LOWER BOUNDS FOR POWER MOMENTS OF L-FUNCTIONS

Transcription

1 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS AMIR AKBARY AND BRANDON FODDEN Abstract. Let π be an irreducible unitary cusidal reresentation of GL d Q A ). Let Lπ, s) be the L-function attached to π. For real k 0 let I k π, ) = Z L π, / + it) k dt be the k-th ower) moment of Lπ, s). Let α π, j) C j d) be the local arameters at rime. We rove that if α π, j) γ for unramified rimes and for a fixed 0 γ < /4, then I k π, ) log ) k for any rational k 0. As a corollary of this result we establish unconditional lower bounds of the conjectured order of magnitude for the fractional k-th moments of Dirichlet L-functions, modular L-functions, twisted modular L-functions, and Maass L-functions. We derive our results as corollaries of more general theorems related to the lower bounds for fractional moments of analytic functions which have Dirichlet series reresentations on a comlex half-lane. We also establish lower bounds for the k-th moments of Artin L-functions and Dedekind zeta functions of number fields.. Introduction a gn) Let s = σ +it denote a oint in the comlex lane. Let gs) = n be a s Dirichlet series absolutely convergent for σ > that can be continued analytically to the region σ /, t. For fixed σ / and non-negative real k, we define the k-th ower) moment of g at σ as I k g, σ, ) = g σ + it) k dt. For σ = /, we denote I k g, /, ) by I k g, ) and we call it the k-th ower) moment of g. We are interested in studying the behaviour of I k g, σ, ) as. If gs)) k has a reresentation on σ > as an absolutely convergent Dirichlet series g k n) n, then from the mean value theorem for Dirichlet series [9, s heorem 7.] we know that g k n).) I k g, σ, ) n σ, for σ >, as. Very little is known about the behaviour of I k g, σ, ) for / σ. However for several classes of Dirichlet series arising in number 000 Mathematics Subject Classification. Primary: M4; Secondary: F66, M06. Key words and hrases. ower moments of L-functions, multilicative functions, automorhic L-functions, Artin L-functions, Dedekind zeta functions. Research of both authors was artially suorted by NSERC. Research of the second author was also suorted by a PIMS ostdoctoral fellowshi.

2 AMIR AKBARY AND BRANDON FODDEN theory it is conjectured that.) holds for / < σ. Another interesting feature is the close connection of the behaviour of I k g, σ, ) with the size of I k g, ). In fact for certain g it is known that if I k g, ) +ɛ for any ɛ > 0, then.) holds for σ > / see [8, Section 9.5]). In this aer, we are interested in finding lower bounds of I k g, ) in terms of for certain number theoretical L-functions g which have Dirichlet series reresentations on the half lane σ >. he secial case when g is the Riemann zeta function ζs) is classical and has been under investigation for a long time. For k = and the asymtotic formulas are known. One of the first results on this subject which treats all integral values of k is due to itchmarsh [9, heorem 7.9], who showed that ) k ζ + it e t/ dt log ) k. o he first general lower bound for the k-th moment of ζs) was given by Ramachandra. In [4, heorem ] he roved that I k ζ, ) log ) k log log ) c k for real k /, where c k is a constant deending ossibly on k. Moreover he roved that under the assumtion of the Riemann Hyothesis this lower bound holds for any real k 0. Later in [5] Ramachandra studied the general roblem of finding lower bounds for the moments of Dirichlet series and in this direction obtained several imortant results. One of his results is the following. heorem. Ramachandra). Let F s) be a Dirichlet series convergent for σ > that can be continued analytically to the region σ /, t, and in this region F s) ex log t) c 0 ) where c 0 is a fixed ositive constant. Moreover assume that F s) has a reresentation in the form F s) = an) n s for σ >, where for / < σ < the series an) n σ σ /) α for some ositive constant α. hen an) n σ ) I / F, ) log ) α. Proof. his is a secial case of [5, heorem ]. σ /) α converges and Letting F s) = gs)) k/ ) = gs)) k in Ramachandra s theorem we get the following. Corollary.. Let gs) be a Dirichlet series convergent for σ > that can be continued analytically to the region σ /, t, and in this region gs) ex log t) c 0 ) where c 0 is a fixed ositive constant. Let k 0 be an integer. Suose that gs)) k/ in σ > can be reresented by a Dirichlet series gs)) k/ g k/ n) = n s.

3 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 3 Moreover suose that for / < σ < the series g k/ n) n σ σ /) α for some ositive constant α. hen g k/ n) n σ I k/ g, ) log ) α. σ /) α converges and Also, if the above conditions hold for real k 0 and in addition gs) does not have any zero in the region σ > /, t, then for any non-negative real number k. I k g, ) log ) α, A secial case of the above corollary is the following. Corollary.3 Ramachandra). For the Riemann zeta function ζs), and for a non-negative integer k we have I k/ ζ, ) log ) k /4. Moreover, under the assumtion of the Riemann hyothesis, for non-negative real k we have I k ζ, ) log ) k. In [4], Heath-Brown extended the above result to the fractional moments of the Riemann zeta function. heorem.4 Heath-Brown). For the Riemann zeta function ζs), and for a non-negative rational number k we have I k ζ, ) log ) k. Under the assumtion of the Riemann hyothesis, Heath-Brown s method also establishes the same lower bound for any non-negative real k. In recent years several authors have emloyed the Heath-Brown method to study the fractional moments of other L-functions. Let χ be a rimitive Dirichlet character mod q and let I k χ, ) denote the k-th moment of the Dirichlet L- function Lχ, s). hen Kacenas, Laurincikas, and Zamarys [6] roved that for k = /n, n, we have I k χ, ) log ) k. In [0] Laurincikas and Steuding roved that if f is a holomorhic cus form of weight l and level, and I k f, ) is the k-th moment of the modular L-function attached to f, then for k = /n, n one has I k f, ) log ) k. Zamarys [0] has generalized this result to the case of the k-th moment of twisted modular L-functions for ositive rational k /. he roofs of the above results closely follow [4]. Our first goal in this aer is to follow the method of Heath-Brown to derive a general theorem in the sirit of Corollary.. More secifically we rove the following:

