Math 213b (Spring 2005) Yum-Tong Siu 1. Explicit Formula for Logarithmic Derivative of Riemann Zeta Function

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1 Math 3b Sprig 005 Yum-og Siu Expliit Formula for Logarithmi Derivative of Riema Zeta Futio he expliit formula for the logarithmi derivative of the Riema zeta futio i the appliatio to it of the Perro formula with error etimate. he goal i to expre <x Λ i term of x with error etimate whih deped o a adjutable parameter omig from applyig the Perro formula to a vertial lie with total legth itead of to a ifiite vertial lie whih orrepod to ettig =. he etimate for <x Λ will be ued later to relate the otaimet of the zero-et of ζ i a vertial trip to the order of the error term of x + <x Λ. It will be applied alo later to give a proof of the Prime Number heorem. Sie we apply the Perro formula oly to a vertial lie with total legth ad keep the error term a a futio of itead of paig immediately to the limit, we will eed the Perro formula with error etimate. We will do it firt for a igle term i the form of the followig lemma. Perro Lemma with Error Etimate. For > 0 ad > 0, +i x d x Ex + O if x i i = log x Ex + O if x =, where 0 if 0 < x < Ex = if x = if x >. Proof. We treat the three ae: i x =, ii x >, ad iii x < eparately. Cae i. Suppoe x =. Coider the brah log z of log o that the agle θ with z = re iθ i ofied to π < θ < π ad log z = log r + iθ. he +i i d = log + i log i. From the hoie of the brah of the logarithmi futio, Re log + i log i = log + i log i = 0

2 Math 3b Sprig 005 Yum-og Siu ad Im log + i log i = arta whih approahe π a. Hee whe x =. i +i i arta = arta x d = Cae ii. Suppoe x >. Chooe a < 0 ad oider the otour itegratio of d x i = log x d e i alog the boudary of the retagle with vertie at Sie the reidue of a i, i, + i,a + i. = e log x i at it oly pole = 0, the otour itegral i equal to. i a+i a i a a, beaue x >. i i a i +i i a+i x d x x d x d where = σ + it. hu, i x d i + i a i +i a+i x a a a x d dt a + t 0 σ +, σ +, π beaue σ + σ +. Fially we have a σ + = a x log x. σ +,

3 Math 3b Sprig 005 Yum-og Siu 3 Cae iii. Suppoe 0 < x <. Chooe b > ad oider the otour itegratio of d x i = log x d e i alog the boudary of the retagle with vertie at Sie the reidue of i, b i, b + i, + i. = e log x i at it oly pole = 0, the otour itegral i equal to. i b+i b i x d x a b, beaue 0 < x <. b i x d i i b+i x d i where = σ + it. hu, b i i i +i x d + b+i i +i beaue σ + σ +. b σ + x b b b x d σ + dt b + t 0 σ +, σ +, π b = σ +, x log x. Appliatio of Perro Lemma to Logarithmi Derivative of Riema Zeta Futio. he expliit formula for the logarithmi derivative of the Riema zeta futio will be derived by applyig reidue alulatio to the otour itegratio of ζ d x i ζ

4 Math 3b Sprig 005 Yum-og Siu 4 over the retagle with vertie at M i, i, + i, M + i, for > ad x equal to a o-iteger >, whe M i allowed to go to. here are four itegral over the four ide of the retagle. For the three ide [ M i, M + i], [ M i, i], ad [ M + i, + i], we are goig to ue ζ d x i ζ for their evaluatio. However, for the ide from i to +i we are goig to ue the formula ζ ζ = Λ N to traform the itegral to +i i i N x d Λ, whih, by the Perro lemma with error etimate, i equal to <x = Λ + O <x Λ +i x d i i > x x Λ log x Λx + O where the value Λx i defied a 0 if x i ot a iteger. he term O o the right-had ide i oly for the ae x =. We have to worry about the deomiator log x i x O Λ log x x, Λx Coider the et of uh that < or > 3x. he for uh we have 3 x 3 log > log

5 Math 3b Sprig 005 Yum-og Siu 5 ad O < 3 or > 3x = O x Λ ζ ζ = O > x = O x Λ x, beaue for > we have ζ ζ = O. We ow etimate O 3 <<3x x Λ log x. I the rage < < 3x, we have < x < 3 ad < x < ad x < 4 x. Let x be the ditae betwee x ad the iteger 3 loet to x o that x. Note that for 0 < λ < we have o that Moreover, we have log + λ = log λ = = log λ λ. λ = λ λ λ = λ λ λ. Whe = x ± x, we have x ± x log = log = log ± x x x 3 x

6 Math 3b Sprig 005 Yum-og Siu 6 ad x log x = O. x For < < 3x x ad x ± x, from 3 x + x log = log = ad log x x 3 = O x x x x it follow that log + x where x i the larget iteger ot exeedig x. Sie two ditit ad would give two ditit iteger x ad x, it follow that hu 3 << 3x, x± x 3 << 3x, x± x log x = O x 3 << 3x, x± x, = O log x. x x = O x log x. Sie for < < 3x we have Λ = O log x ad x = O ad x 3 = O, it follow that O 3 <<3x x Λ = O log x 3 <<3x x log x = O + O log x log x x log x x.

