Prime Geodesic Theorem for the Modular Surface

Home , Prime geodesic

arXiv:1702.01699v5 [math.NT] 23 Mar 2018 classes n eoopiaycniudt h hl ope plane. complex whole the to continued meromorphicaly and h rm ubrtermadtezrso h imn zeta. Riemann the of re zeros the the extent and large a theorem to number resembles prime function the zeta Selberg the of ros pnpolmwehrteerrtr ntepiegoei theorem geodesic prime the in term error the whether problem open PGT. ersnaie of representatives r oeie aldped-rms h egho h rmtv clos primitive the Γ of surface length modular The the pseudo-primes. on called sometimes are Γ ihtehproi erc h norms The metric. hyperbolic the with h imn eai fodr1. order of is zeta Riemann the ae.Tecrnlgclls fipoeet o h oua grou modular the for improvements of list chronological The cases. ros.W oeta t analogue its that note We proofs). ieso famnfl 5 hoe 1]. Theorem [5, manifold a of dimension hyp Riemann the once theorem number prime the in proved. case the be would it yeblcmnflso ihrdmnin,where dimensions, higher of manifolds hyperbolic emi G sstill is PGT in term hsstigcmsfo h atthat fact the from comes setting this RM EDSCTERMFRTEMDLRSURFACE MODULAR THE FOR THEOREM GEODESIC PRIME h eainhpbtentepiegoei hoe n h distrib the and theorem geodesic prime the between relationship The oee,tefunction the However, h anto ntepofo G steSlegzt ucin define function, zeta Selberg the is PGT of proof the in tool main The ntecs fFcsa rusΓ groups Fuchsian of case the In e = Γ Let 2010 h tepst eueteexponent the reduce to attempts The p [13, theorem Koch’s von of analogue an establishing in obstacles The e od n phrases. and words Key P ahmtc ujc Classification. Subject Mathematics ro emo h rm edsctermfrtemdlrsurf modular the for theorem geodesic prime to the of term error Abstract. 0 uhthat such 5 8 SL P + ε usd e ffiielgrtmcmeasure. logarithmic finite of set a outside (2 P , ne h eeaie idlo yohss h xoeti exponent the Lindelöf hypothesis, generalized the Under Z 0 Z O N qaslog( equals , Γ etemdlrgopand group modular the be ) ( ( s P = ) rm edscterm ebr eafnto,mdlrgro modular function, zeta Selberg theorem, geodesic Prime log 0 x Z ) 4 3 H \ Γ x ≤ { Y P aifisteReanhptei.I sa outstanding an is It hypothesis. Riemann the satisfies UAE AVDISPAHI MUHAREM x 0 } bandb adl[8 seas 7,[]frdifferent for [1] [7], also (see [18] Randol by obtained onn w xdpit,wihaetesm o all for same the are which points, fixed two joining for , k Y 1. N ∞ =0 ( ⊂ (1 Introduction P > x Z 0 SL P − ) h ttmn bu h number the about statement The )). Γ O 13,1F2 58J50. 11F72, 11M36, N N samrmrhcfnto fodr2 while 2, order of function meromorphic a is ,i nw stepiegoei theorem, geodesic prime the as known is 0, 1 4 3 x ( ( (2 P P 2 3 nPTwr ucsflol nspecial in only successful were PGT in 0 , 0 d ) R 0 fpiiiecnuayclasses conjugacy primitive of ) − (log ,tebs siaeo h remainder the of estimate best the ), s − H k ,Re( ), x C ´ ) h pe afpaeequipped half-plane upper the d − 0 1 = s ) svldas o strictly for also valid is d > − 2 1 1, c sreduced is ace ainhpbetween lationship and = Γ p is d the n O dgeodesic ed to fze- of ution ≥ SL P tei be othesis ( by d x sthe is 3 π 4 in 84] . up. 1 2 Γ + ( (2 P x ε 0 as ) , of ) Z in ) 2 MUHAREM AVDISPAHIC´

35 7 71 includes 48 + ε (Iwaniec [15]), 10 + ε (Luo and Sarnak [17]), 102 + ε (Cai [8]) and 25 the present 36 + ε (Soundararajan and Young [19]). Iwaniec [14] remarked that the generalized Lindelöf hypothesis for Dirichlet L- 2 functions would imply 3 + ε. 2 We proved [2] that 3 + ε is valid outside a set of finite logarithmic measure. In the present note, we relate the error term in the Gallagherian P GT on PSL(2, Z) to the subconvexity bound for Dirichlet L- functions. This enables us to replace 2 5 3 + ε by 8 + ε under the generalized Lindelöf hypothesis. More precisely, the main result of this paper is the following theorem. Theorem. Let Γ= PSL(2, Z) be the modular group, ε> 0 arbitrarily small and θ be such that 1 A θ+ε L + it,χD (1 + t ) D 2 ≪ | | | | for some fixed A > 0, where D is a fundamental discriminant. There exists a set B of finite logarithmic measure such that x dt 5 + θ +ε π (x)= + O x 8 4 (x ,x / B) . Γ log t → ∞ ∈ Z0 1 Inserting the Conrey-Iwaniec [9] value θ = 6 into Theorem, we obtain Corollary 1.

