How to recover an L-series from its values at almost all positive integers. Some remarks on a formula of Ramanujan

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1 Proc. Indian Acad. Sci. (Math. Sci.), Vol., No. 2, May 2, pp. 2±32. # Printed in India How to recover an L-series from its values at almost all positive integers. Some remarks on a formula of Ramanujan CHRISTOPHER DENINGER WWU, Mathematisches Institut, Einsteinstrasse 62, D-4849 Munster, Germany MS received 25 August 999; revised 5 January 2 Abstract. We define a class of analytic functions which can be obtained from their values at almost all positive integers by a canonical interpolation procedure. All the usual L-functions belong to this class which is interesting in view of the extensive investigations of special values of motivic L-series. A number of classical contour integral formulas appear as particular cases of the interpolation scheme. The paper is based on a formula of Ramanujan and results of Hardy. An approach to the problem via distributions is also presented. Keywords. Interpolation formulas; analytic functions; contour integrals; special values; L-functions. Introduction The purpose of this note is to answer a question, Maur asked me: Is there an interpolation scheme allowing to recover a complex L-series from its values at almost all positive integers? This is interesting for example in view of the extensive investigations of special values of motivic L-series in the last decades, culminating in the Bloch±Kato conjectures [BK]. Note that p-adic L-functions are determined by their values at these points because the set in question is dense in Z p. It turns out that modifying one of Ramanujan's favourite formulas one gets a satisfactory interpolation procedure for a class of analytic functions which in particular comprises all Dirichlet series and hence all L-series. Incidentally the classical representations of certain eta- and L-functions as contour integrals are particular instances of the interpolation scheme. Ramanujan did not specify exactly to which functions his formula applied. A useful class F H was singled out however by Hardy in his commentary on Ramanujan's work [H], ch. XI. We introduce a universal class F of interpoliable functions which is essentially canonical. Hardy's result then implies that F H F. Apart from the general setup and a number of examples we also give a short distribution theoretic proof of a special case of Hardy's result. This uses Schwart' extension of the Paley±Wiener theorem to distributions with compact support. It should be emphasied that this note is essentially a commentary on one aspect of the work of Hardy and Ramanujan. 2. Preliminaries To a sequence of complex numbers a ˆ a, 2 Z we associate the Laurent series: 2

2 22 Christopher Deninger ˆ a x :ˆ X a x : Extend a to a sequence indexed by the integers by setting a ˆ for <. We call `good' if the following conditions are satisfied: converges in a punctured neighborhood of the origin; has a holomorphic continuation to U where U is a neighborhood of ; Š and U ˆ U n fg: 2 For some 2 R we have j j ˆ O jj as jj! in U : 3 If is good, the contour integral I s :ˆ Z s d defines a holomorphic function of s in Re s > for any as in condition (3). Here the integration is along any path within U n ; Š starting at, encircling the origin counterclockwise and returning to. The power s is defined via log ˆ log jj i Arg where < Arg. Note that I does not depend on the choice of the path. Lemma.. In the situation of (4) we have: I ˆ a for > : Proof. For s ˆ > since is single valued the integral reduces to I ˆ I d ˆ Res ˆ d ˆ a jjˆ" 4 by taking " sufficiently small. We need another consequence of the residue theorem: Lemma.2. A rational function of degree d with no poles on ; is good with ˆ d and we have: I s ˆ X Res a s d in Re s > d: a2cn ;Š Finally we require for later use. Lemma.3. Assume that ˆ P a x is good with some < in condition (3). Then we have I s ˆ M x s for < Re s < :

