A p-adic analogue of a formula of Ramanujan

Transcription

1 A -adic analogue of a formula of Ramanuan Dermot McCarthy and Robert Osburn Abstract. During his lifetime, Ramanuan rovided many formulae relating binomial sums to secial values of the Gamma function. Based on numerical comutations, Van Hamme recently conectured -adic analogues to such formulae. Using a combination of ordinary and Gaussian hyergeometric series, we rove one of these conectures. Mathematics Subect Classification 000. Primary: C0; Secondary: S80. Keywords. Gaussian hyergeometric series, Suercongruences.. Introduction In Ramanuan s second letter to Hardy dated February 7, 9, the following formula aears: where is the Gamma function. This result was roved in 94 by Hardy [4 and a further roof was given by Watson [9 in 9. Note that. can be exressed as Other formulae of this tye include π 4,. which is Entry 0, age 5 of [5. It is interesting to note that a roof of. was not found until 987 [8.

2 Dermot McCarthy and Robert Osburn Recently, Van Hamme [7 studied a -adic analogue of.. Namely, he truncated the left-hand side and relaced the Gamma function with the -adic Gamma function. Based on numerical comutations, he osed the following. Conecture.. Let be an odd rime. Then { 4 4 mod if mod 4 where is the -adic Gamma function. The urose of this aer is to rove the following. Theorem.. Conecture. is true. 0 mod if mod 4 Theorem. is one examle of a general hemonena called Suercongruences. This term first aeared in the Ph.D. thesis of Coster [9 and refers to the fact that a congruence holds modulo for some. Other examles of suercongruences have been observed in the context of number theory see [ and the references therein, mathematical hysics [7, and algebraic geometry [6. Van Hamme states other conectures relating truncated hyergeometric series to values of the -adic Gamma function. Motivated by Theorem., one of these conectures has been settled in [0. The remaining include a conectural -adic analogue of. which states mod 4. These conectures were motivated exerimentally and as van Hamme states that we have no real exlanation for our observations, it might be worthwhile to determine whether these congruences arise from considering some aroriate algebraic surfaces see [7 or [5. Finally, if mod 4, then the congruence in Conecture. aears to hold modulo 4. This has been numerically verified for all rimes less than The aer is organized as follows. In Section we recall some roerties of the Gamma function, ordinary hyergeometric series, the -adic Gamma function and Gaussian hyergeometric series. The roof of Theorem. is then given in Section.. Preliminaries We briefly discuss some reliminaries which we will need in Section. For further details see [, [6, or [8. Recall that for all comlex numbers x 0,,,..., the Gamma function x is defined by x : lim! x x

3 A -adic analogue of a formula of Ramanuan where a 0 : and a n : aa + a + a + n for ositive integers n. The Gamma function satisfies the reflection formula x x π sin πx.. We also recall that the hyergeometric series F q is defined by [ a, a, a,..., a F q z : b, b,..., b q n0 a n a n a n a n z n b n b n b q n n!. where a i, b i and z are comlex numbers, with none of the b i being negative integers or zero, and, and q are ositive integers. Note that the series terminates if some a is a negative integer. In [0, While studied roerties of well-oised series where q +, z ±, and a + a + b a + b a + b q. One such transformation roerty of the well-oised series see 6., age 5 in [0 is [ a, + a, c, d, e, f 6F 5 a, + a c, + a d, + a e, + a f [ + a e + a f + a c d, e, f + a + a e f F + a c, + a d.. This is Entry, Chater 0 in Ramanuan s second noteboo see age 4 of [4. Watson s roof of. is a secialization of. combined with Dixon s theorem [0. Let be an odd rime. For n N, we define the -adic Gamma function as and extend to all x Z by setting n : n <n x : lim n x n where n runs through any sequence of ositive integers -adically aroaching x and 0 :. This limit exists, is indeendent of how n aroaches x and determines a continuous function on Z. In [, Greene introduced the notion of general hyergeometric series over finite fields or Gaussian hyergeometric series. These series are analogous to classical hyergeometric series and have layed an imortant role in relation to the number of oints over F of Calabi-Yau threefolds [, traces of Hece oerators [, formulas for Ramanuan s τ-function [4, and the number of oints on a family of ellitic curves [. We now introduce two definitions. Let F denote the finite field with elements. We extend the domain of all characters χ of F to F by defining χ0 : 0.

