1. Preliminaries. In [8] the following odd looking integral evaluation is obtained.

Transcription

1 June, 5. Revied Augut 8th, 5. VA DER POL EXPASIOS OF L-SERIES David Borwein* and Jonathan Borwein Abtract. We provide concie erie repreentation for variou L-erie integral. Different technique are needed below and above the abcia of abolute convergence of the underlying L-erie.. Preliminarie. In [8] the following odd looking integral evaluation i obtained. ) 3 co t log ) ζ / + it) t dt π log. ) + /4 Thi identity turn out formally to be a cae of a rather pretty, and perhap ueful, cla of L-erie evaluation given in Theorem and Corollary cf. []). In Theorem 3 we recover ) entirely rigorouly. Given a Dirichlet erie λ) : n λ n, + iτ, R >, n we conider the integral ι λ ) : λ) dτ a a function of λ. Oberve that when the coefficient λ n are real ι λ ) λ) dτ, but that thi i not necearily o when the coefficient are complex. We refer to [3, 5, 7] for other, largely tandard detail.. Integral Involving with Large Real Part. It i convenient to recall [] that, for u, a >, co at) t + u dt π u e au. ) *Department of Mathematic, Univerity of Wetern Ontario, London, Ontario, 6A 5B7. Reearch upported by SERC. dborwein@uwo.ca Faculty of Computer Science, Dalhouie Univerity, Halifax S, B3H W5. Reearch upported by SEC, the Canada Foundation for Innovation and the Canada Reearch Chair Program. jborwein@c.dal.ca 99 Mathematic Subject Claification. M35, M4, 3B5. Key word and phrae. Dirichlet erie integral, Hurwitz zeta function, Plancherel theorem, L-erie.

2 DAVID BORWEI AD JOATHA BORWEI Theorem. For a) : n a nn, b) : n b nn, and + i τ with fixed R) > uch that both Dirichlet erie are abolutely convergent, we have ι a,b ) : a) b) + τ dτ π A n B n A n B n n, 3) n where A n : n k a k, B n : n k b k, A : B :. Proof. Let a ) : n a nn, b ) : n b nn. Then, in view of ), we have a )b ) + τ dτ + π π n>m> n>m> n>m> n>m> n a n b m nm) a m b n nm) + i n,m> a n b m n +iτ m iτ + τ dτ co τ logn/m) ) + τ dτ a n b m + a m b n nm) co τ logm/n) ) m,n> a n b m + a m b n nm) n/m) + π + τ dτ + a n b m nm) a n B n + A n b n + a n b n n co τ logn/m) ) + τ n π a n b n n n n a n b n n in τ logn/m) ) + τ dτ + π n dτ + τ dτ a n b n n A n B n A n B n n. ote that the imaginary part in the above evaluation vanihed becaue we integrated an odd function over the range < τ <. ext, we oberve that a )b ) where M i independent of τ. Hence n a n b n n M <, a )b + τ M + τ, and 3) follow by Lebegue theorem on dominated convergence on letting. A an immediate conequence we have:

3 EXPASIOS OF L-SERIES 3 Corollary. If λ) : n λ nn with + i τ and fixed R) > uch that the Dirichlet erie i abolutely convergent, then ι λ ) : λ) dτ π Λ n Λ n n, 4) n where Λ n : n k λ k, Λ :. If, in addition, all the coefficient λ n are real, then ι λ ) λ) dτ π n Λ n Λ n n. 5) ote that, by Dirichlet tet [9], the final erie in 5) i convergent for all > when Λ n i bounded, but we cannot automatically guarantee that it i equal to the integral in in thi cae, or even that the integral i finite. Simple continuation argument will not work. Of coure, imilar difficultie arie with regard to 3) and 4). Thi in part motivate the firt example and the following ection. It i, however, eay now to check that a, b : a) b) + τ dτ define an extended-value inner product on the pace of Dirichlet erie with α, α ι α ), which i typically finite for large enough. In the equel, we let L µ ) : n µ n )n denote the primitive L-function correponding to the Kronecker ymbol µ n ), [3]. Below, x i the integer part and x i the truncation of x, o that x { x when x, x when x <. A uual, {x} : x x denote the fractional part of x. Example. For the Riemann zeta function ζl ), and for >, Corollary applie and yield π ι ζ) ζ ) ζ), a λ n and Λ n n. By contrat it i known ee equation 69) of [6] that on the critical line + iτ π ι ζ ) log π) γ. More broadly, for < <, Crandall recorded in [6]) ha found that ι ζ ) π n θ dθ π n + θ) + θ ζ +, θ) dθ, 6)