3 Here Remarks on a formula of Ramanujan 23 MF s ˆ x s F x dx x is the Mellin transform on R. Proof. For Re s > and every " > small enough we have: Z " I s ˆ I ˆ s log jxj i e x s jjˆ" x s " d ; dx x x dx x I Z " s jjˆ" Since j j ˆ O jj as jj! we have the estimate I jjˆ" c" Re s and hence lim ˆ "! Ijjˆ" s log jxj i e x d : dx x for Re s <. Hence the formula. Remark. The theory of the Mellin transform is well developed. In [I], Theorem 3. for example two function spaces are defined which are in bijection via the Mellin transform. Together with Lemma.3, Igusa's result leads to information about the interpolation functional I. Unfortunately the class of functions to which we want to apply I in the next sections is quite different from the one that can be treated in this way. 3. Interpolation We can now set up the interpolation scheme. Consider the C-algebra: A :ˆ fsequences a defined from some onwardsg= where a ˆ a a ˆ a iff a ˆ a for all. If sequences a; a are equivalent, then a is good iff a is good. Hence we can define: A :ˆ f aš 2 A j a is good g: On the other hand let F be the C-vector space of holomorphic functions defined on some half plane Re s > s.t. x ˆ P> x is good. Here the summation is over all integers >. By the principle of analytic continuation we may identify functions in F if they agree for Re s. Theorem 2.. I defines a linear `interpolation' map I : A!F via I aš ˆ I a : It has the `special-values map' S : F!A; S ˆ Š

4 24 Christopher Deninger as a left-inverse: S I ˆ id: Proof. For any sequence a ˆ a such that a is good the function s ˆ I a s is holomorphic in some half plane Re s >. By Lemma. it has the property that > a. Hence x ˆ P> x is good as well and thus 2 F. If a a then a a 2 C x; x Š. By Lemma.2 we therefore have I a ˆ I a in F. Thus the interpolation map is well defined. As we have seen I a > a and hence S I ˆ id on A. We now define: F :ˆ Im I F : Then I and S define mutually inverse C-linear isomorphisms I A! F: 5 S This is clear since I was injective having a left-inverse and we have made it surjective. By construction the functions in F have the property that they are uniquely determined by their values on any set of integers of the form fj g. Moreover given these values for there is an explicit formula for the function, valid in some half plane Re s >. Note that we have a canonical projector: P ˆ I S : F!F; P 2 ˆ P: In these terms we have: PROPOSITION 2.2 () For A; A 2 A form A A 2 A. If A A 2 A, then I A A ˆ P I A I A in F : (2) S and I are equivariant with respect to the Z-action by shift. Proof. () If A A 2 A then I A I A 2 F since I A I A ˆ I A I A ˆ a a for where a ; a are representatives of A; A. Hence S I A I A ˆ A A. Applying I gives the assertion. (2) Shift by one acts on A by T a Š ˆ a Š. The corresponding is x a which is again good. Hence the shift acts on A and by a similar argument also on F. The rest is clear. Remark. There is a convolution product for sequences but it does not pass to A. Before we incorporate the Hardy±Ramanujan theory into the picture let us give some examples. For a sequence a with aš in A we set I a :ˆ I aš. Example 2.3. For 2 C consider a ˆ. Then if =2 ; the class of a is in A and I a ˆ s where arg 2 ; Š. In particular s ˆ s 2 F. The functions s defined using different normaliations of arg lie in F and are mapped via P to the principal one. 6

5 Remarks on a formula of Ramanujan 25 Proof. For =2 ; the function x ˆ a x ˆ X ˆ x ˆ x ; jxj < is good in our sense. By Lemma.2 we have I s ˆ Re s ˆ s d ˆ s : 7 Example 2.4. a ˆ =! defines a class in A and I a ˆ s 2 F. Proof. a x ˆ e x is clearly good and I a s ˆ Z s e d ˆ s is Hankel's representation of the inverse -function. We can also argue as follows: Since a x ˆ e x is good for any 2 R, lemma.3 shows that I a s ˆ M e x s for Re s < : Now by its definition s equals the Mellin transform of e x so that I a s ˆ s ˆ s first in Re s < and then for all s by analytic continuation. Example 2.5. We want to interpolate the values B = of the eta-function at the negative integers. Since they grow so quickly that has radius of convergence ero we renormalie them as follows: B =! for. We expect them to be interpolated by the function s = s and this is indeed the case. More generally consider the sequence: B a =! for < a where B n a is the nth Bernoulli polynomial. Its -function is ˆ X ˆ B a! ˆ X ˆ It is good and j j ˆ O jj for any 2 R. We have Z s e a B a! ˆ ea e : 8 I s ˆ d s; a e ˆ 9 s P by a standard formula from the theory of the Hurwit eta function s; a ˆ ˆ a s c.f. [EMOT].. Thus s;a s 2F is the interpolation of its values B a! for any. It follows that L ; s s 2 F is the interpolation of its values at the integers for any as well.