4 4 AMIR AKBARY AND BRANDON FODDEN a gn) heorem.5. Let gs) = n be a Dirichlet series absolutely convergent s for σ > that has an analytic continuation to the half lane σ / with a ole of degree m at s = with m ossibly equal to 0) and suose that.) gσ + it) t A for a fixed A > 0, σ / and t the imlied constant is indeendent of σ and t). For rational k 0 and σ >, we suose that gs)) k may be reresented by an absolutely convergent Dirichlet series where gs)) k =.3) g k n) n η g k n) n s, for some fixed η with 0 η < /. In addition, assume that there exist fixed constants c > 0 and α 0 such that g k n).4) σ /) α n σ log ) α uniformly for + c log σ and. hen for any rational k 0 we have.5) I k g, ) log ) α. Moreover if the above conditions.3) and.4) hold for real k 0 and in addition gs) does not have any zeros in the half lane σ > / then.5) holds for any non-negative real k. We next emloy the above theorem to establish lower bounds for moments of certain number theoretical L-functions. he L-functions we consider include the following: a) Princial automorhic L-functions Let π be a rincial automorhic L-functions whose local arameters α π, j) at unramified rimes satisfy α π, j) γ, for a fixed 0 γ < /4 see Section for terminology). hen, for rational k, we rove that I k π, ) log ) k. As a corollary of this general result we deduce unconditional lower bounds of the conjectured order of magnitude for the fractional moments of classical Dirichlet L-functions, modular L-functions, and the twisted modular L-functions. Our results generalize the lower bound results of Kacenas-Laurincikas-Zamarys [6] and Laurincikas-Steuding [0] to rational values of k 0, and the lower bound result of Zamarys [0] to rational values of k /. Moreover, we derive an unconditional lower bound for the fractional k-th moment of Maass L-functions. Let Lf, s) be a Maass cus newform of weight zero and level N with nebentyus ψ. hen for any rational k 0 we rove that I k f, ) log ) k. o our knowledge this result is the first established unconditional lower bound for the k-th moment of an L-function for which the truth of the Generalized Ramanujan Conjecture GRC) see Section for terminology) is not known.

5 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 5 Remark.6. Ji [5] has established lower bounds for k-th moments of rincial automorhic L-functions over short intervals. As a consequence of [5, heorem.], for real non-negative k, one has I k π, σ, ) uniformly for σ /. Our results give conditional imrovements of this lower bound for σ = /. b) Artin L-functions Under the assumtion of the Artin holomorhy conjecture we deduce a lower bound for the fractional k-th moment I k K/Q, ρ, ) of the Artin L-function LK/Q, ρ, s) associated to a reresentation ρ of the Galois grou of a number field K/Q. We rove that for any non-negative rational k, I k K/Q, ρ, ) log ) ϕ,ϕ k, where ϕ is the character associated to ρ and ϕ, ϕ is the inner roduct of ϕ and ϕ as defined in Proosition.. As a direct corollary of this result we derive an unconditional lower bound for the fractional k-th moment I k ζ K, ) of the Dedekind zeta function ζ K s) of a number field K. More recisely, for non-negative rational k we rove that I k ζ K, ) log ) nk, where n is the degree of the Galois extension K/Q. Our result imroves the following theorem of Ramachandra [4, heorem ] for rational values of k. heorem.7 Ramachandra). Let K be a degree n Galois extension of Q, and let ζ K s) be the Dedekind zeta function of K. hen for real k / we have where c is a ositive constant. I k ζ K, ) log )nk log log ) c Remarks.8. a) he Generalized Riemann Hyothesis GRH) for an L-function imlies that the L-function does not have any zeros in the half lane σ > /. hus by heorem.5, uon the assumtion of GRH for an L-function g, the result for the fractional k-th moments of g can be extended to non-negative real values of k as long as.3) and.4) hold for gs)) k. b) In 005, Rudnick and Soundararajan [7] devised a method that establishes lower bounds of the conjectured order of magnitude for integral k-th moments of several families of L-functions. hey also commented that their method generalizes to the fractional k-th moments for k. his rocedure is also alicable to the integral k-th ower) moment of the Riemann zeta function. Using this method, Chandee [, heorem.] has established a lower bound of the conjectured order of magnitude for the integral shifted ower) moment of the Riemann zeta function. c) In [4], Heath-Brown also considered the uer bound of the k-th ower) moment of the Riemann zeta function, and by the method similar to the one used in establishing lower bound, he roved that I k ζ, ) log ) k, unconditionally for k = /n > 0, where n is an integer, and under the assumtion of the Riemann Hyothesis for all real values of 0 k. It would be ossible