4 4 Dermot McCarthy and Robert Osburn The first definition is the finite field analogue of the binomial coefficient. For characters A and B of F, define by A B A B : B JA, B where Jχ, λ denotes the Jacobi sum for χ and λ characters of F. The second definition is the finite field analogue of ordinary hyergeometric series. For characters A 0, A,..., A n and B,..., B n of F and x F, define the Gaussian hyergeometric series by n+f n A0, A,..., A n B,..., B n x : A0 χ χ χ A χ B χ An χ χx B n χ where the summation is over all characters χ on F. In [, the case where A i φ, the quadratic character, for all i and B ɛ, the trivial character mod, for all is examined and is denoted n+ F n x for brevity. By [, n n+f n x Z. Before stating the main result of [, we recall that for i, n N, generalized harmonic sums, H n i, are defined by H i n : and H i 0 : 0. For an odd rime, λ F, n Z +, we now define the quantities X, λ, n : φ λ H + n + 0 H + H + n i l l [ l λ + n + n + H H + H +,.4 and Y, λ, n : φ λ 0 + l l [ l λ + n + H Z, λ, n : φ λ 0 n + H H + H +,.5 l 6 l λ,.6 where l n+. The main result in [ rovides an exression for n+f n modulo. Precisely, we have

5 A -adic analogue of a formula of Ramanuan 5 Theorem.. Let be an odd rime, λ F, and n be an integer. Then n n+f n λ φ n+ [ X, λ, n + Y, λ, n + Z, λ, n mod.. Proof of Theorem. Proof of Theorem.. By Theorem 4 in [ or Proosition 4. in [9 and Corollary 5 in [7, we have that F Thus, by Theorem. it suffices to rove { 4 4 mod if mod 4 0 mod if mod φ [ X,, + Y,, + Z,, mod 0. where the quantities X, λ, n, Y, λ, n and Z, λ, n are defined by.4,.5 and.6 resectively. We first show, via the following lemmas, that the terms involving Y,, and X,, in. vanish modulo. Lemma.. Let be an odd rime. Then Y,, 0 mod. Proof. Substituting λ and n in equation.5, we get Y,, 0 + [ + H H H + H +. Noting that u+ + u, we get + mod..

6 6 Dermot McCarthy and Robert Osburn Also, H H r r So we need only show 0 For, note that As gcd r0 r0 4 r + 0 mod. [ + H H mod.. + mod..4!!,, it now suffices to show! + + [ + H H + 0 mod..5 We now use an argument similar to that in Section 4 of [6 see also [9. Let P z : d dz [ P z z + zz + for some integers a. By a comutation, we have [ + z H Combining this with.5, it is enough to show that 0 a z.6 H +z z.! + P 0 mod..7 Note that, for < <, + is divisible by and H i Hi + Z i, so that P 0 mod for such. Hence.7 will hold if we can show! P 0 mod..8 +

7 A -adic analogue of a formula of Ramanuan 7 We now recall the following elementary fact about exonential sums. For a ositive integer, we have { mod if,.9 0 mod otherwise. By.6,.9 and the fact that <, we see that Additionally, by.6 As z + One can also chec that Thus P z a a a 0 a mod. + a z +. has integer coefficients, divides a. Hence a 0 mod. a 0!. P! and.8 holds. This roves the result. mod Now we would lie to show that ord X,, 0 which ensures that the term involving X,, in equation. vanishes modulo. In fact, in the following lemma, we show that ord X,,. Lemma.. Let be an odd rime. Then X,, 0 mod. Proof. Substituting λ and n in equation.4 and alying. and. yields X,, 0 [ + 9 H H + H H + H H + mod.

