Integral Expression of Dirichlet L-Series

Transcription

1 International Journal of Algebra, Vol,, no 6, Integral Expression of Dirichlet L-Series Leila Benferhat Université des Sciences et de la technologie Houari Boumedienne, USTHB BP 3 El Alia ALGER (ALGERIE) lbenferhat@hotmailcom Abstract In this paper, we study the Mahler measure of a family of reciprocal polynomials depending of a parameter l These polynomials do not vanish on the torus and define curves of genus except for the singular values of the parameter Using Mahler measure, we prove that Picard- Fuchs equation associated to this family admits explicit solutions at ± and express certain Dirichlet L-series L (χ, ) as integrals of functions related to F (,, ; z) Mathematics Subject Classification: C8, F66, F66 eywords: Mahler measure, Dirichlet L series Introduction In a previous paper Villegas [] proved the formula where = ( )dk k = log d, () dx ( x )( k x ) and d := 3/ L(χ, ) = L (χ, ) = G The constant G is the Catalan s constant G = L(χ, ) =

2 78 L Benferhat The formula dk = d () can be proved similarly The formulae () and () are also mentioned without demonstration in the book of Byrd and Friedman [3] There is also another proof of the second one in Borwein s book [] In a recent paper [], I proved the formulae and L (χ 8, ) = log( + ) + log L (χ 8, ) = ( u + )du k k dk In this article, we study the logarithmic Mahler measure of a family of elliptic curves given by reciprocal polynomials depending of a parameter l : P l (x, y) =y (x +) + y(x +(l +)x +)+(x +), l R We prove that the derivative of the Mahler measure of P l (x, y) with respect to the parameter is a period of the curve An integration with respect to k, the knowledge of the Mahler measure for the singular values of the parameter and a passage to the limit allows to get the formulae : L (χ, ) = 3 u +3 du L (χ, ) = 3 log(3) 3 L (χ 3, ) = 3 3 k 3k dk 3t + dt L (χ 3, ) = 6 5 k k dk + 3 log( + 3)

3 Integral expression of Dirichlet L-series 79 Definitions and nown results If P denotes a Laurent polynomial in n variables, P C[X ±,,X± n ] and T n = {(x,,x n ) C n ; x = = x n =} is the n torus, then the logarithmic Mahler measure is defined by m(p ):= (i) n and the Mahler measure M(P )by log P (x,,x n,,, ) T x n x n M(P ) := exp(m(p )) dx x dx n x n If n = and P (x) =a d j= (x α j), applying Jensen s formula, we get ie log P (exp(it) dt = log a + log M(P )= a d (max( α j, ), j= d (max( α j, ) j= We recall the formula deduced from the functional equation of Dirichlet s series, L (χ f, ) = f 3/ L(χ f, ), ( ) f where χ f (n) = is the real odd Dirichlet character with conductor f n Definition We define the hypergeometric series by where It verifies F (z) =F (, ), ; z =+ (k) = ( ) z + (k) = F (k ) ( x )( k x ) dx ( ) 3 z +

4 8 L Benferhat 3 Proof of the main theorem Theorem 3 Consider the family P l (x, y) =y (x +) + y(x +(l +)x +)+(x +), l R In what follows, we use the notations m(p l )=m(l) and m (l) = d dl m(l) If l>, then m (l) = l +l l + is a solution of Picard-Fuchs differential equation of the family P l (x, y) in a vicinity of + If l>, then 3 We have m (l) = 8 k ( l k 3k dk log + log ) ++ l m() = log(3) 8 k 3k dk If l<, then m (l) = l l l is a solution of Picard-Fuchs differential equation of the family P l (x, y) in a vicinity of 5 If l<, then 6 We have m (l) = 8 k m( ) = 8 l + l k dk + log k k k dk + log( + 3)

5 Integral expression of Dirichlet L-series 8 7 If <l<, then 8 We have m (l) = l m() = l + u +3 du 9 If <l<, then m (l) = l l We have m( ) = 3 3t + dt Proof The Mahler measure of P l is given by m(l) = log (i) (y + y )x +(y + l + y )x + y + y x = y = We set y = exp(is); so with f (s) = i x = m(l) = f(s)ds dx dy x y log x ( cos s +)+x ( cos s + l + ) + ( cos s +) dx x and P l,s (x) =x ( cos s +)+x ( cos s + l + ) + ( cos s +) The discriminant Δ de P l,s is equal to Δ = l (l + 8 cos s +); as a result, using Jensen s formula, we get

6 8 L Benferhat if Δ >, then f (s) = log ( cos s + ) max ( X, X ) where X,X are roots of P l,s (x), if Δ, then f (s) = log cos s + If l> then Δ >, so, using Jensen formula, we have m (l) = log cos s + l ++ Δ ds By differenciating with respect to l, we obtain m (l) = ds l (l + 8 cos s +) Let X = sin s and k = l +, then m (l) = l +l l + We put T = l so we get m ( l) =Tf(T ) and f(t ) admits entire series expansion in a vicinity of Let be the set of complex numbers l for which P l (x, y) vanishes on the torus T = {(x, y), x = y =} For l/, log P l (x, y) is an analytic function of l and if we define m(p l )= log P (i) l (x, y) dx dy x y, then so d m(p l ) dl x = y = m(p l )=R( m(p l )), = For the previous family, we get T P l d m(p l ) dl (x, y) P l (x, y) = dx dy x y

