Zeros of Derivatives of Sum of Dirichlet L-Functions

Transcription

1 Zeros of Derivatives of Sum of Dirichlet L-Functions Haseo Ki and Yoonbok Lee Deartment of Mathematics, Yonsei University, Seoul 0 749, Korea haseo@yonsei.ac.kr leeyb@yonsei.ac.kr July 7, 008 Abstract A linear combination L(s) of two Dirichlet L-functions has infinitely many comlex zeros in Re s < 0. In this note we rove an infinity of comlex zeros of L (k) (s) in the same reion. Introduction We define Dirichlet s L-functions (mod 5) by L j (s) = n= χ j(n)n s where χ 0 is the rincial character, χ 3 is non rincial real character, and χ () = i, and χ () = i. All these Dirichlet characters are comletely multilicative and have the eriod 5. We denote a Dirichlet series L(s) by L(s) = sec θ ( e iθ L (s) + e iθ L (s) ), where θ is the root of sin π + tan θ sin 4π = 5 between 0 and π/4. Then we have the 5 5 functional equation ( 5 π ) s Γ ( + s ) L(s) = ( ) 5 s Γ ( ) π s L( s) (.) This work was suorted by the Korea Science and Enineerin Foundation (KOSEF) rant funded by the Korea overnment(most) (No. R )

2 and the followin theorem in [8, 0.5]. Theorem A. L(s) has an infinity of zeros in the half-lane Re s >, and that the number of such zeros between 0 and T is reater than AT as T. The roof of Theorem A follows from the method due to Davenort and Heilbronn [3]. Thus, an analoue of the Riemann hyothesis(rh) for L(s) is false. By (.) Theorem A also holds for the half-lane Re s < 0. Seiser [7] roved that RH is equivalent to the derivative of the Riemann zeta function ζ (s) havin no zeros in 0 < σ < /, and Levinson and Montomery [5] showed that under RH, ζ (k) (s) has at most a finite number of comlex zeros in σ < / for k. For a quantitative behavior for zeros of ζ(s), Selber [6] roved ( β ) = O(T ). T γ T β>/ On the other hand, Levinson and Montomery [5] showed ( π β (k) ) T γ (k) T = kt lo lo T π + T lo lo + O ( ) T, lo T where β (k) +iγ (k) denote the comlex zeros of the kth derivative of ζ(s). Note that the number of zeros in 0 < Im s < T for ζ (k) (s) (k 0) is aroximately T lo T (see [8, Theorem 9.4] π and []). Thus, we easily observe that zeros of the derivatives of ζ(s) tend to move to the riht whenever we differentiate the derivatives. Based on this roensity for the derivatives, Levinson [4] was able to rove at least /3 of zeros of ζ(s) are on the critical line, and Conrey [] imroved it u to /5. Theorem. Let k be a nonneative inteer. Then, L (k) (s) has an infinity of zeros in the halflane Re s < 0, and that the number of such zeros between 0 and T is reater than AT as T. Note that the roof of Theorem follows from the same method as in that of Theorem A. As contrasted with the behavior of zeros of the derivatives of ζ(s), Theorem rovides an interestin asect of zeros of the derivatives of a Dirichlet series for we enerally exect that comlex zeros of the derivatives of a Dirichlet series move to the riht whenever we differentiate them.

3 3 Proof of the Theorem We define α by α() = ( + i)χ () + ( i)χ () for any rime, and α(mn) = α(k)α(n) for any ositive inteers m, n. Let α(n)χ j (n) M j (s) = and N(s) = n s sec θ ( e iθ M (s) + e iθ M (s) ). n= Since M (s) = ( α()χ () s ), we have lo M (σ) = for σ >. Hence we et = + i = + i = i α()χ () χ () N(σ) = sec θre [ e iθ M (σ) ] = χ 3 () + O() + i + i lo(σ ) + O() χ ()χ () χ 0 () + O() + O() sec θ ( ) cos lo(σ ) + O() e O() σ Thus N(σ) has zeros at σ = + e (n+)π+o() for each inteer n. Let s > be a real zero of N(s). Choose a small value 0 < η < s such that N(s) 0 for s s = η. Let ɛ = inf s s =η N(s) > 0, and 0 < δ < s η. Lemma. Let D be the differential oerator d ds and l be a nonneative inteer. Let ɛ > 0. Then, for some A > 0, there are at least AT values of t in the oen interval (0, T ) such that D l [L(s + it) N(s)] < ɛ for Re (s) + δ and Im (s) η. This lemma follows from [8, 0.5] for l = 0, and by Cauchy s interal formula for the ositive inteer l. Lemma. On any fixed vertical stri a Re (s) b as t, we have ( ) ψ(s) = Γ Γ (s) = lo t + O() and ψ(m) (s) = O (m =,, 3,...). t m

4 4 Stirlin s formula for the Gamma function imlies Lemma. We set (s) = ( ) s 5 Γ(s) sin πs 5 π. By (.), we have L( s) = (s)l(s). Then, we obtain (j) (s + it) = loj t + O(lo j t) (j =,, 3,...). (.) To rove (.), we use Lemma and induction. From Lemma and the formula 5 (s + it) = lo π + ψ(s + it) + π cos π(s + it)/ sin π(s + it)/, we have the case j =. Now, suose (.) is true for j n. Then, we have [ ] (n+) (s + it) = (n) (n) (s + it) (s + it) + D (s + it) = lo n+ t + O(lo n t) by the induction assumtion and Cauchy s Interal formula. Thus we rove (.). By Lemma and (.), we have D k [(s + it)(l(s + it) N(s))] = D k [(s + it)n(s)] (s + it)n(s) k j=0 ( ) k j D j [(s + it)]d k j [L(s + it) N(s)] ɛ lo k t (s + it) ( + O(lo t)); k = (k) (s + it) + j= = lo k t + O(lo k t) ( ) k (k j) (s + it) N (j) j N (s) on s s = η. By these and takin ɛ < ɛ/ in Lemma, we have on s s = η ( ) k L (k) ( s it) lo k t(s + it)n(s) = D k [(s + it){l(s + it) N(s)}] + D k [(s + it)n(s)] lo k t(s + it)n(s) < lo k t (s + it)n(s) for sufficiently lare t. Thus, by Rouché s theorem and Lemma, there are at least AT values of t in (0, T ) such that L( s it) and (s+it)n(s) has the same number of zeros in s s < η. Since s is a zero of N(s) there, L(s) has at least one zero in s ( s it) < η. Hence, we comlete the roof of Theorem. Acknowledment. We thank to D. Farmer for encourain us to this work.

5 5 References [] B. C. Berndt, The number of zeros for ζ (k) (s), J. London Math. Soc.() (970), [] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. reine anew. Math. 399 (989), 6 [3] H. Davenort and H. Heilbronn, On the Zeros of Certain Dirichlet Series I, II, J. London Math. Soc., (936), 8 85 and 307 [4] N. Levinson, More than one-third of zeros of Riemann s zeta function are on σ =, Adv. Math. 3 (974), [5] N. Levinson and H. L. Montomery, Zeros of the Derivatives of the Riemann Zetafunction, Acta Math., 33 (974), [6] A. Selber, Contributions ot the Theory of the Riemann zeta-function, Arch. for Math. o Naturv. B., 48 (946), No. 5, [7] A. Seiser, Geometrisches zur Riemannschen Zetafunktion, Math. Ann. 0 (935), 54 5 [8] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, nd ed., revised by D. R. Heath-Brown, Oxford University Press, Oxford, 986