8 8 Dermot McCarthy and Robert Osburn By.4 and as gcd + [!,, it suffices to rove that H H H H Similar to the roof of Lemma., we now let H + Qz : z d [ dz zz + H + 0 for some integers a. One can chec that it now suffices to show 0 mod..0 a z. Q 0 mod.. By.9,. and the fact that <, we have Q 0 0 a a a mod. Here we have used that a 0 0 as z Qz. One can chec that z + + a z +. As z + has integer coefficients, divides a. Hence a 0 mod. Thus. holds and the result is roven Via., Lemmas. and., the roof of Theorem. is comlete uon roving the following Proosition. Proosition.. Let be an odd rime. Then φ Z,, mod. 0 Proof. Substituting λ and n in equation.6, we get Z,, 0 6..

9 A -adic analogue of a formula of Ramanuan 9 Noting that, it suffices to rove φ 0 0 mod..4 Letting a, c + i, d i, e + and f [ 6F, 5 4, + i 5 i, +, 4, i, + i,, + [ + F +, i, + i in., we get..5 By. and the fact that [ 6F, 5 4, + i 5 0 is a negative integer, i, +, 4, i, + i,, i i + i + i +!..6 Now,!, ,.8 and + i i i + i mod 4..9

10 0 Dermot McCarthy and Robert Osburn Therefore, substituting.7,.8 and.9 into equation.6, we get [ 6F, 5 4, + i, i, +, 5 4, i, + i,, mod 4..0 Next we examine the right hand side of.5. By., [ + F +, i, + i i + i 0!.. Now, via. and the fact that x + xx and π, we have +. sin π φ. Also, we have + i + i mod.. Using.7 and substituting.,. into., we get + [ F +, i, + i φ mod..4 Finally, combining.5,.0 and.4 yields.4 and hence the result follows. Remar.4. We would lie to mention another aroach, indly ointed out to us by Eric Mortenson, which confirms Theorem.. By [7, the right hand side in Conecture. is equal to a where a is the -th Fourier coefficient of 0

11 A -adic analogue of a formula of Ramanuan η 6 4z. Here ηz is the Dedeind eta-function. Thus, in conunction with.4, Conecture. follows from φ a mod. 0 This congruence has been roven in [, [5, [9, and [8. Acnowledgements The first author would lie to than the UCD Ad Astra Research Scholarshi rogramme for its financial suort. The second author thans the Institut des Hautes Études Scientifiques for their hositality and suort during the rearation of this aer. References [ S. Ahlgren, Gaussian hyergeometric series and combinatorial congruences, Symbolic comutation, number theory, secial functions, hysics and combinatorics Gainesville, Fl, 999,, Dev. Math., 4, Kluwer, Dordrecht, 00. [ S. Ahlgren, K. Ono, A Gaussian hyergeometric series evaluation and Aéry number congruences, J. Reine Angew. Math , 87. [ G. Andrews, R. Asey, R. Roy, Secial functions, Encycloedia of Mathematics and its Alications, 7, Cambridge University Press, Cambridge, 999. [4 B. Berndt, Ramanuan s noteboos. Part II, Sringer-Verlag, NewYor, 989. [5 B. Berndt, Ramanuan s noteboos. Part IV, Sringer-Verlag, New Yor, 994. [6 B. Berndt, R. Evans, K. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monograhs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New Yor, 998. [7 F. Beuers, C. Peters, A family of K surfaces and ζ, J. Reine Angew. Math , [8 J. Borwein, P. Borwein, Pi and the AGM. A study in analytic number theory and comutational comlexity, Canadian Mathematical Society Series of Monograhs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New Yor, 987. [9 M. Coster, Suercongruences, Ph.D. thesis, Universiteit Leiden, 988. [0 A. Dixon, Summation of a certain series Proc. London Math. Soc. 5 90, [ S. Frechette, K. Ono, and M. Paaniolas, Gaussian hyergeometric functions and traces of Hece oerators, Int. Math. Res. Not. 004, no. 60, 6. [ J. Fuselier, Hyergeometric functions over finite fields and relations to modular forms and ellitic curves, Ph.D. thesis, Texas A&M University, 007.

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13 A -adic analogue of a formula of Ramanuan Dermot McCarthy and Robert Osburn School of Mathematical Sciences University College Dublin Belfield Dublin 4 Ireland dermot.mc-carthy@ucdconnect.ie robert.osburn@ucd.ie