where U π = {z C sup Re( ze iθ ) < π} θ [ π 2, π 2 ]

35 8 CHAPTER. RAMANUJAN SUMMATION Then if z U π \{} we can write e zx e e e z(x+) = e zx z z this gives R f (x) = e zx e 2 e zx z e dx, thus we have z and we get for z U π \{} R f (x) = e zx e z e z z e zn = e z e z e z z (.5) For z = we have R e zn = R = 2 = lim z ( e z e e z z z ). If z = it with t ] π, π[\{}, taking real and imaginary part of (.5) we get cos(nt) = sin(t) (.6) t 2 sin(nt) = 2 cot( t 2 ) cos(t) t (.7) 2) Let k be a positive integer. By the definition of the Bernoulli polynomials we verify that B k+ (x + ) k + and since B k+(x)dx = we have 2 B k+ (x) k + dx = B k+(x) k + = x k B k+ (x + ) B k (x) dx = k + k + dx + Thus if f(x) = x k where k is an integer then x k dx = k + R x k(x) = B k+(x) k + (.8) thus we get n k = B k+ k + if k and = 2 (.9)

36 .4. THE SUMMATION 9 3) Take f(x) = x with Re(x) >. z a) For Re(z) > and Re(x) > the series n and defines the Hurwitz zeta function: (n+x) z is convergent Since ζ(z, x) = n= (n + x) z ζ(z, x) ζ(z, x + ) = x z (.2) we have Since the series + n= 2 n= 2 R f (x) = ζ(z, x) (n+x) z (n + x) z dx = z n= Thus for Re(z) > we have n= (n + x) z dx is uniformly convergent for x [, 2] we have ( (n + ) z (n + 2) z ) = z R (x) = ζ(z, x) xz z b) For every z the integral formula (.3) for the function R f gives x z R (x) = xz z z + 2 x z + 2 (x 2 + t 2 ) z/2 sin(z arctan(t/x)) e 2πt dt since this last integral defines an entire function of z we see that the Hurwitz zeta function, previously defined by the sum + n= (n+x), can be continued z analytically for all z by ζ(z, x) = x z z + 2 x z + 2 (x 2 + t 2 ) z/2 sin(z arctan(t/x)) e 2πt dt And by analytic continuation the equation (.2) remains valid, thus for all z we have R (x) = ζ(z, x) (.2) xz z If we define for z the Riemann zeta function by ζ(z) = ζ(z, ) then we have n z = ζ(z) z (.22)

37 2 CHAPTER. RAMANUJAN SUMMATION We use the notation ζ R (z) = n z (.23) 4) If f(x) = x then the series + n= n+x is divergent and to get a solution of the difference equation R(x) R(x + ) = x we replace it by the series + n= ( n+x n+ ). Thus we have R f (x) = This last integral is 2 n= n= ( n + x 2 n + ) n= ( n + x n + )dx ( n + x + n + )dx = (Log(n + ) Log(n) n ) = γ n= where γ is the Euler constant. Thus and we get R (x) = + ( x n + x n + ) + γ n= n = γ (.24) 5) Another way to get (.24) is to use the the digamma function ψ = Γ /Γ that verifies ψ(x + ) ψ(x) = 2 x with ψ(x)dx = Thus we have Note that we have γ + ψ(x) = R = ψ(x) (.25) x n= ( n + n + x ) (.26) and ψ() = γ. By derivation we have for any integer k > k ψ(x + ) k ψ(x) = ( ) k k! x k+ with 2 k ψ(x)dx = k ψ(2) k ψ() = ( ) k (k )!

38 .4. THE SUMMATION 2 Thus we get R x k+ = ( )k k ψ k! k (.27) Note that for any integer k > we have R = ζ(k + ) n k+ k, thus we have k ψ() = ( ) k+ ζ(k + ) k! 6) Take f(x) = Log(x), where Log is the principal determination of the logarithm. Then the relation Γ(x + ) = xγ(x) gives thus Log Γ(x + ) Log Γ(x) = Log(x) R Log = Log Γ + 2 Log Γ(t)dt since it is known that 2 Log Γ(t)dt = + Log( 2π) cf. Srivastava and Choi) we get R Log (x) = Log(Γ(x)) + Log( 2π) (.28) and for x = we have Log(n) = Log( 2π).4.2 The fractional sum We will now explain how the Ramanujan summation is related to the function that Ramanujan writes ϕ(x) = f() f(x) Consider f O π and suppose we have a function ϕ analytic for Re(x) > a with < a < and of exponential type < π which satisfies ϕ(x) ϕ(x ) = f(x) with ϕ() = This gives ϕ(n) = f() f(n) for every positive integer n, thus the function ϕ is an interpolation function of the partial sums of the series f(n). If we set R(x) = ϕ(x ) + ϕ(x)dx then R is an analytic function for Re(x) > a + of exponential type < π, which satisfies R(x) R(x + ) = f(x) with 2 R(x)dx =, thus R = R f and we get ϕ(x) = ϕ(x)dx R f (x + )