6 26 Christopher Deninger 4. Invoking the Hardy, Ramanujan theory. Further examples The problem is of course to give good criteria as to when an analytic function defined in some right half plane belongs to F. For this we take up ideas of Hardy. We first require a formula of Hardy, [H], (.4.4) whose proof is omitted in [H]. For the convenience of the reader we give a proof below. Actually, in the following proposition, we show a slightly stronger result since this requires no extra effort and may be useful for extending the theory. PROPOSITION 3. Assume that is holomorphic in Re s > and satisfies an estimate of the form: j it j f t e P jtj for > where P 2 R and f 2 L R is such that lim t! f t ˆ. Fix an integer > and choose r > such that < r <. Then for any real e P < x < we have the integral representation: Z r i x ˆ X x ˆ r i s x s ds: Here the series is absolutely convergent and the integral is in the Lebesgue sense. Proof. Consider the contour C ˆ C C 2 C 3 C 4 : where L 2 2 Z. By the residue theorem: X x ˆ Z s x s dx: <L C We have e jim sj for jim sj : Using periodicity of sin we get that for R large enough c e jim sj holds on C for all L: Hence Z c L r e R e max Pr;PL R max x r ; x L f R : C 2

7 Remarks on a formula of Ramanujan 27 R Thus for fixed L, we have lim R! C 2 ˆ. Similarly lim R! RC 4 ˆ. Next Z L i c 2 e jtj e PL e jtj f t x L dt c 3 e L P log x f t dt: L i Hence the integral exists and tends to ero for L! by our assumption e P < x < i.e. P log x <. Similarly the integral from r i to r i exists. Hence the formula. One now uses the integral representation for of the proposition to show that which a priori is holomorphic only in < jj < e P extends to a holomorphic function in some punctured neighborhood U as in (2) above which is bounded by a power of jj as in (3). More can be done but let us stay with a class of functions introduced by Hardy. For A < set: ( ) 's analytic in Re s > for some 2 R such that there F H A ˆ : exists P 2 R with j it j e P Ajtj in Re s > Any such is called allowable for. Set F H ˆ SA< F H A. Then we have the following result which follows from the preceeding considerations and those in [H],.4: Theorem 3.2. F H F. More precisely, if is allowable for 2 F H let be the analytic continuation of P > x to a punctured neighborhood U of ; Š. Then we have j j ˆ O jj as jj! in U for every > and the interpolation formula I s ˆ therefore holds in Re s >. Z s d ˆ s Remark. The example of s ˆ shows that the condition A < is not unnatural. Proof. By assumption j it j e P Ajtj in > for some P 2 R; A <. Hence Proposition 3. is applicable. Let be the least integer > and choose r > such that < r <. Then by (3.) we have for any e P < < : ˆ Z r i r i s s ds: Choose < < A. Then for < arg < we have: j s j ˆ jj e t arg jj e jtj : Thus Z r i r i c e jtj e Pr Ajtj jj r e jtj dt c 2 jj r e A jtj dt ˆ O jj r :

