On the Modulus of the Selberg Zeta-Functions in the Critical Strip

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1 Mahemaical Modelling and Analysis Publisher: Taylor&Francis and VGTU Volume X Number x, xx 0xx, 5 hp:// hp://dx.doi.org/0.3846/3969.xxxx.xxxxxx ISSN: c Vilnius Gediminas Technical Universiy, 0xx eissn: On he Modulus of he Selberg Zea-Funcions in he Criical Srip Andrius Griguis a and Darius Šiaučiūnas b a Faculy of Mahemaics and Informaics, Vilnius Universiy Naugarduko sr. 4, LT-035 Vilnius, Lihuania b Insiue of Informaics, Mahemaics and E-Sudies, Šiauliai Universiy P. Višinskio sr. 9, LT-7756 Šiauliai, Lihuania corresp.: siauciunas@fm.su.l andrius.griguis@mif.vu.l Received June 5, 05; revised June 5, 05; published online Mxx, 0xx Absrac.We invesigae he behavior of he real par of he logarihmic derivaives of he Selberg zea-funcions Z PSL,Z s and Z Cs in he criical srip 0 < σ <. The funcions Z PSL,Z s and Z Cs are defined on he modular group and on he compac Riemann surface, respecively. Keywords: zea-funcion, criical srip. Selberg zea-funcion, modular group, compac Riemann surface, Riemann AMS Subjec Classificaion: M36. Inroducion Le s = σ + i denoe a complex variable. We sar wih definiion and some properies of he Riemann zea-funcion. For σ >, he Riemann zea-funcion is given by a he series ζs = n= n s, and can be analyically coninued o he whole complex plane, excep for a simple pole a s = wih residue. Trivial zeros of ζs are locaed a he negaive even inegers. The remaining, he so-called nonrivial zeros, lie in he criical srip 0 < σ <. The Riemann zea-funcion saisfies he funcional equaion ζs = s π s ζ sγ s sin πs or ξs = ξ s,

2 A. Griguis and D. Šiaučiūnas where ξs = ss π s/ Γ s/ζs, where Γ s denoes he Euler gamma-funcion. The funcion ξs is an enire funcion whose zeros are he nonreal zeros of ζs, see [0, II]. In he paper [], i was proved he following relaion beween funcions ζs and ξs. Theorem. The funcions ζ and ξ saisfy, for 8 and σ < /, he inequaliy Re ζ s ζs < s Reξ ξs. Sondow and Dumirescu proved in [8] he following heorem for he funcion ξ. Theorem. The xi funcion is increasing in modulus along every horizonal half-line lying in any open righ half-plane ha conains no xi zeros. Similarly, he modulus decreases on each horizonal half-line in any zero-free, open lef half-plane. In he same paper, he following reformulaion for he Riemann hypohesis was given. Theorem 3. The following saemens are equivalen. I. If is any fixed real number, hen ξσ + i is increasing for / < σ <. II. If is any fixed real number, hen ξσ+i is decreasing for < σ < /. III. The Riemann hypohesis is rue. Theorem 3 laer was reproved in [] in a slighly differen way. Relaed properies of funcions ζs and ξs in he criical srip were also invesigaed in [6]. In his paper, we ask wheher Selberg zea-funcions have similar properies as he Riemann-zea funcion has in Theorems - 3. Noe ha, for Selberg zea-funcions, he analogue of he Riemann hypohesis is usually valid. We consider Selberg zea-funcions for cocompac and modular subgroups. Le H be he upper half-plane, and Γ be a subgroup of PSL, R. Le Γ \H be a hyperbolic Riemann surface of finie area. The Selberg zea-funcion Zs is defined [6], for σ >, by Zs = {P } NP s k, k=0 where {P } runs rough all he primiive hyperbolic conjugacy classes of Γ, and NP = α if he eigenvalues of P are α and α, α >. The Selberg zea-funcion has a meromorphic coninuaion o he whole complex plane, C [6]. If Γ \H is a compac Riemann surface of genus g, we use he noaion Zs = Z C s.