39 22 CHAPTER. RAMANUJAN SUMMATION Since ϕ() = we have ϕ(x)dx = R f () = f(n) and the function ϕ is simply related to the function R f by ϕ(x) = R f () R f (x + ) Inversely this last equation defines a function ϕ analytic for Re(x) > a with < a < and of exponential type < π which satifies ϕ(x) ϕ(x ) = f(x) with ϕ() = We are thus led to the following definition: Definition 3 Let f O π, there is a unique function ϕ f analytic for Re(x) > a with < a < and of exponential type < π which satisfies ϕ f (x) ϕ f (x ) = f(x) with ϕ f () = We call this function the fractional sum of f, it is related to R f by ϕ f (x) = R f () R f (x + ) = f(n) R f (x) + f(x) (.29) and we have f(n) = ϕ f (x)dx (.3) Example If k is an integer then ϕ x k(x) = B k+ k + B k+(x) k + If k = we have ϕ (x) = B (x) + 2 = x. If k is an integer then For k = we have ϕ x k+ = ( )k k! + x k = B k+(x) B k+ k + k Ψ + + ζ(k + ) xk+ ϕ x (x) = Ψ(x) + x + γ + x k

40 .4. THE SUMMATION 23 More generally for z we have ϕ (x) = R (ζ(z, x) x z nz z ) + x z = ζ(z) ζ(z, x) + x z Integral expression of ϕ f The relation (.29) between ϕ f and R f gives by (.3) ϕ f (x) = f(n) + f(x) 2 + x f(t)dt i f(x + it) f(x it) e 2πt dt If we replace x by a positive integer n we get the classic Abel-Plana formula f() f(n) = f() 2 Note that the condition is sufficient to obtain + f(n) 2 lim n + + i + n f( + it) f( it) e 2πt dt f(t)dt i f(n + it) f(n it) e 2πt dt = [ f(n) f(n) = lim f() f(n) n + 2 f(n + it) f(n it) e 2πt dt n f(t)dt ] Examples ) Take the function f = x integers. We have p+xq where p and q are strictly positive p + nq = lim ( n + p + q p + nq q Log(nq)) + Log(p + q) q Thus the generalized Euler constant (cf. Lehmer) defined by γ(p, q) = ( lim n + p + p + q p + nq q Log(nq)) is related to the Ramanujan summation by γ(p, q) = p + nq + p Log(p + q) q

41 24 CHAPTER. RAMANUJAN SUMMATION 2) Take f(x) = Log(x) then ϕ Log (x) = Log(Γ(x + )) and by the integral expression of ϕ Log we have Log(Γ(x + )) = Log( 2π) + 2 Log(x) + xlog(x) x + 2 Remark In his Notebook Ramanujan uses for the function ϕ f the expression ϕ f (x) = n= arctan( t x ) e 2πt dt (f(n) f(n + x)) (.3) when this series is convergent. Note that if the series (f(n) f(n + x)) is convergent and if n= lim f(x) = x + then the relation N N (f(n) f(n + x)) (f(n) f(n + x )) = f(x) f(x + N) gives when N + n= ϕ f (x) ϕ f (x ) = f(x) Thus if we assume that the function defined by (.3) is analytic for Re(x) > a with < a < and of exponential type < π then it verifies the requirements of definition Relation to usual summation. Consider a function f O π, such that By the definition of R f we have lim f(n) = n + n R f () R f (n) = f(k). Thus the series f(n) is convergent if and only if R f (n) has a finite limit when n + and in this case k= f(n) = f(n) + lim R f (n) n +

42 .4. THE SUMMATION 25 Now let us see how this last limit is simply related to the integral of the function f. Recall the integral formula R f (n) = f(n) 2 and assume that n lim n + f(t)dt + i f(n + it) f(n it) e 2πt dt f(n + it) f(n it) e 2πt dt = Then the convergence of the integral f(t)dt is equivalent to the fact that R f (n) has a finite limit when n + and in this case lim R f (n) = n + Thus we have proved the following result : f(t)dt Theorem 2 Consider f O π such that and lim n + lim f(n) = n + f(n + it) f(n it) e 2πt dt = (.32) Then the series f(n) is convergent if and only if the integral f(t)dt is convergent and we have f(n) = f(n) f(x)dx (.33) From now on, we will say in this case that we are in a case of convergence. Example For z C\{, 2, 3,...} we have ( n n + z ) = ( n n + z ) and by (.26) we get R n = γ + ψ(z + ) n + z ( x x + z )dx ( x x + z )dx

43 26 CHAPTER. RAMANUJAN SUMMATION Thus for z C\{, 2, 3,...} we have = ψ(z + ) + Log(z + ) (.34) n + z If p and q are strictly positive integers this gives p + nq = q ψ(p q ) p + q Log(p q + ) By Gauss formula for ψ (c.f. Srivastava and Choi) we get for < p < q p + nq = q γ p + q Log(p + q) + π 2q cot(π p q ) q cos(2πk p q q )Log(2 sin(π k q )) k= Remark Take a function f O π such that the function g : x n= f(x + n) is well defined for Re(x) > and assume that g O π. Then we have Thus we deduce that If in addition we have then it follows that 2 n= R f (x) = g(x) g(x + ) = f(x) R f (x) = g(x) f(x + n)dx = n= 2 2 n= f(x + n) g(x)dx f(x + n)dx f(x)dx. (.35) Thus we see that in this case the function R f is simply related to the usual remainder of the convergent series f(n).