6 6 AMIR AKBARY AND BRANDON FODDEN to generalize this method to obtain uer bounds for the k-th ower) moments of general L-functions. However, other than the case of the Dirichlet L-functions, the results would be conditional uon GRH and only true for some secific values of k <. See [0, heorem ] for an examle of such a result. he structure of the aer is as follows. In Section we describe some alications of heorem.5. o do this we first observe that if the sequence { a g ) } has regular distribution then the technical condition.4) holds. We rove this result in Section 3. Using this observation in Section, we establish a new version heorem.3) of our main heorem. Our results on lower bounds for moments of rincial automorhic L-functions, Artin L-functions and Dedekind zeta functions are simle consequences of heorem.3. Finally in Sections 4 and 5 we rove heorem.5. Notation.9. s = σ + it denotes a oint in the comlex lane and denotes a rime number. We use the notation, O.), and with their usual meaning in analytic number theory. All constants imlied by the notation may deend on k and the function g. However, the constants are indeendent of the variables σ and. By abuse of notation and for simlicity we denote the k-th moment of an L-function Lf, s) by I k f, ) instead of I k Lf,.), ).. Alications of heorem.5 We first identify some classes of Dirichlet series for which the technical condition.4) holds. o do this we emloy a classical theorem on the average values of multilicative functions and derive the following roosition. Proosition.. Let hn) be a non-negative multilicative function. Suose that for some fixed real α > 0 we have h) α x log x as x, and Suose also that h) = α log log x + O). h r ) ) rθ for some 0 θ < / and for all r. Let and let 0 < c log be fixed. hen we have hn) σ /) α log )α nσ uniformly in the range + c log σ. he receding roosition is roved in Section 3. As a consequence of this roosition we rove a version of heorem.5 for Dirichlet series with certain Euler roducts. o simlify our exosition, from now on we consider the class C of Dirichlet a gn) n s series gs) = that are absolutely convergent for σ > and satisfy the following: Excet for a ole of degree m at s = with m ossibly equal to 0), gs) extends to an entire function for σ / and gσ + it) t A for a fixed A > 0, σ / and t.

7 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 7 For any real k we define the multilicative function τ k.) by τ k r ) = Γk + r) r!γk) = kk + )k + ) k + r ). r! Note that τ k n) 0 when k 0. he following lemma summarizes some roerties of τ k n). Lemma.. i) For fixed k 0 and ɛ > 0 we have τ k n) n ɛ. ii) If j is a ositive integer then τ kj n) = τ k n )τ k n ) τ k n j ). n=n n n j Proof. See [4, Lemma ] heorem.3. Let gs) C. Suose gs) has an Euler roduct of the form.) gs) = a g n) n s for σ >, where α g, j) C and = d j= α ) g, j) s α g, j) ) γ, for a fixed 0 γ < /4. Suose we have.) a g ) β x log x as x, and.3) a g ) = β log log x + O) for some fixed constant β, where a g ) = d j= α g, j). hen for any rational k 0 we have I k g, ) log ) βk. Proof. Since gs) is reresented by an absolutely convergent Euler roduct and the Euler factors are nonvanishing on σ >, we have gs) 0 for σ >. Letting b g r ) = d α g, j) r, j= for σ > 3/ we define a branch of log gs) by log gs) = r= b g r ) r rs, and a branch of g k s) by g k s) = exk log gs))

8 8 AMIR AKBARY AND BRANDON FODDEN for any real k. For σ > 3/ we have g k s) = ex k r= = d ex k j= α g, ) s = Using the aylor exansion on the region z < we have g k s) = z) k = = j= ) b g r ) r rs log α ) g, j) s ) k α ) g, d) k s. r=0 r=0 Γk + r) r!γk) zr d ) τ k r ) α g, j) r rs r=0 g k r ) rs = g k n) n s. Here g k n) is the multilicative function given by g k r ) = τ k j )α g, ) j τ k j )α g, ) j τ k j d )α g, d) j d. r = j j d By emloying α g, j) ) γ and Lemma. we have.4) g k r ) ) rγ τ k j ) τ k j d ) = ) rγ τ kd r ) ) rγ+ɛ), r = j j d for any ɛ > 0. Alying this bound we have g k r ) rσ <, r= for σ >. On the other hand g k ) σ = k g k n) n s a g ) σ kgσ) <, for σ >. hus is absolutely convergent for σ >, which together with.4) imlies that.3) holds. It remains to show that.4) holds with α = βk. Let hn) = g k n), so that h) = k a g ). We note that.),.3), and.4) show that the conditions of Proosition. hold for α = βk, and thus we may use Proosition. to see that.4) holds. herefore for rational k heorem.5 holds with α = βk, which comletes the roof of the theorem. Next we describe several alications of heorem.3 in the cases of automorhic L-functions, Artin L-functions, and Dedekind zeta functions.

9 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 9 Princial Automorhic L-functions Let π be an irreducible unitary cusidal reresentation of GL d Q A ). he global L-function at finite laces attached to π is given by the Euler roduct of local factors for σ > :.5) Lπ, s) = where L π, s) = a π n) n s d j= = L π, s), α ) π, j) s, and α π, j) C j d) are the local arameters at rime. It is known that [6, Formula.3)]).6) α, j) d +. It is conjectured that for all rimes excet finitely many α, j) =. o each π there is a finite set of rimes associated that are called the ramified rimes. he local arameters for rimes outside the set of ramified rimes i.e. unramified rimes) can be give as eigenvalues of a matrix in GL d C). We say that an irreducible unitary cusidal reresentation π of GL d Q A ) satisfies the Generalized Ramanujan Conjecture GRC) if for local arameters α π, j) j d) at unramified rimes we have α π, j) =. Our aim is to aly heorem.3 to Lπ, s). We call such an L-function a rincial automorhic L-function. It is known that Lπ, s) C, and we clearly have an Euler roduct of the form.). We next investigate sufficient conditions under which formulas analogous to.) and.3) hold. o do this, for rimes and integers r, we define { d b π r j= ) = α π, j) r unramified, 0 ramified. Note that for an unramified rime, b π ) = a π ). We define b π n) = 0 if n is not a rime ower. We consider the following hyothesis on the size of the coefficients b π r ) for r. Hyothesis H 0 here is ɛ > 0 such that for any fixed r, log ) b π r ) x ɛ. r x H 0 is a mild hyothesis and exected to be true for all π in general. It is true under the assumtion of GRC. More generally if the local arameters satisfy α π, j) γ, for a fixed 0 γ < /4 then H 0 holds. Emloying.6) we have log ) b π r ) x ɛ, r α r x where 0 < ɛ < /d + ) and α/d + ) ɛ) >. Hence, assuming H 0, there is ɛ > 0 such that.7) log ) b π r ) x ɛ. r r x