4 4 DAVID BORWEI AD JOATHA BORWEI where ζ, a) : n + a) n i the Hurwitz zeta function which i eay to compute. Thi devolve from van der Pol repreentation ζ) e ω e ω e ω ) e iτω dω, + iτ with < <. 7) One way to obtain identity 7) i to note that e ω e ω e ω ) dω t t t ) dt θ ζ +, θ) dθ lim on integrating once by part. ow et : n n n n n ) n n n+ t t t ) dt θ θ + n) dθ and oberve that we have hown that converge to ome number. Further n n n) ) n+ n n n ) ζ), which implie that ζ), a required. Another way to etablih 7) kindly uggeted by the David Bradley) i to tart with the tandard identity ζ) x x x ) dx which can eaily be derived under the retriction R > and then extended by analytic continuation to the punctured half-plane {, R >, }. Thi together with the identitity x x x ) dx which i valid for R < yield 7) via the change of variable x e ω. We now write 7) a a Fourier tranform: ζ) F τ e ω {e ω } )

5 and o obtain, for < <, ι ζ ) EXPASIOS OF L-SERIES 5 ζ) from the L Plancherel theorem [9]. It follow that e ω e ω e ω dω n+ n n θ n dτ π t t t dt t t t dt θ + n) dθ, e ω e ω e ω dω n θ θ + n) dθ a required to yield 6). ote that all term are abolutely convergent which legitimate the operation. We have alo etablihed, inter alia, that, for < <, π θ ζ +, θ) dθ dτ θ ζ +, θ) dθ. Moreover, revering the order of integration and ummation above lead to ι ζ ) π n π ι ζ) θ ) π ζ ) dθ + ζ) n + θ) + which in the limit a recapture the evaluation quoted above. Recapitulating, we have ζ ) ζ), < < ; 8) ζ) + ζ ), < <. There are imilar formulae for ζ k) with k integral. applying the reult in 5) with ζ : t ζt + ) yield π ζ3/ + iτ) /4 + τ dτ π ι ζ on uing Euler reult ee [4]) that ζ, ) : ) ζ, ) + ζ3) 3 ζ3), n n n k k ζ3). For intance, Example. For the alternating zeta function, α : )ζ), we recover, a in [7], that π ι α) α), a λ n ) n+, Λ n ) n )/ and Λ n Λ n ) n+ /.

6 6 DAVID BORWEI AD JOATHA BORWEI Set : /. Then α/ + it) /4 + t / i t ζ/ + i t) /4 + t 3 ) ζ/ + i t) co t ln )) /4 + t i preciely the integrand in ). Thu, ince α) log we ee that ι α /) π log, i the evaluation in [7]. ote that to jutify the exchange of um and integral implicit in 5) we hould have to more carefully analye the integrand, ince / i below the abcia of abolute convergence of the erie, and note that thi would not have been legitimate in Example becaue of 8). Thi motivate the approach in the ection below on the Hurwitz zeta function. Example 3. a) For the Catalan zeta function βl 4 ), and for > : π ι β) β), a λ n, λ n+ ) n and again Λ n Λ n λ n. b) For L 3, the ame pattern hold, in that π ι L 3 ) L 3), but not for L 5, L 7, and o on. c) In general the erie L ±d doe not lead to output which i again a primitive L-erie modulo d. For example, π ι L 5 ) 5 n ) n mod 5 n, and π ι L 8 ) L 8 ) L 4). Thee are not character um, though alway the coefficient repeat modulo d. d) Finally, let ϑ : + ). Then again π ι ϑ) ϑ). For each of thee example, which are dicued further in [], the defect of Theorem and Corollary i that, a we have een, they only directly apply when i large enough. The van der Pol approach offer a nice alternative, epecially in the critical trip. 3. Van der Pol Approach to Hurwitz Zeta. Recall that ζ, a) : n n + a), a >, i initially defined for R > and appropriately analytically continued for R <. Thu, ζ) ζ, ). We firt work out a Fourier tranform repreentation for ζa, ). Propoition. For a > and > R >, we have e ω e ω e ω a ) dω ζ, a + ). 9)