44 Chapter 2 Properties of the Ramanujan summation In this chapter we give some properties of the Ramanujan summation in comparison to the usual properties of the summation of convergent series. In the first section we begin with the shift property which replaces the usual translation property of convergent series. This has important consequences in the use of the Ramanujan summation, especially for the classical formula of summation by parts and of summation of a product. General functional relations for the fractional sum ϕ f are also deduced. In the second section we examine the relation between the Ramanujan summation and derivation, we see that the fractional sums ϕ f and ϕ f are simply related. We deduce some simple formulas for the evaluation of the Ramanujan summation of some series, which constitutes the content of theorems 5 and 6. In the third section we show that in the case of the Ramanujan summation of an entire function the sum R f(n) can be expanded in a convergent series involving the Bernoulli numbers. This very easily gives some formulas involving classical constant or trigonometric series. 2. Some elementary properties 2.. The unusual property of the shift. Let f O π and consider the function g defined by g(u) = f(u + ), we have by definition of R f R f (u + ) R f (u + 2) = f(u + ) = g(u) 27

45 28 CHAPTER 2. PROPERTIES OF THE RAMANUJAN SUMMATION we deduce that the fractional remainder function R g is given by Thus we have R g (u) = R f (u + ) 2 = R f (u) f(u) R f (u + )du 2 R g (u) = R f (u) f(u) + And for u = this gives the shift property f(n + ) = (R f (u) f(u))du 2 f(n) f() + 2 f(u)du f(x)dx (2.) Therefore we see that the Ramanujan summation does not satisfy the usual translation property of convergent series. More generally let f O π and for Re(x) let g(u) = f(u + x). Then we have R f (u + x) R f (u + x + ) = f(u + x) = g(u) we deduce that R g (u) = R f (u + x) thus taking u = in this equation we get x+2 x+ f(n + x) = R f (x + ) This last integral can be evaluated by using which gives x+ R f (v)dv x+2 x+ Therefore we have f(n+x) = R f (x+)+ x+ x+ R f (v)dv = x+2 R f (v)dv x+ R f (v + )dv = x+ R f (v)dv x+ f(v)dv f(v)dv = R f (x) f(x)+ f(v)dv x+ Since R f (x + ) = R f () ϕ f (x) we get the general shift property f(n + x) = f(n) ϕ f (x) + x+ f(v)dv (2.2) f(v)dv (2.3)

46 2.. SOME ELEMENTARY PROPERTIES 29 If m is a positive integer then the shift property is simply m f(n + m) = f(n) f(n) + n= m+ f(x)dx Remark Let us consider a function f such that x f(x ) is in O π, we define R n f(n) by f(n) = f(n ) n By the shift property applied to x f(x ) we have f(n) = f() + f(n) + n n p f(x)dx Examples ) Let f(x) = x and H m = m k= k, we have R n + m = m+ n H m + x dx = γ H m + Log(m + ) (which is a special case of (.34) since H m = ψ(m + ) + γ). As an application, consider a rational function g(x) = M m= c m x + m such that M c m = m= then we are in a case of convergence for the series g(n) and we have Note that and m= M n= g(n) = M c m n + m = M m= M m= m= c A m x + m dx = lim A + c + m n + m + M c m n + m = M m= m= M m= c m H m + c m x + m dx M c m Log(m + ) m= c M m x + m dx = c m Log( + m) m=

47 3 CHAPTER 2. PROPERTIES OF THE RAMANUJAN SUMMATION Thus we get n= g(n) = M c m H m m= By summation by parts we obtain the classical result n= g(n) = c + c + c c c m m 2) For f(x) = Log(x) we get by (2.2) Log(n + x) = Log(Γ(x + )) + Log( 2π) + (x + )Log(x + ) x Let s have t >, since then we get arctan( t n ) = (Log(n + it) Log(n it)) 2i arctan( t n ) = R 2i ( Log(n + it) Log(n it)) With t = we get arctan(n) = = + it) Log(Γ( 2i Γ( it) ) + tlog( t 2 + ) + arctan(t) t R π 2 arctan( n ) = + i) Log(Γ( 2i Γ( i) ) + Log( 2) + π 2 Note also that we have arctan( t n ) t n = O( n ), thus we are in a case of 2 convergence and by (.33) we deduce that n= (arctan( t n ) t R n ) = arctan( t n ) γt + (arctan( t x ) t x )dx thus we get n= (arctan( t n ) t n ) = 2i Log(Γ( + it) Γ( it) ) γt