10 0 AMIR AKBARY AND BRANDON FODDEN Finally observe that H 0 imlies the hyothesis H of Rudnick and Sarnak [6, Page 8]. hat is, under the assumtion of H 0 we have.8) r r x log ) b π r ) r <. In the roof of the following lemma we use some roerties of the Rankin-Selberg L-function Lπ π, s). See [6, Section.4] for a review of these roerties. Lemma.4. Under the assumtion of H 0 we have i) b π ) x log x as x, and ii) b π ) = log log x + O). Proof. i) Let S be the roduct of all ramified rimes for π and L S π π, s) = S L π π, s) be the artial Rankin-Selberg L-function. hen it is known that for σ >, fs) = L S L S π π, s) = Λn) b π n) where Λn) = log if n = r and zero otherwise see [6, Page 8, formula.)]). From the roerties of the Rankin-Selberg L-functions we deduce that fs) has a meromorhic continuation to σ with a simle ole of residue at s =. By the Wiener-Ikehara auberian theorem we have.9) Λn) b π n) x. n x For simlicity in our exosition and in a similar fashion to the classical number theoretical functions we define Ψx) = n x We have Λn) b π n), Θx) = n s log ) b π ), and Πx) =.0) Θx) Ψx) log x)πx) + C ɛ x ɛ, b π ). where C ɛ is a fixed constant here we have used.7)). On the other hand, from.9), for large x we have Πx) Ψx) + ɛ)x. Emloying this uer bound for Πx), for 0 < α < we have.) Θx) log ) b π ) Πx) Πx α )) log x α x α α Πx) + ɛ)x α ) log x. Since we can choose α arbitrary close to, from.0),.), and.9) we have lim x Πx) x/ log x = lim Θx) Ψx) = lim =. x x x x

11 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS ii) Under the assumtion of.8) Rudnick and Sarnak [6, Proosition.3] have shown that Λn)b π n) log x) = + Olog x). n n x Emloying.8) and artial summation in the above formula yields the desired result. Proosition.5. Let Lπ, s) be a rincial automorhic L-function. Suose that for the local arameters at unramified rimes we have.) α π, j) γ, for a fixed 0 γ < /4. hen for rational k 0. I k π, ) log ) k Proof. Let S be the roduct of all ramified rimes and gs) = L S π, s) = S L s, π ) be the artial L-function. It is clear that if gcdn, S) then the n-th coefficient of L S π, s) is zero; otherwise a π n) is the n-th Dirichlet coefficient of L S π, s). We have L S π, s) C see [, Section 5.] for details). Moreover.) imlies that H 0 in Lemma.4 is satisfied, and so.) and.3) hold for β =. hus the conditions of heorem.3 hold and therefore we deduce the desired lower bound for the fractional k-th moment of L S π, s). Since L S π, s) and Lπ, s) differ only by finitely many factors bounded from below on s = /), the same lower bound holds for I k π, ). Corollary.6. he conclusion of Proosition.5 remains true if GRC holds for π. Since Dirichlet L-functions associated to rimitive characters, modular L- functions associated to newforms, and twisted modular L-functions associated to newforms and rimitive Dirichlet characters are all examles of rincial automorhic L-functions that satisfy GRC see [, Sections 5. and 5.] for details), as an immediate corollary of the above roosition we have the following extensions of the lower bound results of [6], [0], and [0]. Corollary.7. For the Dirichlet L-function Lχ, s) attached to a rimitive Dirichlet character and for any non-negative rational k we have I k χ, ) log ) k. Corollary.8. Let Lf, s) be the L-function associated to a newform of weight l, level N, and nebentyus ψ. hen for any non-negative rational k we have I k f, ) log ) k. Corollary.9. Let Lf, χ, s) be the L-function associated to a newform of weight l, level N, nebentyus ψ, and a rimitive Dirichlet character χ mod q where q, N) =. hen for any non-negative rational k we have I k f, χ, ) log ) k, where I k f, χ, ) denotes the k-th ower) moment of Lf, χ, s). he following is also a direct corollary of Proosition.5.