2 +ε π (x)= li (x)+ O x 3 (x ,x / B) . Γ → ∞ ∈ Any improvement of θ immediately results in the obvious improvement of the error term in PGT. Taking into account that the Lindel¨of hypothesis allows θ = 0, we get Corollary 2. Under the Lindel¨of hypothesis,

5 +ε π (x)= li (x)+ O x 8 (x ,x / B) . Γ → ∞ ∈ 7 Remark 1. The obtained exponent for strictly hyperbolic Fuchsian groups is 10 +ε outside a set of ﬁnite logarithmic measure [3] and coincides with the above men- tioned Luo-Sarnak unconditional result for Γ = PSL(2, Z). In the case of a co- compact Kleinian group or a noncompact congruence group for some imaginary 13 quadratic number ﬁeld, the respective Gallagherian bound is 9 + ε [4]. 2. Preliminaries. The motivation for Theorem comes from several sources, including Gallagher [11], Iwaniec [15] and Balkanova and Frolenkov [6]. 5 + θ +ε 5 + θ +ε Recall that πΓ (x)= li (x)+O x 8 4 is equivalent to ψΓ (x)= x+O x 8 4 , where ψΓ (x) = log N (P0) is the Γ analogue of the classical Chebyshev k N(P0) x ≤ function ψ. P Under the Riemann hypothesis, Gallagher improved von Koch’s remainder term 1 2 in the prime number theorem from ψ(x) = x + O x 2 (log x) to ψ(x) = x +

1 2 O x 2 (log log x) outside a set of ﬁnite logarithmic measure. Following Koyama [16], we shall apply the next lemma [10] due to Gallagher to our setting. PGT ON MODULAR SURFACE 3

Lemma A. Let A be a discrete subset of R and η (0, 1). For any sequence c(ν) C, ν A, let the series ∈ ∈ ∈ S (u)= c (ν) e2πiνu ν A X∈ be absolutely convergent. Then 2 U 2 + πη ∞ U S (u) 2 du c (ν) dt. U | | ≤ sin πη η η Z− Z−∞ t ν t+ ≤ X≤ U

Iwaniec [15] established the following explicit formula with an error term for ψ Γ on Γ= PSL(2, Z).

1 x 2 Lemma B. For 1 T 2 , one has ≤ ≤ (log x) xρ x ψ (x)= x + + O (log x)2 , Γ ρ T γ T | X|≤ 1 where ρ = 2 + iγ denote zeros of ZΓ. Recently, O. Balkanova and D. Frolenkov have proved the following estimate. Lemma C. iγ 1 + θ 1 θ 3 x max x 4 2 Y 2 , x 2 Y log Y , ≪ γ Y | X|≤ 1 + 7 θ x 2 6 xiγ Y log2 Y if Y > , ≪ κ (x) γ Y | X|≤ 1 where ρ = 2 + iγ are the zeros of ZΓ, θ is the subconvexity exponent for Dirichlet L functions, and κ (x) is the distance from √x + 1 to the nearest integer. − √x 3. Proof of Theorem. 1 x 2 Inserting T = (log x)2 into Lemma B, we obtain ρ x 1 4 (1) ψ (x)= x + + O x 2 (log x) . Γ ρ γ T | X|≤ xρ We would like to bound the expression ρ , where Y (0,T ) is a parameter γ T ∈ | |≤ to be determined later on. P

iγ 1 n n+1 x ε 2 Let n = log x and Bn = x e ,e : ρ > x Y . Looking at ⌊ ⌋ ( ∈ γ T ) | |≤ the logarithmic measure of Bn, we get P

e n+1 2 iγ dx 2ε dx x dx (2) µ∗Bn = = x Y x x1+2εY ≤ ρ x1+2εY Z Z Zn γ Y Bn An e | X|≤ n+1 e 2 1 xiγ dx . ≤ e2nεY ρ x Zn γ Y e | X|≤

4 MUHAREM AVDISPAHIC´

2π u+ 1 After substitution x = en e ( 4π ), the last integral becomes · 1 4 2 π n+ 1 iγ e( 2 ) 2π e2πiγu du. ρ Z 1 γ T 4π | X|≤ − n+ 1 iγ 1 e( 2 ) Applying Lemma A, with η = U = and cγ = for γ T , cγ = 0 4π ρ | | ≤ otherwise, we get 2 1 2 + 4π n 1 iγ 1 2 + ( + 2 ) ∞ ∞ e 2πiγu 4 1 (3) e du 1   dt. ρ ≤ sin 4 ρ Z 1 γ T Z t<γ t+1 | | 4π | X|≤ −∞  γX≤ Y  −  | |≤  1   Note that t<γ t+1 ρ = O (1) since # γ : t< γ t +1 = O (t) by the Weyl ≤ { | |≤ } law. | | P Thus, 2 + + Y ∞ ∞ 1 (4)   dt = O dt = O (Y ). ρ   Z t<γ t+1 | | Z0 −∞  γX≤ Y   | |≤      Y 1 The relations (2), (3) and (4) imply µ∗Bn 2nε = 2nε . Hence, the set ≪ e Y e B = Bn has a ﬁnite logarithmic measure. ∪ iγ 1 x ε 2 For x / B, we have ρ x Y , i.e. ∈ γ Y ≤ | |≤ P ρ x 1 +ε 1 (5) x 2 Y 2 . ρ ≤ γ Y | X|≤

