The Laplace transform of Dirichlet L-functions

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1 Nonlinear Analysis: Modelling and Control, 1, Vol. 17, No., The Lalace transform of Dirichlet L-functions Aidas Balčiūnas a, Antanas Laurinčikas b a Institute of Mathematics and Informatics, Vilnius University Akademijos str. 4, Vilnius LT-8663, Lithuania a.balciunui@gmail.com b Faculty of Mathematics and Informatics, Vilnius University Naugarduko str. 4, Vilnius LT-35, Lithuania antanas.laurincikas@mif.vu.lt Received: 17 August 11 / Revised: 17 Aril 1 / Published online: 1 May 1 Abstract. In the aer, a formula for the Lalace transform of the square of Dirichlet L-functions on the critical line is obtained. Keywords: Dirichlet L-function, functional equation, Lalace transform, Riemann zeta-function. 1 Introduction Let s = σ + it be a comlex variable. The Lalace transform Ls) of the function fx) is defined by Ls) = fx)e sx dx rovided the integral exists for σ > σ with some σ. The function Ls) can be alied for the investigation of the asymtotics for the mean value T M T f) def = fx) dx. Suose that fx) for all x [, ) and, for a given m >, as δ. Then [1, Cha. VII], Lδ) 1 δ logm 1 δ M T f) T log m T c Vilnius University, 1

2 18 A. Balčiūnas, A. Laurinčikas as T. The latter examle shows that Lalace transforms can be used in the moment roblem of zeta and L-functions. Let χ be a Dirichlet character mod q, and let Ls, χ) denote the corresonding L-function which is defined, for σ > 1, by Ls, χ) = χm) m s. Denote by χ the rincial character mod q. Then it is well known that the function Ls, χ ) is analytically continued to the whole comlex lane, excet for a simle ole at s = 1 with residue 1 1 ), where denotes a rime number. If χ χ, then Ls, χ) is analytically continued to an entire function. In the theory of Dirichlet L-functions, usually the moments T χ=χ mod q) Lσ + it, χ) k dt, k, σ 1, are considered, see, for examle, []. This corresonds to the Lalace transform χ=χ mod q) Lσ + it, χ) k e sx dx. The aim of this note is an exlicit formula for the individual Lalace transform Ls, χ) of L1/ + ix, χ), i.e., for the function Ls, χ) = ) 1 e L + ix, χ sx dx. In forthcoming aers, this formula will be alied for the investigation of the Mellin transform of L1/ + ix, χ), for the case of ζρ + ix) with 1/ < ρ < 1, see [3]. In a series of aers, A. Ivič, M. Jutila and Y. Motohashi develoed the method of Mellin transforms for the moment roblem in the theory of zeta-functions, see, for examle, [4]. These remarks show the motivation of the aer. For the statement of the formula, we need some notation. Let Gχ) denote the Gauss sum, i.e., Gχ) = q χl)e l/q. l=1

3 The Lalace transform of Dirichlet L-functions 19 Moreover, let where { if χ 1) = 1, a = 1 if χ 1) = 1, { ɛχ) if a =, Eχ) = ɛ 1 χ) if a = 1, ɛχ) = Gχ) q, ɛ 1 χ) = Gχ) q. As usual, denote by dm) the divisor function dm) = d m 1. Theorem 1. Suose that χ is a rimitive character mod q, q > 1. Then Ls, χ) = a e is/ { dm)χm) ex m } e is + λs, χ), qeχ) q where the function λs, χ) is analytic in the stri {s C: σ < π}, and, for σ θ, < θ < π, the estimate λs, χ) = O 1 + s ) 1) is valid. Let µm) be the Möbius function, and let γ denote the Euler constant. Theorem. Suose that χ is the rincial character mod q, q > 1. Then Ls, χ ) = ie is/ +πe is/ n q 1 1 ) ) π γ log π s i + m q µm)µn) m k=1 { dk) ex kn m e is log 1 + ) µm) log m m q } + λs, χ ), where the function λs, χ ) has the same roerties as λs, χ) in Theorem 1. The case q = 1 corresonds the Riemann zeta-function. Let L ζ s) be the Lalace transform of ζ1/ + ix). Corollary 1. We have L ζ s) = ie is/ γ log π +πe is/ ) ) π s i where function λs) has the same roerties as λs, χ). dm) ex { me is} + λs), Nonlinear Anal. Model. Control, 1, Vol. 17, No.,

4 13 A. Balčiūnas, A. Laurinčikas Lemmas First we remind the functional equation for Ls, χ). We reserve the notation used in introduction. Let ) s+a)/ ) π s + a ls, χ) = Γ Ls, χ). q Lemma 1. Suose that χ is a rimitive character mod q, q > 1. Then ls, χ) = Eχ)l1 s, χ). Proof of the lemma is given, for examle, in [5]. Lemma. For σ > 1, Proof. For σ > 1, we have that L s, χ) = L s, χ) = χm) m s n=1 dm)χm) m s. χn) n s = am) m s, where am) = ) m χd)χ = χm)dm). d d m Denote by ϕq) the Euler totient function. Lemma 3. Let σ be arbitrary real number. Then, for σ σ, Ls, χ) = E ϕq) qs 1) + O σ q c t + ) c), where c = cσ ) >, and E = { 1 if χ = χ, if χ χ. The lemma is Theorem 7.3. from [6]. Lemma 4. The estimate ) 1 L + it, χ = O q c1 t + ) c 1 ) holds with some c 1 >. Proof. The lemma is a corollary of Lemma 3 and the integral Cauchy formula.