3 On he Modulus of he Selberg Zea-Funcions in he Criical Srip 3 If Γ = PSL, Z, hen we denoe Zs = Z PSL,Z s. Similarly, as he Riemann zea-funcion, he Selberg zea-funcion Z PSL,Z s has a meromorfic coninuaion o he whole complex plane, and saisfies he symmeric funcional equaion [9] where and Ξs = Ξ s,. Ξs = Z PSL,Z sz id sz ell sz par s, π s /6 Z id s = Γ s /3, Γ s s / / s + s /3 /3 s + Z ell s = Γ Γ Γ Γ,. 3 3 π s Z par s = Γ sζsγ s + / s. The funcion Γ s is called he double Barnes gamma-funcion, and is defined by he canonic produc Γ s + = π s exp { s γ 0 + s } k= { + s } k exp s + s,.3 k k where γ 0 denoes he Euler consan. The funcion Γ s saisfies he relaions Γ =, Γ s + = Γ s Γ s, Γ n + = n n! n, see, for example [], [7] or []. The funcion Ξs is an enire funcion of order and has zeros a he poin s = / + ir n n 0, where r n = λ n 4, and 0 = λ 0 < λ λ λ 3 are he eigenvalues of he Laplace operaor [4], [8]. The funcion Z PSL,Z s has poles and zeros a he following poins [7]: Poles of Z PSL,Z s: s = 0; order, s = / k, k 0; order. Zeros of Z PSL,Z s: s = ; order, s = 6k j, k 0, j =,, 3, 4, 6; order k +, Mah. Model. Anal., Xx: 5, 0xx.

4 4 A. Griguis and D. Šiaučiūnas 3 s = 6k 5, k 0; order k + 3, 4 s = ρ/, where ρ are non-rivial zeros of ζs, 5 s = / ± ir n, n 0. We prove he following heorem. Theorem 4. There exiss a posiive number C such ha, for > C and 0 < σ < /, Re Ξ s Ξs < 0. Furher more, if we assume he Riemann hypohesis for ζs, hen here exiss a posiive number C such ha Re Z PSL,Z s Z PSL,Z s < s ReΞ Ξs for > C and 0 < σ < /4. Conversely, if Re Z PSL,Z s Z PSL,Z s < 0 for > C and 0 < σ < /4, hen he funcion ζs, for > C, has zeros only for σ = /. Theorem 4 is proved in he nex secion. Below we formulae a couple of corollaries of Theorem. We also wan o menion ha he par of asserion of Theorem can be obained following he proof of Theorem 6. in [3]. Corollary. For a fixed sufficienly large, he funcion Ξσ+i is decreasing for 0 < σ < /, and is increasing for / < σ < wih respec o σ. Corollary. If he Riemann hypoesis is rue for ζs, hen, for a sufficienly large fixed, he funcion Z PSL,Z σ + i is decreasing for 0 < σ < /4 wih respec o σ. Proofs of Corollaries and follows from Lemma, funcional equaion Ξs = Ξ s and equaliy Ξs = Ξs. We urn o Selberg zea-funcions aached o compac Riemann surfaces. The funcion Z C s is an enire funcion of order [5,.4, Theorem.4.5] and saisfies a funcional equaion [5,.4, Theorem 4.] Z C s = fsz C s,.4 where fs = exp 4πg s / 0 v anπv dv. The above funcional equaion is equivalen o Ms = M s, where Ms = Z C s exp πg / s 0 v an πv dv..5

5 On he Modulus of he Selberg Zea-Funcions in he Criical Srip 5 The Selberg zea-funcion Z C s has rivial a zeros a s =, 0,,,..., non-rivial zeros on he criical line σ = / and also, possibly, on he inerval 0, of he real axis, see [5,.4, Theorem.4.] and [4]. In his sense he analogue of he Riemann hypohesis holds for Z C s. Moreover, he following saemen is rue. Theorem 5. There exiss a posiive number B such ha he funcions Z C s and Ms, for > B, 0 < σ < /, saisfy he inequaliy Re Z C s Z C s < ReM s Ms < 0. Noe ha Theorem 5 is proved in Luo [0], namely Re Z C s Z C s < 0 for c σ / and 0 > 0, where c > 0 is an arbirary consan, and 0 is a consan depending on c. A couple of corollaries follow from Theorem 5 for funcions Z C s and Ms. Corollary 3. For a fixed and sufficienly large, he funcion Mσ + i is decreasing for 0 < σ < /, and is increasing for / < σ <. Corollary 4. For a fixed and sufficienly large, he funcion Z C σ + i is decreasing for 0 < σ < /. Proofs of Corollaries 3 and 4 are he same as proofs of Corollaries and, and Theorem 5 is proved in Secion 3. Proof of he Theorem 4 Before he proof of Theorem 4, we sae one lemma. Lemma. a Le f be a holomorphic funcion in an open domain D and no idenically zero. Le us also suppose Re f s fs < 0 for all s D such ha fs 0. Then fs is sricly decreasing wih respec o σ in D, i.e., for each s 0 D, here exiss δ > 0 such ha fs is sricly monoonically decreasing wih respec o σ on he horizonal inerval from s 0 δ o s 0 + δ. b Conversely, if fs is decreasing wih respec o σ in D, hen Re f s fs 0 for all s D such ha fs 0. The proof of he Lemma is given in []. Remark. Of course, he analogous resuls hold for monoonically increasing fs and Re f s fs > 0. Now we prove Theorem 4. Mah. Model. Anal., Xx: 5, 0xx.