7 EXPASIOS OF L-SERIES 7 Proof. i) Oberve that the formula ue the truncation of e w a and not the floor, and that the two only differ in the interval < w < log a. We have log a e ω {e ω a} dω n n++a n+a a t t a t a ) dt t t a t a ) dt θ ζ +, θ + a) dθ lim n n θ θ + n + a) dθ ) n + a) + a) a ζ, a + ) a ), ) thi evidently being true for R > and o for < R < by analytic continuation both ide of ) clearly being analytic for R >. Incidentally, we have hown that ) i valid for R >, a >. The additional retriction R <, a i needed for the next part of the proof. It follow from ) that e ω e ω e ω a ) dω log a e ω {e ω a} + a) dω + log a a ζ, a + ) ) + a + a ζ, a + ), e )ω dω and thi etablihe 9). ii) Another proof of equation ) uggeted by the David Bradley) i to proceed from Theorem. in Apotol [, p. 69] which tate that for < a, R >,,,..., ζ, a) : n + a) {x} dx. ) n + a) x + a) + Putting in ) and making the change of variable x e ω a give equation ). Propoition. For R > and a, we have, a) : lim n Another form of the limit i, a) ) + a) ζ, a + ). n + a) tζ +, a + t) dt,

8 8 DAVID BORWEI AD JOATHA BORWEI and indeed, ) ζ). Proof. The cae a > follow immediately from the proof of ) in part i) above, and the cae a wa treated in the earlier proof of 7). We can alo derive the cae a > by letting in ) and oberving that ζ, a) a ζ, a + ). Mot of what follow concern Dirichlet erie having coefficient which repeat modulo. Theorem. Let λ) : n λ nn, + iτ, with coefficient, λ n, repeating modulo i.e., λ +n λ n ). Then, for < <, e ω Proof. Oberve firt that λ) k m k λ k e ω e ω ) + k λ k m + k) k e iτω dω λ). ) λ k ζ, k ) ; 3) trictly thi i true for > in the firt place and the equating of the extreme term for > follow by analytic continuation. It follow from 9) that, for k,,...,, e ω e ω e ω k ) e iτω dω ζ, k ) k. We now change variable w ω log to obtain e e ω ω e ω ) k e iτω dω ζ, k ) k, o that ince e e ω ω e ω + k e ω ) + k e iτω dω ζ, k ) e ω { k when w log k, when w < log k. Summing 4) for k,,...,, and applying 3) yield ), a deired. We are now in poition to prove the following companion to Theorem in which we ue the notation: given a Dirichlet erie λ) : n λ nn with coefficient repeating modulo, we define an aociated kernel W λ, t) : k λ k t + k. 4)

9 EXPASIOS OF L-SERIES 9 Theorem 3. Suppoe that the coefficient of the two Dirichlet erie a) : n a nn, b) : n b nn repeat modulo, that A n : n k a k, B n : n k b k, A : B :, and that ι a,b ) : a) b) dτ with + iτ. + τ Suppoe further that A B. Then, for < <, ι a,b ) Moreover, for all >, π ι a,b ) π ζ +, t) W a, t) W b, t) dt. 5) n A n B n A n B n n. 6) Proof. Suppoe firt that > >. Since A B, it follow from Theorem that a) e e ω ω + k a k e iτω dω, k with a correponding formula for b)/. Hence, by the L Plancherel theorem [9], π π a) b) + τ dτ n π π e ω k e ω ) + k e ω ) + k a k b k dω k u n + u) + a k + k ) k u b k + k ) du k ζ +, t) a k t + k ) k b k t + k ) dt ζ +, t) W a, t) W b, t) dt. Thi etablihe 5). We now denote the characteritic function of the interval k/, k + )/) by χ k and oberve that, ince A B, we can ue ummation by part to re-expre the kernel a follow: W a, t) k k A k χ k t), W b, t) k B k χ k t). 7)