5 The Lalace transform of Dirichlet L-functions 131 Lemma 5. Suose that a >. Then 1 a+i a i Γs)b s ds = e b. The lemma is the well-known Mellin formula, see, for examle, [7]. 3 Proof of Theorems Define the function λs, χ) by λs, χ) = ) 1 e L + ix, χ sx dx e is/ i 1 a 1/+i 1/ i First suose that a =. Then the formulae Lz, χ)l1 z, χ)e izπ/ s) cos πa πz ) dz. 1) give cos s = eis + e is and Ls, χ) = L s, χ) e is/ i 1/+i 1/ i = e is/ = + Lz, χ)l1 z, χ)e izπ/ s) dz cos πz L 1 + ix, χ)l 1 ex{ ix, χ) ex{ L 1 + ix, χ) ex{ xs} ex{ } + ex{ } dx + is xs} } + ex{ } dx L 1 + ix, χ) ex{ + xs} ex{ } + ex{ } dx. ) Since 1 ex{ ex{ } } + ex{ } = ex{ ex{ } } + ex{ }, Nonlinear Anal. Model. Control, 1, Vol. 17, No.,

6 13 A. Balčiūnas, A. Laurinčikas we find from 1) and ) that λs, χ) = L 1 + ix, χ) e xs ex{ } ex{ } + ex{ } dx L 1 + ix, χ) e xs ex{ ex{ } + ex{ Now let a = 1. In this case, in view of the formula we find that e is/ 1/+i 1/ i = e is/ = Clearly, we have 1 + ex{ sin s = eis e is, i Lz, χ)l1 z, χ)e izπ/ s) dz sin πz L 1 + ix, χ)l 1 ex{ } ix, χ) ex{ L 1 + ix, χ) ex{ ex{ xs} } ex{ } dx L 1 + ix, χ) ex{ ex{ ex{ } } ex{ thus, from 1) and 4), it follows that λs, χ) = + } dx. 3) + is xs} } ex{ } dx + xs} } ex{ } dx. 4) } = ex{ ex{ } L 1 + ix, χ) e xs ex{ } ex{ } ex{ } dx } ex{ }, L 1 + ix, χ) e xs ex{ } ex{ dx. 5) } ex{ } By estimates for L1/ + ix, χ) of Lemma 3, we have that the integrals in 3) and 5) converge uniformly in comact subsets of the stri {s C: σ < π}, thus, the function λs, χ) is analytic in that stri.

7 The Lalace transform of Dirichlet L-functions 133 It remains to estimate the function λs, χ). Suose that σ θ, where < θ < π. First let s is small. Then the integrals in 3) and 5) are bounded by a constant. If s is large, then integrating by arts with resect to e ±sx and using the estimate ) 1 ) L + ix, χ ) 1 1 = L + ix, χ L ) 1 1 = il )L + ix, χ ix, χ = O q x + )) c )) ix, χ ) 1 1 il )L + ix, χ ix, χ with some c >, which follows from Lemmas 3 and 4, we obtain that λs, χ) = O s 1). So, in all cases, we have that, for σ θ, < θ < π, Equality 1) shows that Ls, χ) = e is/ i 1 a λs, χ) = O 1 + s ) 1). 1/+i 1/ i Lz, χ)l1 z, χ)e izπ/ s) cos πa πz ) dz + λs, χ), 6) where the function λs, χ) is analytic in the stri {s C: σ < π}, and, for σ θ, < θ < π, the estimate λs, χ) = O 1 + s ) 1) is valid. It remains to calculate the integral in 6). Using Lemma 1, we find ) z+1/ π L1 z, χ) = E 1 Γ z χ) + a ) q Γ 1 + a z χ). 7) )Lz, Taking into account the formulas, see [8], and Γz)Γ1 z) = π, z / Z, sin πz Γz)Γ z + 1 ) = π z Γz), Nonlinear Anal. Model. Control, 1, Vol. 17, No.,

8 134 A. Balčiūnas, A. Laurinčikas we obtain that and Γ z ) Γ 1 z ) = Γ z )Γ 1 + z ) Γ 1 z )Γ 1 + z ) = Γ z )Γ 1 + z π = π z Γz) cos πz π ) cos πz = 1 z π 1/ Γz) cos πz, Γ 1 + z ) Γ1 z ) = Γ 1 + z )Γ 1 z ) Γ1 z )Γ 1 z ) = π π 1+z Γ1 z) cos πz = π 1/ Γz) sin πz z cos πz = 1 z π 1/ Γz) sin πz. Since a = or a = 1, hence we have that Γ z + a ) πa Γ 1 + a z ) = 1 z π 1/ Γz) cos πz ), and, in view of 7), this gives L1 z, χ) = E 1 χ) 1 z π z q z 1/ Γz) cos Hence, we find that e is/ i 1 a 1/+i 1/ i e is/ = i 1 a qeχ) Next we rove Theorem 1. Lz, χ)l1 z, χ)e izπ/ s) cos πa πz ) dz 1/+i 1/ i Γz)L z, χ) πa πz ) Lz, χ). q e is ) z dz. 8) Proof. Let χ be a non-rincial character. Then the integrand in 8) is a regular function in the stri {z C: 1/ < Rz < }. Therefore, e is/ i 1 a qeχ) e is/ 1/+i 1/ i = i 1 a qeχ) Γz)L z, χ) +i i q e is Γz)L z, χ) ) z dz q e is ) z dz. 9)