12 AMIR AKBARY AND BRANDON FODDEN Corollary.0. Let Lf, s) be a Maass cus newform of weight zero and level N with nebentyus ψ. hen for any rational k 0 we have I k f, ) log ) k. Proof. Let α f, ) and α f, ) be the local arameters of f at an unramified rime. From a result of Kim and Sarnak [7, Proosition ] we know that α f, ) 7/64 and α f, ) 7/64. he result follows from Proosition.5 as 7/64 < /4. Artin L-functions Let K/Q be a Galois extension with the Galois grou GalK/Q) and ρ : GalK/Q) GL d C) be a reresentation of GalK/Q). Let LK/Q, ρ, s) be the Artin L-function associated to ρ, which is defined as an Euler roduct L K/Q, ρ, s) of the local factors L K/Q, ρ, s) for σ >. For unramified rimes the local factor of the Artin L-function is defind by L K/Q, ρ, s) = deti d s ρσ ))), where I d denotes the identity matrix, σ is the Frobenius conjugacy class, and ρσ ) is the value of ρ at any element belonging to σ. Letting α ρ, j) j d) be the eigenvalues of the unitary matrix ρσ ), we have d L K/Q, ρ, s) = α ) ρ, j) s, j= where α ρ, j) =. As a consequence of the Brauer induction theorem it is known that any Artin L- function has a meromorhic continuation to the whole comlex lane. Moreover Artin s conjecture asserts that LK/Q, ρ, s) has an analytic continuation for all s excet ossibly for a ole at s = of order equal to the multilicity of the trivial reresentation in ρ. Proosition.. Assume Artin s conjecture for the Artin L-function LK/Q, ρ, s). hen for any non-negative rational k, we have I k K/Q, ρ, ) log ) ϕ,ϕ k, where I k K/Q, ρ, ) denotes the k-th moment of LK/Q, ρ, s). character associated to ρ and ϕ, ϕ = ϕg) G where G = GalK/Q). Proof. Set L ur K/Q, ρ, s) = unramified g G L K/Q, ρ, s) = a ρ n) n s. Here ϕ is the Under the assumtion of the Artin conjecture we know that L ur K/Q, ρ, s) C see [, Section 5.3] for details), and for σ >, L ur K/Q, ρ, s) has an Euler roduct satisfying the conditions given in heorem.3. Note that for unramified we have a ρ ) = ϕ). o aly heorem.3 we need to study the asymtotic

13 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 3 behavior of a ρ). By decomosing this sum according to the conjugacy class C of G that contains σ, we have.3) a ρ ) = ϕ) unramified = ϕg C ) #{unramified x; σ C}), C where g C is any element of C. From the version of the Chebotarev density theorem with the remainder [8, heorem.3] we get #{unramified x; σ C} = C G Lix) + O ) x ex c 0 log x/n), where Lix) = dt/ log t x/ log x, c 0 is an absolute constant and n is the degree of K over Q. Alying this in.3), we get ) ) a ρ ) = ϕ, ϕ Lix) + O ϕg C ) x ex c 0 log x/n). C It is clear that this formula imlies.) for β = ϕ, ϕ. Moreover we can derive.3) from this formula by artial summation. hus the conditions of heorem.3 are satisfied and we get the desired lower bound for the k-th moment of L ur K/Q, ρ, s). Since L ur K/Q, ρ, s) and LK/Q, ρ, s) differ only by finitely many factors bounded from below on s = /), the same lower bound holds for I k K/Q, ρ, ). As a direct corollary of Proosition. we have the following unconditional imrovement of Ramachandra s heorem.7 for non-negative rational values of k. Corollary.. Let K be a degree n Galois extension of Q, and let ζ K s) be the Dedekind zeta function of K. hen for non-negative rational k we have I k ζ K, ) log ) nk. Proof. Let ρ be the regular reresentation of GalK/Q). In this case we have LK/Q, ρ, s) = ζ K s) and Artin s conjecture holds. hus the result follows since ϕ, ϕ = n, where ϕ is the character of the regular reresentation of GalK/Q). Remark.3. R. Murty [3, heorem 3.] has shown that the Selberg orthogonality conjecture imlies the Artin conjecture. hus in Proosition. we can relace the Artin conjecture with the Selberg orthogonality conjecture. 3. Proof of Proosition. We start by recalling a classical theorem on the average values of multilicative functions. Lemma 3. Levin and Fainleib). Let hn) be a comlex multilicative function for which h) α x log x as x, h) = O ) x, log x

14 4 AMIR AKBARY AND BRANDON FODDEN and hen, as x, h r ) ) rθ, for some 0 θ < /, and for all r. n x hn) e γ 0α Γα) x log x Π hx), where α is a fixed number, γ 0 = is Euler s constant and Π h x) = ) h r ) + r. r= Proof. See [9, heorem 3]. We are ready to rove Proosition.. Proof. Proof of Proosition.) With the notation of Lemma 3. we have Π h x) = + h) ) h r ) + + h) r. r= Since hn) is non-negative and h) α x log x as x, we have h) for large enough. his together with the bound given on h r ) for r imly that + h) ) Π h x) + h) ). Moreover, by emloying the bound h) for large ) we can deduce that ex h) + O) Π h x) ex h) + O). Combining h) with the revious inequality results in = α log log x + O) log x) α Π h x) log x) α. hus by alying these bounds in Lemma 3., we have 3.) xlog x) α n x hn) xlog x) α. Next note that by artial summation we have 3.) hn) n σ = σ n hn) + σ hn) n u du u σ+.

15 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 5 By alication of the uer bound of 3.) in 3.) and considering that σ we deduce that + c log hn) n σ e c log ) α + log u) α du u σ σ ) log log ) α + σ ) α e t t α dt log ) α + σ /) α log ) α. Similarly by emloying the lower bound of 3.) in 3.), and considering that σ we have + c log 0 hn) n σ σ ) α σ ) log c σ ) α e t t α dt σ /) α. he roof of the roosition is now comleted. 0 0 e t t α dt 4. Proof of heorem.5 In this section, we aim to rove heorem.5. he roof closely follows the method devised by Heath-Brown in [4]. We assume that.) holds for g, and moreover.3) and.4) hold for rational k 0. Note that from.4) follows that 4.) log ) α g k n) log ) α. n For and < t <, let wt, ) = ex t τ) ) dτ. he next lemma summarizes some roerties of wt, ) that will be used later. Lemma 4.. Let and < t <. We have wt, ) for all t and, wt, ) ex t + )) 36 for t 0 and t 3, and for all 4 3 t 5 3. wt, ) Proof. See [4].