7 Integral expression of Dirichlet L-series 83 T (xy + xy )+(y x + x y )+(y + y )+(x + )+l + x dx dy x y According to Poincaré residue theorem, d m(p l) is a period of P l (x, y) = dl We know that periods of first species differentials on the compact Riemann surface associated with the curve P l (x, y) = verify a linear differential equation of second order given by Gauss-Manin s connexion It is called Picard-Fuchs differential equation Ω + P Ω + QΩ =, where P and Q are rational functions of l We determine the equation verified by m (l) by using Stiller s method So we get for the family, the differential equation g l +3l l (l + ) (l ) g + l +l + l (l + ) (l ) g = By Frobenius [6] this differential equation admits, in a vicinity of infinity, a logarithmic solution and a solution with entire series expansion of the form Tf(T ) with T =, and f(t ) with entire series expansion in a l vicinity of T = We check easily that the family of polynomials ( do ) not vanish on the torus for l> So m (l) = l +l is a l + regular solution of the differential equation in a vicinity of + Accordind to (), by integration of m ( l) between l and s we have s l so m (l) dl = 8 k s k 3k m(s) m(l) = 8 k dk + k s s k 3k dk avec s = k 3k dk +α l(s), with α l (s) = log + ( s s + log + log + ) s + ( l ) log ++ l + log( 3) s +,

8 8 L Benferhat When s +, m(s) log s, then m (l) = 8 k 3 According to (), put l =, we have If l< then Δ >, so then ( l k 3k dk log + log ) ++ l m () = log 3 8 m (l) = m (l) = Set X = cos s, it comes m (l) = Set k =, we have l k 3k dk log ( cos s + l +)+ Δ ds l l m (l) = ds l (l + 8 cos s +) dx ( X ) ( X) l l l l As m ( l) admits an entire series expansion in a vicinity of, it is the regular solution of the Picard-Fuchs differential equation associated to the family 5 According to (), we have k = 3 then dl = l k dk 3 So m (l) dl = 8 k k dk By integration between s and l, we have l s m (l) dl = 8 k s k k dk avec s = s,

9 Integral expression of Dirichlet L-series 85 then m(l) m(s) = 8 k k s k k dk s k k dk We prove that ( ( k ) ( )) s k k dk = log k + k log s + s When s,s and m(s) log( s), then m (l) = 8 k (k ) l + l k dk + log k 6 For l<, we have m (l) = 8 (k ) l + l k k dk + log k So m( ) = 8 ( k k dk + log + ) 3 ( ) l 7 If <l<, put a = arccos ; then Δ > if<s<aand 8 Δ, if a s< So m (l) = [ a log cos s + l ++ ] Δ ds + log cos s + ds By differentiating with respect to l, Put X = sin s m (l) = l, it comes a ds (l + 8 cos s +) a m (l) = l vut l + + dx (( l + (X ) ) ), X

10 86 L Benferhat and if k = l +, then So m (l) = l + k m (l) = l dx ( X )( k X ) l + 8 If <l<, we have k = l + then dl =3kdk, so m (l) dl = l 3kdk By integration between and, we obtain Put u = k 3, so As m () =, then m (l) dl = 8 m () m () = m () = 3 k 3 kdk u +3 du u +3 du ( 9 If <l<, put a = arccos +l ) ; then Δ > ifa<s<and 8 Δ, if <s a So m (l) = [ a log cos s + ds + log cos s l + ] Δ ds a By differentiating with respect to l, we obtain m (l) = a ds l(l + 8 cos s +)

11 Integral expression of Dirichlet L-series 87 Put X = cos s, we have m (l) = l + k dx ( ( ) ), l ( X ) X and if k = l, it comes m (l) = l l For <l<, we have k = l and dl = 3kdk By integration between and, we obtain Put u = k, we have Put u = 3t, it comes As m() =, then m (l) dl = 8 m (l)dl = m (l) dl = 3 m( ) = 3 3 k k dk u + du 3t + dt 3t + dt Corollary 3 L (χ, ) = 3 u +3 du

12 88 L Benferhat L (χ, ) = 3 log(3) 3 L (χ 3, ) = 3 3 k 3k dk 3t + dt L (χ 3, ) = 6 5 k k dk + 3 log( + 3) Proof According [5], the explicit values m() and m( ) are and m() = 8 3 L (χ, ) m( ) = 3 L (χ 3, ) Then L (χ, ) = 3 u +3 du L (χ, ) = 3 log(3) 3 L (χ 3, ) = 3 3 k 3k dk 3t + dt L (χ 3, ) = 6 5 k k dk + 3 log( + 3)

13 Integral expression of Dirichlet L-series 89 References [] LBenferhat, Integral expression of Dirichlet L-series, JP Journal of Algebra, Number Theory and Applications, (8), 9-59 [] JBorwein and PBorwein, Pi and AGM, JWiley and Sons, 986 [3] PFByrd and MDFriedman, Handbook of elliptic integrals for engineers and physicists, Springer-Verlag, 95 [] FRodriguez-Villegas, Modular Mahler Measures, preprint, 996 [5] NTouafek, Mahler s measure : proof of two conjectured formulae, An St Univ Ovidius Constanta, (9), 7-36 [6] MYoshida, Fuchsien Differential Equations, Aspects of Mathematics, A Publication of the Max-Planck-institut, Vieweg, 987 Received: September, 9