9 The Lalace transform of Dirichlet L-functions 135 Moreover, in view of Lemmas and 5, +i i Γz)L z, χ) q e is ) z dz = This, 8), 9) and 6) rove Theorem 1. Next we rove Theorem. dm)χm) = +i i ) z m Γz) e is dz q { dm)χm) ex m e }. is q Proof. Let χ be the rincial character mod q. Then we have that Ls, χ ) = ζs) 1 1 ) s, where ζs) is the Riemann zeta-function. Using the functional equation ) ) s 1 s π s/ Γ ζs) = π 1 s)/ Γ ζ1 s), we find that Therefore, L1 z, χ ) = ζ1 z) 1 1 ) 1 z = ζz) 1 z π z Γz) cos πz Lz, χ )L1 z, χ )e izπ/ s) = ζ z) 1 z π z Γz) cos πz and, similarly to the case of non-rincial character, we obtain Ls, χ ) = e is/ i 1/+i 1/ i Γz)ζ z) e is) z 1 1 ) 1 z. 1 1 ) z 1 1 ) 1 z, 1 1 ) z 1 1 ) 1 z dz + λs, χ ). 1) The roduct q z) = 1 1 ) 1 z Nonlinear Anal. Model. Control, 1, Vol. 17, No.,

10 136 A. Balčiūnas, A. Laurinčikas has zeroes lying on the line σ = 1. Therefore, we write q z) as a sum. Using the Möbius function µm), we obtain that z) = µm)m z 1. q The function Γz)ζ z) e is) z 1 1 z ) m z 1 has the ole of order at z = 1. Moreover, since and Γ1) = 1, Γ 1) = γ, we find that Res z=1 Γz)ζ z) e is) z Therefore, ζz) = 1 z 1 + γ + γ 1 z 1) +..., = lim z 1 z 1) Γz)ζ z) e is) z) = e is Res z=1 Γz)ζ z) e is) z 1 1 z ) m z 1 ) ) π γ log π s i. = lim z 1) Γz)ζ z) e is) z 1 1 ) )m z 1 z 1 z = lim z 1 z 1) Γz)ζ z) e is) z) + lim z 1 z 1) Γz)ζ z) e is) z + lim z 1 z 1) Γz)ζ z) e is) z log m ) = eis π γ log π s i + e is ) ) = eis 1 1 ) γ log π ) 1 1 z ) m z z ) 1 1 ) 1 log z 1 1 z 1 1 ) z m z 1 log 1 + e is ) 1 log m ) π s i + log 1 + log m m z ) ).

11 The Lalace transform of Dirichlet L-functions 137 Thus, in view of 1), Ls, χ ) = ie is/ ie is/ 1 1 ) ) π γ log π s i + +i i Γz)ζ z) e is) z log 1 + ) µm) log m m q 1 1 ) z 1 1 ) 1 z dz +λs, χ ). 11) We have that 1 1 ) z 1 1 ) 1 z = n q m q µm) m µn) ) z m. n Moreover, for σ > 1, ζ s) = k=1 Therefore, the alication of Lemma 5, leads to Γz)ζ z) e is) z = µm)µn) m n q m q k=1 +i i This together with 11) roves the theorem. References dk) k s. 1 1 ) z 1 1 ) 1 z dz { dk) ex kn m e is 1. E.C. Titchmarsh, The Theory of Riemann Zeta-Function, nd edition revised by D.R. Heath- Brown, Clarendon Press, Oxford, H.L. Montgomery, Toics in Multilicative Number Theory, Lect. Notes Math., Vol. 7, Sringer, Berlin, Heidelberg, New York, A. Laurinčikas, The Mellin transform of the square of the Riemann zeta-function in the critical stri, Integral Transforms Sec. Funct., 7), , A. Ivič, M. Jutila, Y. Motohashi, The Mellin transform of owers of the zeta-function, Acta Arith., 95, ,. 5. K. Chandrasekharan, Arithmetical Functions, Sringer-Verlag, New York, Berlin, 197. }. Nonlinear Anal. Model. Control, 1, Vol. 17, No.,

12 138 A. Balčiūnas, A. Laurinčikas 6. K. Prachar, Primzahlverteilung, Sringer-Verlag, Berlin, Götingen, Heidelberg, E.C. Titchmarsh, Theory of Functions, Oxford University Press, Oxford, A. Laurinčikas, R. Garunkštis, The Lerch Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London,.