16 6 AMIR AKBARY AND BRANDON FODDEN Define J k g, σ, ) = gσ + it) k wt, )dt. he next lemma shows that when finding a lower bound for I k g, ) it is enough to find a lower bound for J k g, /, ). Lemma 4.. We have I k g, 3 ) + e /37 J k g, /, ). Proof. By emloying.) and Lemma 4. we have J k g, /, ) = 3 g / + it) k wt, )dt g / + it) k dt 0 0 ) + + t ) ka ex g / + it) k dt + e /37. t + ))) dt 36 Let 0 < λ < 3/4π be fixed. We now define 4.) s k g, s, ) = and let S k g, σ, ) = λ Observe that, by.3), for σ / we have 4.3) s k g, s, ) λ g k n) n / g k n) n s, s k g, σ + it, ) wt, )dt. λ We find uer and lower bounds for S k g, σ, ). n η / η+/. Lemma 4.3. Let σ 3 4 and 0 < λ < 3/4π. hen λ g k n) n σ S k g, σ, ) λ g k n) n σ. Proof. Recall the following mean-value theorem for Dirichlet olynomials. We have N a n n it dt = + ξ 4π ) N N a n, 3 0 where ξ see [, Page 50]).

17 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 7 We use Lemma 4., 4.3), and the above mean value theorem for N = λ to deduce that S k g, σ, ) = s k g, σ + it, ) wt, )dt + O) λ g k n) n σ+it λ = 3 + O )) λ g k n) n σ. dt + g k n) We again use Lemma 4. and the above mean value theorem for N = λ to deduce that 5 3 λ g k n) S k g, σ, ) dt = n σ n σ+it ) λ + λξ 4π 3 λ g k n) n σ. g k n) Note that in the above formula ξ. his comletes the roof of the lemma. Next we obtain an uer bound on J k g, σ, ). Lemma 4.4. Let σ 3 4 and and. hen n σ J k g, σ, ) σ Jk g, /, ) 3 σ + e /7. he roof of this lemma is given in Section 5. Write k = u v where u and v are ositive corime integers. We define D k g, σ, ) = d k g, s, ) = g u s) s v k g, s, ) We find an uer bound for D k g, σ, ). d k g, σ + it, ) v wt, )dt. Lemma 4.5. Let σ 3 4, k = u v, and. hen D k g, σ, ) D k g, /, ) 5+η 4σ 3+η 4σ +ɛ)/v) 3+η + D k g, /, ) 7+η 8σ 3+η e σ 9+6η, where 0 η < / is given in.3) and 0 < ɛ < / η. he roof of this lemma is given in Section 5. We now use the bounds found in Lemmas 4.4, 4.5 and 4.3 to find a lower bound for J k g, /, ).

18 8 AMIR AKBARY AND BRANDON FODDEN Proosition 4.6. We have J k g, /, ) log ) α. Proof. Notice that by the definition of d k g, s, ) we have hus s k g, s, ) = s v k g, s, ) v = g u s) d k g, s, ) v gs) k + d k g, s, ) v. 4.4) S k g, σ, ) J k g, σ, ) + D k g, σ, ). We also have hus d k g, s, ) v = gs) u s v k g, s, ) v gs) k + s k g, s, ). 4.5) D k g, /, ) J k g, /, ) + S k g, /, ). We now consider two cases, and show that the lemma follows from either case. We first consider the case D k g, /, ). In this case, Lemma 4.3, 4.4) with σ = /, and 4.) imly log ) α λ g k n) n J k g, /, ) +, from which the lemma follows. We next consider the case D k g, /, ) >. hen Lemma 4.5 yields ) 4.6) D k g, σ, ) D k g, /, ) +ɛ) 4σ) v3+η) + e σ 9+6η D k g, /, ) σ v, the last line following because η < / by.3) and σ 0. We see that 4.4) followed by 4.6) and 4.5) imlies S k g, σ, ) J k g, σ, ) + J k g, /, ) + S k g, /, )) σ v. So there exists a constant ck) > 0 such that for any given σ and we have either ) 4.7) S k g, σ, ) ck) S k g, /, ) σ v or 4.8) S k g, σ, ) ck) ) J k g, σ, ) + J k g, /, ) σ v. We claim that 4.7) cannot hold for all σ and. he reason is that 4.7) together with Lemma 4.3 and.4) yield σ /) α log ) α λ λ g k n) n σ g k n) n S kg, σ, ) S k g, /, ) σ v