ρ x Now, we rely on Lemma C to estimate ρ . Let us put S (x, T ) = Y< γ T | |≤ xiγ . By Abel’s partial summation, we have P γ T | |≤ P T xiγ S (x, T ) S (x, Y ) S (x, u) = + i du. ρ 1 + iT − 1 + iY 1 2 Y< γ T 2 2 Z 2 + iu X| |≤ Y 1 iγ Multiplying the last relation by x 2 and recalling that Lemma C yields x γ Y ≪ 1 | |≤ 1 θ 1 2 P 4 + 2 +ε 2 x x Y for Y

T ρ 3 + θ +ε 3 + θ +ε 3 + θ +ε 1 3 + θ +ε x x 4 2 x 4 2 x 4 2 u 2 x 4 2 (6) 1 + 1 + 2 du 1 . ρ ≪ T 2 Y 2 u ≪ Y 2 Y< γ T Z X| |≤ Y

Combining (5) and (6), we see that the optimal choice for the parameter Y is 1 θ ρ 1 1 5 θ 4 + 2 x 2 +ε 2 8 + 4 +ε Y x . Then, ρ = O x Y = O x for x / B. ≈ γ T ∈ | P|≤ PGT ON MODULAR SURFACE 5

The relation (1) becomes

5 + θ +ε ψ (x)= x + O x 8 4 (x , x / B), Γ → ∞ ∈ as asserted.

References

[1] Avdispahić, M. “On Koyama’s refinement of the prime geodesic theorem.” Proc. Japan Acad. Ser. A 94, no. 3 (2018), 21–24. [2] Avdispahić, M. “Gallagherian P GT on PSL(2, Z).” Funct. Approximatio. Comment. Math. doi:10.7169/facm/1686 [3] Avdispahić, M. “Prime geodesic theorem of Gallagher type.” arXiv:1701.02115. [4] Avdispahić, M. “On the prime geodesic theorem for hyperbolic 3-manifolds.” Math. Nachr. (to appear; cf. arXiv:1705.05626). [5] Avdispahić, M., and Dˇz. Guˇsić. “On the error term in the prime geodesic theorem.” Bull. Korean Math. Soc. 49, no. 2 (2012), 367–372. [6] Balkanova, O., and D. Frolenkov. “Bounds for the spectral exponential sum.” arXiv:1803.04201. [7] Buser, P. Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, Vol. 106, Birkhäuser, Boston-Basel-Berlin, 1992. [8] Cai, Y. “Prime geodesic theorem.” J. Théor. Nombres Bordeaux 14, no. 1 (2002), 59–72. [9] Conrey, J. B. and H. Iwaniec. “The cubic moment of central values of automorphic L- functions.” Ann. of Math. (2) 151, no. 3 (2000), 1175–1216. [10] Gallagher, P. X. “A large sieve density estimate near σ = 1.” Invent. Math. 11 (1970), 329–339. [11] Gallagher, P. X. “Some consequences of the Riemann hypothesis.” Acta Arith. 37 (1980), 339–343. [12] Hejhal, D. A. The Selberg trace formula for PSL(2, R). Vol I, Lecture Notes in Mathematics, Vol 548, Springer, Berlin, 1976. [13] Ingham, A. E. The distribution of prime numbers, Cambridge University Press, 1932. [14] Iwaniec, H. “Non-holomorphic modular forms and their applications.” In Modular forms (Durham, 1983), 157-âe“196, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984. [15] Iwaniec, H. “Prime geodesic theorem.” J. Reine Angew. Math. 349 (1984), 136–159. [16] Koyama, S. “Refinement of prime geodesic theorem.” Proc. Japan Acad. Ser A Math. Sci. 92, no. 7 (2016), 77–81. 2 [17] Luo, W. and P. Sarnak. “Quantum ergodicity of eigenfunctions on PSL2(Z)\H .” Inst. Hautes Etudes´ Sci. Publ. Math. no. 81 (1995), 207–237. [18] Randol, B. “On the asymptotic distribution of closed geodesics on compact Riemann surfaces.” Trans. Amer. Math. Soc. 233 (1977), 241–247. [19] Soundararajan, K. and M. P. Young. “The prime geodesic theorem.” J. Reine Angew. Math. 676 (2013), 105–120.

University of Sarajevo, Department of Mathematics, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovina E-mail address: [email protected]