6 6 A. Griguis and D. Šiaučiūnas Proof.of Theorem 4. Firs we prove ha Re Z PSL,Z s Z PSL,Z s < ReΞ s Ξs, > C > 0, 0 < σ < /4. From he equaliy Ξs = Z PSL,Z sz id sz ell sz par s, we find ha Ξ s Ξs = Z PSL,Z s Z PSL,Z s + Z id s Z id s + Z ell s Z ell s + Z pars Z par s =: Z PSL,Z s + Us.. Z PSL,Z s Hence, o complee he proof i is sufficien o show ha By. we obain Us =a ReUs > 0, > C > 0, 0 < σ < /4. s Ψ + s Ψ Ψ s 7 6 Ψs Ψ s + Ψ s 3 + s Ψ 3 3 ζ s,. ζ where a 0 = 6 log π + log π = , Ψs = Γ s/γ s and Ψ s = Γ s/γ s. To prove he inequaliy ReUs > 0, we need o invesigae he behavior of he funcions Ψs, Ψ s and ζ s/ζs in he region 0 < σ < /4 and > C > 0. For he funcion Ψs he esimae [8] Ψs = log s s + O s, for s, arg s π δ < π, holds. From his we deduce ha ReΨs = log + O I is known []? ha, for s N? Γ s + Γ s + = Ψ log π s + =,, arg s < π..3 + γ 0 + s + log π = + s sψs, s / N. This and.3 show ha 3 + log π ReΨ s = 3 + log π = + σ + σ log log π = + σ + σ log + arcan k= k k + s + s k + σ + σreψs + ImΨs π arcan + O σ + O σ,.4

7 On he Modulus of he Selberg Zea-Funcions in he Criical Srip 7 for 0 < σ < /4 and > C > 0. From expressions of Riemann s ξ funcion [3] ξs = ξ0 ρ s, ρ we obain ξ ξ s = ρ s ρ, where he summaion runs over all non-rivial zeros of he Riemann zeafuncion aken in conjugae pairs and in order of increasing imaginary pars. If ρ = β + iγ, hen Re ξ ξ s = σ β σ β + γ, β+iγ If we assume he Riemann hypohesis, i.e, β = /, hen Reξs /ξs > 0 for σ > /, and Reξs /ξs < 0 for σ < /. On he oher hand, from he equaion ξs = s π s/ Γ s/ + ζs we ge ξ ζ s = ξ ζ s + s Ψ + log π + s. This yields ha, for σ < /, Re ζ s ζs > log log π + O,.5 and, for σ < /, Re ζ s ζs > log log π + O..6 In view of.,.3,.4 and.5 we find ha, for, ReUs = a log + 3 ReΨ s Re ζ ζ s + O = log π + σ 3 σ + log arcan Re ζ σ ζ s + O > 3 arcan 5 + σ log + cσ + O σ 6,.7 Mah. Model. Anal., Xx: 5, 0xx.

8 8 A. Griguis and D. Šiaučiūnas where a 0 = 6 log π + log π and cσ = log + σ 3. This shows ha here exiss a consan C > 0 such ha ReUs is posiive for > C and 0 < σ < /4. This proves ha, for > C, and 0 < σ < /4, Re Z PSL,Z s Z PSL,Z s < ReΞ s Ξs..8 We noe ha he resricion of σ < /4 is due o he zeros of he funcion ζs. Now we prove ha Ξ s Re < 0 Ξs for > C and 0 < σ < /. The funcion Ξs is an enire funcion of order wo. I has a canonical produc expansion [5], [9] Ξs = e as +bs+c s n ˆρ sˆρ e s/ˆρ+/s/ˆρ,.9 where ˆρ runs over he nonzero roos of Ξs, and a, b, c, and n are consans. This implies Ξ s Ξs = as + b + n s + ˆρ = as + b + n s + ˆρ s ˆρ s ˆρ s ˆρ + ˆρ +. s ˆρ If ˆρ = / + ir n, n 0, hen he laer sum splis ino wo pars: for hose ˆρ for which he numbers / + ir n are real, and for hose ˆρ for which numbers / + ir n are complex. There is only a finie number of real numbers / + ir n. Then Ξ s Re = aσ + b + nσ Ξs σ + + σ/4 r n + r n /4 r n>n 0 n + rn + /4 + rn + σ / σ / + r n>n n 0 n>n 0 / + 0 n n 0 We see ha he sum σ/4 r n + r n /4 r n>n 0 n + rn σ / + ir n + + σ / ir n / + ir n σ / ir n +.0 = σ/4 r n 0+ + r n0+ /4 rn 0+ + rn + σ/4 rn + r n 0+ /4 r n>n 0+ n + rn is posiive and unbounded as. Then, from equaion.0, i follows ha here exiss a number C > 0, such ha Re Ξ s Ξs > 0