19 uniformly for + inequality yields LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 9 c log λ σ 3 4, λ. Letting σ = + δ log δ α e δ v in the above which cannot be true for large values of δ. hus 4.7) is false for a value δ 0, and so 4.8) must hold for σ 0 = + δ 0 log and large enough. Using these values of σ and in 4.8) and alying Lemma 4.4 yields 4.9) S k g, σ 0, ) σ 0 / J k g, /, ) 3 σ 0 + σ 0 v J k g, /, ) + e /7. hen by the lower bound for S k g, σ 0, ) given in Lemma 4.3, 4.9), and 4.), we deduce that log ) α his is the desired result. λ g k n) n S k g, σ 0, ) J k g, /, ). We are ready to rove our main result. Proof. Proof of heorem.5) We assume that.3) and.4) hold for rational k 0. Lemma 4. and Proosition 4.6 yield I k g, ) + e /3) /37 J k g, /, /3) λ /3 g k n) n log ) α. Next assume that.3) and.4) hold for real k 0 and gs) does not have any zero in the half lane σ > /. hen it is clear that Lemmas 4., 4.3, and 4.4 extend to real k 0. Moreover since gs) has no zero in the half lane σ > /, gs)) k is analytic on the half lane σ > / excet ossibly at s = ) and so Lemma 4.5 holds for u = k and v =. hus the theorem follows similar to the case for rational k. 5. Proofs of Lemmas 4.4 and 4.5 We state a result which we will utilize in the roofs of this section. Lemma 5. Gabriel). Let fs) be analytic in the infinite stri α < σ < β, and continuous for α σ β. Suose fs) 0 as Im s) uniformly for α σ β. hen for α γ β and any q > 0 we have fγ + it) q dt fα + it) q dt Proof. See [3, heorem ]. ) β γ β α ) γ α fβ + it) q β α dt. Lemma 4.4 Let σ 3 4 and. hen J k g, σ, ) σ Jk g,, ) 3 σ + e /7.

20 0 AMIR AKBARY AND BRANDON FODDEN Proof. Let fs) = s ) m gs) ex k s iτ)), γ = σ, α = /, β = 3/, q = k and / σ 3/4. We use Lemma 5. to see that 5.) 5.) Next we consider f σ + it) k dt f / + it) k dt = f / + it) k dt f 3/ + it) k dt Since e /+it iτ) e 4 e t τ), by emloying.) we have τ + Using this with 5.) yields 5.3) 3τ f / + it) k dt τ mk 3τ ) 3 σ ) σ. / + it mk g / + it) k e /+it iτ) dt. ) f / + it) k) dt e τ /5. In a similar fashion as above, we may show that 5.4) Using 5.3) and 5.4) with 5.) yields 5.5) f σ + it) k dt Moreover, since e t τ) 5.6) τ f 3/ + it) k dt τ mk. τ mk e σ+it iτ), we have g σ + it) k e t τ) dt g / + it) k e t τ) dt + e τ /5. ) 3 g / + it) k e t τ) σ dt + e τ /6. 3τ τ g σ + it) k e t τ) dt + e τ /5 τ mk f σ + it) k dt + e τ /5. 5.5) and 5.6) together imly g σ + it) k e t τ) dt g / + it) k e t τ) dt ) 3 σ + e τ /6. Integrating over τ and using Holder s inequality comletes the roof of the lemma.

21 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS In the subsequent roof, we will require the use of another theorem due to Gabriel. Lemma 5. Gabriel). Let R be a rectangle with vertices s 0, s 0, s 0 and s 0. Let F s) be continuous on R and analytic on the interior of R. hen ) / ) / F s) q ds F s) q ds F s) q ds L P P for any q 0, where L is the line segment from s 0 s 0 ) to s 0 s 0 ), P consists of the three line segments connecting s 0 s 0 ), s 0, s 0 and s 0 s 0 ), and P is the mirror image of P in L. Proof. See [3, heorem ]. We also need the following mean-value theorem for Dirichlet series. Lemma 5.3 Montgomery-Vaughn). If n a n <, then a n n it dt = a n + On)), 0 the imlied constant being absolute. Proof. See [, Corollary 3]. We are ready to rove Lemma 4.5. Lemma 4.5 Let σ 3 4, k = u v, and. hen D k g, σ, ) D k g, /, ) 5+η 4σ 3+η 4σ +ɛ)/v) 3+η + D k g, /, ) 7+η 8σ 3+η e σ 9+6η, where 0 η < / is given in.3) and 0 < ɛ < / η. Proof. Using Lemma 5. with fs) = d k g, s, ) ex v s iτ)), γ = σ, α = /, 3/4 < β <, and q = /v, we get 5.7) By.) and 4.3), we have f σ + it) v dt f / + it) v dt ) β σ β / ) σ / f β + it) v dt 5.8) d k g, s, ) v + t Au + t A) u+v. hus 5.8) yields β /. 5.9) f β + it) v dt = 3τ + O τ +k) τ f β + it) v dt + 3τ ) + t A ) +k e t τ) ) dt ).

22 AMIR AKBARY AND BRANDON FODDEN We observe that τ 5.0) + 3τ ) + t A Putting 5.9) and 5.0) together, we have 5.) f β + it) v dt = 3τ τ ) +k e t τ) ) dt e τ 5. f β + it) v dt + O +k) e τ 5 We will now use Lemma 5. with F s) = f s + β + iτ), s 0 = β / + /)iτ and q = v. his allows us to avoid the ossible ole of gs). On one hand, we have 3τ 5.) F s) q ds = f β + it) v dt. We also have 5.3) F s) q ds = P and 5.4) L 3τ + P F s) q ds = τ τ f β / + it) v dt β / β 3τ + τ β f µ + iτ ) v + f f / + it) v dt f µ + iτ ) v + f ). µ + 3 ) ) iτ v dµ µ + 3 ) ) iτ v dµ. However, 5.8) yields f µ + ) iτ = d k g, µ + ) iτ, v ex µ ) ) 5.5) iτ + τ A) u+v e vτ /8 and similarly 5.6) f µ + 3 ) iτ + τ A) u+v e vτ /8. hus 5.5) and 5.6) show that the second integrals in 5.3) and 5.4) are +k) e τ /5. Using this, along with Lemma 5., 5.), 5.3) and 5.4), we have 5.7) 3τ τ ) / f β + it) v dt f / + it) v dt 3τ τ f β / + it) v dt ) / + +k) e τ /. From 5.) and 5.7) together with 5.7) and the definition of fs) we have