8 28 Christopher Deninger Since we know that the series for converges in < jj < e P it follows that extends to an analytic function in some region U as in (2) where it satisfies ˆ O jj r as jj! for any r >. Thus is good and hence 2 F. Moreover I s defines a holomorphic function in Re s >. It remains to prove that I ˆ. Unfortunately this cannot be checked by substituting the above integral representation for into the contour integral I since the former does not converge for the s on the loop around ero. Instead we reduce the claim to a formula of Hardy and Ramanujan ± the last equality in [H],.4± which itself is an application of Mellin- or Fourier-inversion: Formula of Hardy±Ramanujan. For < <, let H be holomorphic in Re s and satisfy the estimate H s e P Ajtj there for some P and A <. Setting H x ˆ X H x ˆ we have that x w H x dx x ˆ sin w H w for < Re w < : By Lemma.3 we have for < Re s < : I s ˆ x s x dx x : Now choose < < so that in particular <. Set H s ˆ s. Then the Hardy±Ramanujan formula applied to H s ˆ s gives the equality: x w x dx x ˆ sin w w : Thus for < Re s < we find that x s x dx x ˆ s : Together with the above formula for I s it follows by analytic continuation that I s ˆ s for Re s > as claimed. Remark 3.3. Our interpolation functional I has two advantages over the one of Hardy± Ramanujan: I HR : 7! Z x s x dx x which requires convergence at and whereas I needs convergence at only. As a consequence interpolation formulas involving I HR are valid at most in some region < Re s < whereas those using I hold in a half plane Re s >. Moreover only in I is it possible to add to an arbitrary Laurent polynomial without changing its value. This is crucial for interpolating elements of A i.e. sequences which are only given up to equivalence.

9 In the rest of this section we use distributions to give a different and more conceptual proof of the assertion 2 F in Theorem 3.2 for a restricted class of functions : Let be an entire function which satisfies an estimate of the form j s j jsj N AjIm sj e in C for some A <. By Schwart' extension of the Paley±Wiener theorem to distributions [Y], VI.4 the function is the Fourier±Laplace transform of a distribution T with compact support in ;. Choose some " > such that supp T is disjoint from the set C " of y in R with je iy j < ". Let be a smooth function on R which is on C " and equal to on supp T. We have: Hence Remarks on a formula of Ramanujan 29 s ˆ ^T s ˆ 2 =2 ht y ; e isy i: x :ˆ X ˆ for jxj <. The formula x ˆ 2 =2 ht y ; xe iy i ˆ 2 =2 ht y ; y e iy i gives the analytic continuation of to a neighborhood of ; Š. Since jt h j C X sup jd h y j jyjl jjn for some constants C; N; L and all smooth functions h on R it follows that j j ˆ O jj as! : Hence is good and thus 2 F. Now Z I s ˆ s 2 =2 ht y ; y e iy i d * Z + ˆ 2 =2 T y ; s y e iy d 2:3 ˆ 2 =2 ht y ; e isy i ˆ s : Since supp T ;. Hence 2 F. If more generally T has compact support in R n Z then is still good by the identical argument, so that 2 F. However we now have, again using (2.3), that: P ˆ I s ˆ 2 =2 ht y ; y e isy i; where y 2 ; Š is such that y y mod Z. Note that e isy is not smooth but y e isy is. Writing T as a finite sum T ˆ X T

10 3 Christopher Deninger of distributions T with compact support in ; it follows that ˆ P where ˆ ^T; ˆ ^T. By () we see that P ˆ e is and hence P ˆ X e is 2 F: Incidentially this is also a consequence of Theorem 3.2 applied to e is. 5. Applications In this section we illustrate the preceeding theory by interpolating certain interesting classes of functions. We are mostly interested in L-series and their completed versions by -factors. Set F H 8 < ˆ : 's analytic in Re s > for some 2 R s.t. for every > there exist a and some P 2 R with j it j e P jtj in Re s > 9 = ; and ( ) F H ˆ 's analytic in Re s > for some `associated' s.t. j it j e P : in Re s > for some P 2 R Clearly F H F H F H. Moreover F H and F H are C-algebras and F H is a module under them. Note that if f 2 F H and 2 F H have f and associated to them, then max f ; is associated to f. Clearly every Dirichlet series P a n s n with n > and abscissa of absolute convergence < belongs to F H with being admissible. In particular L-series and their inverses belong to F H. On the other hand L-series completed by -factors do not even belong to F since the associated power series has radius of convergence ero. The reciprocal function however has a better behaviour if the -factor is simple. To see this we require the following fact: PROPOSITION 4. For every < a < 2; b 2 R the function as b belongs to F H with associated ˆ a 2 b. Proof. For given > the complex Stirling asymptotics for s implies that s ˆ e s e =2 log s 2 =2 O s in j arg sj as jsj!. Hence this estimate holds for all s with Re s 2 ; jsj >. Thus we also have and hence: s ˆ e s e =2 s log s 2 =2 O s in Re s 2 ; jsj > j s j e e =2 log jsj e jtj=2 e jtj=2