10 DAVID BORWEI AD JOATHA BORWEI Thu, Conequently, by 5), a) b) + τ dτ W a, t)w b, t) π π π k n k n k k+ A k B k k k A k B k χ k t). ζ +, t)dt ) A k B k n + k) n + k + ) A k B k A k B k n + k) π m A m B m A m B m m, and thi how that 6) hold when < <. By Theorem, 6) alo hold when >, ince the Dirichlet erie defining a) and b) are abolutely convergent in thi range. Further, ince A n B n A n B n i bounded, by Dirichlet tet, the erie in 6) i abolutely convergent and thu continuou a a function of for >. Finally, it i eay to how, by mean of partial ummation, that a) and b) are analytic in the dik {, < 4 }, and hence bounded therein by a contant M, ay. It follow, by Lebegue theorem on dominated convergence, that ι a,b ) i continuou for 3 4 < < 5 4, and hence that 6) hold for all >. A an immediate conequence we have the following companion to Corollary. Corollary. Suppoe that the coefficient of the Dirichlet erie λ) : n λ nn repeat modulo, and that Λ n : n k λ k, Λ :. Suppoe further that Λ. Then, for + iτ with >, we have ι λ ) : λ) dτ π n where Λ n : n k λ k, Λ :. If, in addition, all the coefficient λ n are real, then ι λ ) λ) dτ π n Λ n Λ n n, Λ n Λ n n. ote that for, λ λ we reobtain from Corollary, a rigorou form of the original evaluation in the MAA Monthly [8]. Example 4. Recall that ζu, v) : ) n n n u n k k v, while ζu, v) : ζu, v) : n n u n k ) n n n u ) k, and k v n k ) k k v.

11 and π EXPASIOS OF L-SERIES The ame approach a in Example applied to 5) and 6) produce π a companion to α3/ + iτ) α3/ + iτ) /4 + τ dτ ζ, ) + ζ3) 3 ζ) log 9 4 ζ3), α3/ + iτ) ζ3/ + iτ) /4 + τ dτ ζ, )+ζ, )+α3) 9 8 ζ) log 3 4 ζ3), π ince, by technique dicued in [4,5], ζ3/ + iτ) ζ3/ + iτ) /4 + τ dτ 3 ζ3), ζ, ) 8 ζ3), ζ, ) ζ3) 3 ζ) log, ζ, ) 3 ζ) log 3 8 ζ3). 4. Final Remark. Many other imilar reult obtain. For example: n++a e ω {maxe ω a, )} dω t {t a} dt n n+a n θ θ + n + a) dθ θ ζ +, θ + a) dθ. While 4) and 6) give an effective way of evaluating the integral, directly evaluating the integral numerically to high preciion preent a greater challenge. Thi i largely becaue of the evere ocillation of the integrand. The iue appear to lie in etimating the integrand well and o i intrinically non-trivial. Acknowledgement. We hould like to thank David Bradley for the many ueful uggetion he made during the completion of thi reearch. Reference. Tom M. Apotol, Introduction to Analytic umber Theory, Springer-Verlag, ew York, Jonathan Borwein, A cla of Dirichlet erie integral, MAA Monthly in pre, Jonathan Borwein and David Bailey, Mathematic by Experiment. Plauible Reaoning in the t Century, AK Peter, Jonathan M. Borwein and David M. Bradley, On two fundamental identitie for Euler um, February, Jonathan Borwein, David Bailey and Roland Girgenohn, Experimentation in Mathematic. Computational Path to Dicovery, AK Peter, Jonathan M. Borwein, David M. Bailey and Richard E. Crandall, Computational trategie for the Riemann zeta function, J. Comp. and Appl. Math, pecial volume umerical Analyi in the th Century. Vol. : Approximation Theory. ), [CECM Preprint 98:8]. 7. Godfrey H. Hardy and Edward M. Wright, Introduction to the Theory of umber, Oxford Univerity Pre, Oxford, Solution to MAA Problem 939, propoed by Ivić, with hi olution, MAA Monthly, ovember 3), Karl R. Stromberg, Introduction to Claical Real Analyi, Wadworth, 98.