9 On he Modulus of he Selberg Zea-Funcions in he Criical Srip 9 for > C and / < σ <. By noe afer Lemma, for fixed > C, he funcion Ξσ +i is monoonically increasing as a funcion of σ, 0 < σ < /. In view of he funcional equaion Ξs = Ξ s and Ξs = Ξs, he funcion Ξσ + i is monoonically decreasing for > C as a funcion of σ, / < σ <. So, he real par of is logarihmic derivaive is negaive, and he heorem is proved. The saemen ha if Re Z PSL,Z s Z PSL,Z s < 0 for > C and 0 < σ < /4, hen he Riemann hypohesis is rue, follows sraigh forward from Lemma and he fac ha he funcion Z PSL,Z s has zeros s = ρ/, where ρ are non-rivial zeros of ζs. Recall ha Us =a s Ψ + s Ψ Ψ s 7 6 Ψs Ψ s + Ψ ζ ζ s, where a 0 = 6 log π + log π = , eq... Corollary 5. If 0 < σ < /4, hen Re Z PSL,Z s Z PSL,Z s > ReUs holds. If / < σ <, hen > 3 arcan σ Re Z PSL,Z s Z PSL,Z s < ReUs < 3 arcan σ holds, where cσ = log + σ 3 s 3 + s Ψ σ log + cσ + O, σ log + cσ + O, 6 and. Proof. The fis par of corollary follows from he fac Re Ξ /Ξs < 0, 0 < σ < /, and inequaliy.7. The second par is obained analogically. 3 Proof of Theorem 5 Proof. of Theorem 5. Recall ha he Selberg zea-funcion aached o compac Riemann surfaces saisfies he funcional equaion Ms = M s, where Ms = Z C s exp πg / s 0 v an πv dv. Mah. Model. Anal., Xx: 5, 0xx.

10 0 A. Griguis and D. Šiaučiūnas The funcion Ms is an enire funcion of order wo, and i has he same form of canonical produc expansion.9 as he funcion Ξs. So, for > 0 > 0, he funcion Ms is monoonically decreasing wih respec o 0 < σ < /. Le / s ls = exp v an πv dv. To complee he proof, we need o show ha l s Re > 0, for 0 < σ < /, > ˆ 0. ls 0 By elemenary calculaion, we obain l { s Re = Re s } an π ls s e 4π = e 4π e π cos πσ + + σ e π sin πσ e 4π e π cos πσ + = + o, as T. Taking B = max 0, ˆ 0 complees proof. In he same way, he following corollary follows. Corollary 6. If 0 < σ < /, hen Re Z C s Z C s > πg + O e π,, holds. If / < σ <, hen holds. Re Z C s Z C s < πg + O e π,, Proof. Proof is he same as for Corollary 5. 4 Some remarks on he Riemann zea-funcion In his secion, we presen some remarks on he Riemann zea-funcion ζs, which could have been obained proving heorems and 4. Le, as above, ρ = β + iγ be non-rivial zeros of ζs. Recall ha ξ ξ s = ρ s ρ, where he summaion is over all non-rivial zeros of he Riemann zea-funcion aken in conjugae pairs in order of increasing imaginary pars. Also, ξ ζ s = ξ ζ s + s Ψ + log π + s.