11 Remarks on a formula of Ramanujan 3 in Re s =2; jsj > and hence in Re s =2. Thus j as b j e a jtja=2 for Re s a 2 b : Examples. () It follows again that s 2 F. (2) Since 2s 2 2 F H with ˆ =2 (abscissa of absolute convergence) and since s 2 F H with ˆ =2 by the proposition we find that s 2s 2 2 F H with ˆ =2. Hence theorem 3.2 gives us: Z X s ˆ d! 2 2 ˆ s 2s 2 Note that the series in the integral converges everywhere. Setting ^ s ˆ s=2 s s 2 we get similarly that ^ 2s 2 2 F H with ˆ =2 and that: for Re s > 2 : Z X s ˆ d ^ 2 2 ˆ Similarly ^ s 2 F H with ˆ and hence: ^ 2s 2 in Re s > 2 : Z X s ˆ2 d ^ ˆ ^ s in Re s > : (3) A similar formula holds for the completed L-series ^L E; s ˆ L E; s 2 s s of an elliptic curve E over Q: Z X s ˆ2 d ^L E; ˆ ^L E; s in Re s > 3 2 : (4) For s ˆ 2s 2 F H F and ˆ =2 the corresponding function in Theorem p 3.2 is given by ˆ f where f is the even function f w ˆ w e w w e w 2 e w e w : After some calculation which we leave to the reader the formula of theorem 3.2 leads to the functional equation of s. This example was suggested by the discussion of Ramanujan's formula in [E],.. Remark. A variant of the first formula was first given by Ries as mentioned by Hardy: X s x ˆ valid for =2 < Re s <. x dx! 2 2 x ˆ s 2s 2

12 32 Christopher Deninger The case 2s b is not covered by the proposition. We close by noting that a direct computation gives: Fact s n 2 F for all n 2 Z. Proof. Since F is shift-invariant we may restrict to 2s. The associated function is x ˆ P ˆ x = 2! which is entire. We have w 2 ˆ 2 ew e w. The mapping w7!w 2 transforms any strip Re w into a neighborhood U of ; Š. In Re w the function 2 ew e w is bounded and hence is bounded in U. Thus 2s 2 F. For Re s > we have: Z s d ˆ ˆ Z Z s d s s Š i i s d : Using Lemma (.3) we see that for < Re s < this equals x s x dx ˆ 2 x 2s cos x dx x x Z ˆ x 2s sin x dx s by integration by parts. Substituting the formula in [EMOT],.5. (38) x sin x dx ˆ sin in < Re < : 2 We arrive after some calculation at the desired formula: Z s Acknowledgements d ˆ 2s : I would like to thank B Maur for his question and the Harvard mathematics department for its hospitality. I would also like to thank the referee for suggestions to improve the exposition. References [BK] Bloch S and Kato K, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift, vol., Prog. Math. 86 (99) 333±4 [E] Edwards H M, Riemann's eta function (Academic Press) (974) [EMOT] Erdelyi A et al, Higher transcendental functions. The Bateman Manuscript Project (McGraw-Hill) (953) vol. [H] Hardy G H and Ramanujan S, Twelve Lectures on Subjects Suggested by His Life and Work (Chelsea) (978) [I] Igusa J-I, Lectures on forms of higher degree. (Bombay: Tata Institute of Fundamental research) (978) [Y] Yosida K, Grundlehren Bd. 23, (Springer: Functional Analysis) (97)