7 Math 3b Sprig 005 Yum-og Siu 7 Fially we ue Λx log x to get Λx log x O = O. Puttig everythig together, we get +i ζ d x i ζ = Λ +i x d i = i = x x log x Λ + O + O + <x x log x log x +O + O. x Itegral over the wo Horizotal Lie Segmet. Whe we oider the itegral of ζ d x i ζ over the two horizotal lie-egmet [ M i, M + i] ad [ M i, i], we have to worry that the oe of the two lie egmet may otai ome zeroe of ζ. o avoid uh a ituatio we are goig to replae by ome other appropriate with < by uig the followig reult o the root deity of the Riema zeta futio o the ritial trip. Root Deity for Riema Zeta Futio o the Critial Strip. For ay give C > 0 the umber of root ρ of ζ with 0 Re ρ ad Im ρ C i o more tha O log. From thi tatemet o root deity it follow that there exit ome uh that <, Im ρ > log for ome poitive otat idepedet of. For our ue of the otour itegratio we aume alway that uh a replaemet of by ha bee made o that we alway have Im ρ > log.

8 Math 3b Sprig 005 Yum-og Siu 8 o hadle the otour itegral over the two horizotal lie-egmet ad the left vertial lie-egmet, we eed the followig etimate ζ ζ = O log whih we are goig to derive from for large, i the expaio with ii the futioal equatio Γ z Γz + z = γ + z = γ = lim N log N, N ζ = χζ, where ad χ = π Γ Γ, iii the relatio betwee the ie futio ad the Gamma futoi ΓΓ = π i π. where ζ = χζ, χ = π χ χ = log π + ζ ζ = log π + Γ Γ. Γ Γ Γ Γ Γ Γ Γ Γ ζ ζ.

9 Math 3b Sprig 005 Yum-og Siu 9 From it follow that ad Γ Γ + Γ Γ ΓΓ = π i π o π = π i π = π eiπσ πt + e iπσ+πt + eiπσ+πt = π e iπσ πt eiπσ+πt e iπσ+πt Γ Γ + Γ Γ π eπ t + e π t For σ > 0 very large, to etimate for t 0. Γ z Γz + z = γ + z = we a ue the ompario with x= x dx = log x log x + z x + z x = log x + z = log = O log z. + z x= x= From we get ζ ζ = O log for large. Etimate of hree Itegral. We firt oider the itegral M+i M i ζ d x ζ, whoe limit vaihe a M beaue of the otributio from the fator x ad beaue of the etimate. he etimate of i M i ζ d x ζ a M

10 Math 3b Sprig 005 Yum-og Siu 0 i ompletely aalogou to that of +i M+i ζ d x ζ a M ad o we will do the etimate oly for the latter itegral. From we get it domiatio by log λ + xλ dλ x λ + = O log. λ= Computatio of Reidue. We ow ompute the reidue. he otributio to the reidue of ζ x ζ ome from i the pole of ζ at =, whih give rie to x, ii the deomiator at = 0, whih give rie to ζ 0 ζ0, iii the trivial zeroe of ζ at N where N i the et of all poitive iteger, whih give rie to j N x j j, iv the zeroe ρ of ζ with Im ρ <, whih give rie to ρ Z, Im ρ < he um of all reidue i We implify to x ζ 0 ζ0 + x j j j N ogether with the reidue we get <x Λ = x ζ 0 ζ0 j N x j j log x. ρ Z, Im ρ < x ρ ρ log ρ Z, Im ρ < x ρ ρ. x + O x x ρ ρ.

11 Math 3b Sprig 005 Yum-og Siu x log x +O + O o hadle the deomiate i we hooe = + log x ad o that x log x + O x O log x x x = x + log x = e + log xlog x = e log x+ = ex x ex log x =. x log + O. With the hoie of = +, the error term O x log log x a be rewritte a O x log. Now our fial reult i the followig. Expliit Formula. <x Λ = x ζ 0 ζ0 +O x log x x ρ Z, Im ρ < x ρ ρ + O x log x x log + O.