11 On he Modulus of he Selberg Zea-Funcions in he Criical Srip Comparing he laer equaliies wih ζ ζ s = b s Ψ s + + ρ s ρ +, ρ where b = log π γ/, we have [0], ha ρ ρ = + γ log 4π. The inequaliies.5 and.6 give he bounds for he real par of he logarihmic derivaive of he Riemann zea-funcion in he half-planes σ < / and σ > /, respecively. Assuming he Riemann hypohesis, allows o consruc more precise bounds. For his we need some lemmas. Lemma. Le NT be he number of zeros of ζs in he recangle 0 < σ <, 0 < < T. Then, as T, log T log log T NT = T π log T π T + RT, π where RT = Olog T. If we he Riemann hypohesis is rue, hen RT = O. The proof of he lemma can be found, for example, in [0]. Lemma 3. For > he inequaliy holds. Proof. we have ha π = 0 dx + x = γ>0 0 arcan < π dx + x + dx + x > arcan + dx x + x = arcan +. Lemma 4. Le ρ = / + γ, γ = be he firs non-rivial zero of ζs. Then /4 + γ > log + O γ and γ>0 /4 + γ + > 8π log π + O as. Mah. Model. Anal., Xx: 5, 0xx.

12 A. Griguis and D. Šiaučiūnas Proof. By Lemma, summing by pars, we ge /4 + γ γ>0 u = γ /4 + u d π log u π u π + Ru + O = logu/π du π γ /4 + u + O log u d γ /4 + u = logu/π du π /4 + u + logu/π du π /4 + u + O γ > π log γ π γ du /4 + u + π log π = arcan γ log γ π π + log π + O > log + O γ + O du /4 + u + O,, 4. where he las inequaliy was obained using ha π/ arcan v π/. This proves he firs par of he lemma. Similar argumens and Lemma 3 show ha /4 + γ + = logu/π du π γ>0 γ /4 + u + + π + O > π log γ π arcan 4 arcan γ + π + O > 8π log π + O,. I is well known ha Re ζ ζ s = ρ logu/π du /4 + u + π arcan 4 log π σ β σ β + γ σ σ + s + + log π. 4. Re Ψ Assume he Riemann hypohesis. Then in view of Lemma 4, we obain σ / + γ > /4 + γ γ γ = /4 + γ + /4 + + γ γ>0 γ>0 > log + γ 8π log π + O,. 4.3

13 On he Modulus of he Selberg Zea-Funcions in he Criical Srip 3 Using his and.3 we find ha for 0 < σ < /, and Re ζ ζ s > Re ζ ζ s < σ 3 log π + σ / log σ / 8π log γ + O σ 3 log σ / 8π log π + σ / log π, log π log γ + O for / < σ < as. Nebeciuojami du šaliniai: [] ir []! Ar juos išmesi? References []G.A. Andrews, R. Askey and R. Roy. Special funcions. Cambridge universiy press, 999. []R.E.W. Barnes. The heory of he G-funcion. Quar. J. Mah., 3:64 34, 899. [3]H.M. Edwards. Riemann s Zea-Funcion. Academic Press, New York, 974. Reprined by Dover Publicaions, Mineola, N.Y. 00. [4]J. Fischer. An approach o he Selberg race formula via he Selberg zea-funcion. Lecure Noes in Mahemaics, 53 Springer-Verlag, Berlin, 987. [5]D.A. Hejhal. The Selberg race formula for P SL, R. Vol.. Lecure Noes in Mahemaics, 548, Springer-Verlag, 976. [6]D.A. Hejhal. The Selberg race formula for P SL, R. Vol.. Lecure Noes in Mahemaics, 00, Springer-Verlag, 983. [7]H. Iwaniec. Prime geodesic heorem. J. Reine Angew. Mah., 349:36 59, 984. [8]S. Koyama. Deerminan expression of Selberg zea-funcions. I. Trans. Amer. Mah. Soc., 34:49 68, 99. [9]N. Kurokawa. Parabolic componens of zea-funcions. Proc. Japan Acad. Ser. A Mah. Sci., 64: 4, 988. [0]W. Luo. On zeros of he derivaive of he Selberg zea-funcion. American Journal of Mahemaics, 75:4 5, 005. []W. Magnus, F. Oberheinger and R.P. Soni. Formulas and heorems for he special funcions of mahemaical physics. Third ediion, Springer, New York, 966. []Yu. Maiyasevich, F. Saidak and P. Zvengrowski. Horizonal monooniciy of he modulus of he Riemann zea-funcion and relaed funcions. arxiv:05.773v [mah.nt]. [3]M. Minamide. On zeros of he derivaive of he modified Selberg zea-funcion for he modular group. Journal of he Indian Mah. Soc., 803-4:75 3, 03. Mah. Model. Anal., Xx: 5, 0xx.

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