23 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 3 5.8) d k g, σ + it, ) v e t τ) dt ) β σ/ /4 d k g, / + it, ) v e t τ) β / dt 3τ τ d k g, β / + it, ) v e t τ) dt + d k g, / + it, ) v e t τ) dt +k) e τ / ) σ / β /. ) β σ β / ) σ/ /4 β / Integrating 5.8) for τ and alying Hölder s inequality yields 5.9) D k g, σ, ) D k g, /, ) β σ/ /4 β / Since wt, ), we have 3τ τ d k g, β / + it, ) v e t τ) dtdτ + D k g, /, ) β σ ) σ/ /4 β / β / e σ β ). 3τ τ d k g, β / + it, ) v e t τ) dtdτ 3 d k g, β / + it, ) v dt. Now choose β > 3/4 so that β / >. By this choice of β, d k g, β / + it, ) is reresented by an absolutely convergent Dirichlet series. Also note that hus, for σ >, we have g u n) = g kv n) = n=n n v g k n ) g k n v ). 5.0) d k g, s, ) = g u s) s v k g, s, ) λ ) v g u n) g k n) = n s n s = λ an) n s

24 4 AMIR AKBARY AND BRANDON FODDEN where 5.) an) = g u n) g k n ) g k n v ) n=n n v n j λ for j=,...,v = g k n ) g k n v ) n=n n v j such that n j >λ g k n ) g k n v ) n=n n v n η τ v n) n η+ɛ 0. In the last inequality ɛ 0 is a ositive constant that will be chosen aroriately later. hus, 5.0) and 5.) allow us to use Lemma 5.3 to get 3 5.) d k g, β / + it, ) an) ) 5 dt = n 4β + On) λ λ η+ɛ 0) 4β+3. t η+ɛ 0 4β+ dt + λ t η+ɛ 0 4β+ dt he last inequality holds if β > η + ɛ 0 )/ + 3/4. We now use Holder s inequality and 5.) to see that 5.3) 3 d k g, β / + it, ) v dt 5 ) v η+ɛ 0) 4β+3 ) v +η+ɛ 0) 4β+)/v. Alying the bound 5.3) in 5.9) and choosing ɛ 0 = /)/ η) ɛ/ and β = 7/8) + /4)η comletes the roof of the lemma. Acknowledgements he authors would like to thank Nathan Ng for several helful discussions related to this work and for his comments on an early draft of this aer. References [] V. Chandee, On the correlation of shifted values of the Riemann zeta function, o aear in the Quarterly Journal of Mathematics, 4 ages. [] H. Iwaniec and E. Kowalski, Analytic Number heory, Colloquium ublications American Mathematical Society); v. 53, 004. [3] R. M. Gabriel, Some results concerning the integrals of moduli of regular functions along certain curves, J. London. Math. Soc. 97), 7. [4] D. R. Heath-Brown, Fractional moments of the Riemann zeta-function, J. London. Math. Soc. ) 4 98), [5] G. Ji, Lower bounds for moments of Automorhic L-functions over short intervals, Proc. Amer. Math. Soc ), [6] A. Kacenas, A. Laurincikas, S. Zamarys, On fractional moments of Dirichlet L-functions. Russian), ranslation in Lithuanian Math. J ), [7] H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, Aendix to H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL, J. Amer. Math. Soc ),

25 LOWER BOUNDS FOR POWER MOMENS OF L-FUNCIONS 5 [8] J. Lagarias and A. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic Number Fields, A. Fröhlich ed.), 977), [9] B. V. Levin and A. S. Fainleib, On a method of summation of multilicative functions. Russian), Izv. Akad. Nauk SSSR Ser. Mat ), [0] A. Laurincikas and J. Steuding, On fractional ower moments of zeta-functions associated with certain cus forms, Acta Al. Math ), [] H. L. Montgomery, oics in multilicative number theory, Lecture Notes in Math. 7, Sringer Verlag, 97. [] H. L. Montgomery and R. C. Vaughan, Hilbert s inequality, J. London Math. Soc. ) 8 974), [3] M. R. Murty, Selberg s conjectures and Artin L-functions, Bull. Amer. Math. Soc ), 4. [4] K. Ramachandra, Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series-i, Hardy-Ramanujan J. 978), 5. [5] K. Ramachandra, Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series-ii, Hardy-Ramanujan J ), 4. [6] Z. Rudnick and P. Sarnak, Zeros of rincial L-functions and random matrix theory, Duke Math. J ), [7] Z. Rudnick and K. Soundararajan, Lower bounds for moments of L-functions, Proc. Natl. Acad. Sci. USA 0 005), [8] E. C. itchmarsh, he theory of functions, second edition, Oxford University Press, 939. [9] E. C. itchmarsh, he theory of the Riemann Zeta-function, Clarendon Press, Oxford, 95. [0] S. Zamarys, On fractional moments of twists of L-functions attached to cus forms, Siauliai Math. Semin. 009), 3. Deartment of Mathematics and Comuter Science, University of Lethbridge, Lethbridge, AB K 3M4, Canada address: amir.akbary@uleth.ca Deartment of Mathematics and Comuter Science, University of Lethbridge, Lethbridge, AB K 3M4, Canada address: brandon.fodden@uleth.ca