Riemann Hypothesis: a GGC representation

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1 Riemann Hypohesis: a GGC represenaion Nicholas G. Polson Universiy of Chicago Augus 8, 8 Absrac A GGC Generalized Gamma Convoluion represenaion for Riemann s reciprocal ξ-funcion is consruced. This provides a proof of RH based on Thorin s condiion. Keywords: Riemann Hypohesis RH, GGC, Zea, Xi, Thorin s Condiion. Inroducion Riemann 859 defines he ζ-funcion via he analyic coninuaion of n= n s on he region Res > and he ξ-funcion by ξs = ss π s Γ s ζs. Theorem provides a GGC Generalized Gamma Convoluion represenaion for Riemann s reciprocal ξ-funcion. The moivaion for he GGC represenaion is known as Thorin s condiion which is relaed o he Riemann Hypohesis RH see Bondesson, 99, p.8, Polson, 7. This also provides an alernaive mehod o Riemann s analyical coninuaion of he ζ-funcion.. GGC Preliminaries Bondesson 99 defines he GGC class of probabiliy disribuions on [, whose Laplace ransform LT akes he form, wih lef-eremiy a, for s >, E e sh = ep as + log z/z + s Udz, Here Udz a non-negaive measure on, wih finie mass on any compac se of, such ha, log Udz < and, z Udz <. The σ-finie measure Nicholas Polson is Professor of Economerics and Saisics a ChicagoBooh: ngp@chicagobooh.edu. I would like o hank Lennar Bondesson for many helpful conversions and Jianeng Xu for his commens.

2 U on, is chosen so ha he eponen φs = log + s/zudz =,,, e sz e z Udz <. U is ofen referred o as he Thorin measure and can have infinie mass. The corresponding Lévy measure is, e z Udz. A key propery of he class of GGC disribuions is ha i is equivalen o he class of generalized convoluions of miures of eponenials Ken, 98. Hence, we can wrie H D = τ> Y τ where Y τ Epτ. This is cenral o characerizing he zeros of he zea funcion and consrucing is Hadamard facorizaion. The res of he paper is oulined as follows. Secion derives he GGC represenaion of Riemann s reciprocal ξ-funcion based on he resuls of Polson 7. Riemann s ξ-funcion and GGC represenaion The following Theorem is based on Polson 7, Theorem and provides an GGC represenaion of Riemann s reciprocal ξ-funcion. Firs, by definiion, ξα + s = + sπ α+s Γ + α + s ζα + s 3 ξ α = π α Γ + α ζ α. 4 Theorem. Riemann s reciprocal ξ-funcion saisfies, for α > and s >, ξα ξα + is = ep α ξ is + e is νξ α d 5 where ν ξ α = ν Γ α + ν ζ α + ν α and ν Γ α = π ν ζ α = π n e / e e α+ d 6 Λn n α e log n/ 7 να = e α erfc 8 Moreover ν α µ = e z U αdz µ can be calculaed via U αz µ given by U αz µ = sin z/e α µd. 9 π πz

3 Proof. From he decomposiion??, for α > and Res >, ξα + s ξα e s α ξα = + s e s Γ + α + s α Γ + α e s ψ+ α where aking derivaives of log ξs a s = α, wih ψs = Γ s/γs, gives ζα + s ζ s e ζ α. ζ α ξ α = ζ log π + ζ α + ψ + α. Euler s produc formula, for α > and Res >, gives ζ α + s = p prime and ζα = p ζ pα, yields p α s = p prime ζ p α + s where ζ p s := p s /p s log Hence ζα + s ζα = p = log ζα + s ζα p α p α s = p r= e s e α µζ d r p αr e sr log p 3 where µ ζ d = p log pδ r log p d. 4 r= e ζ α ζα s = ep e s + s e α µζ d µ ζ d = p µ ζ pd = n 5 Λn log n δ log nd 6 where Λn is he von Mangold funcion. For Res >, he Gamma funcion can be represened as Γ + α + s Γ + α e s ψ+ α = ep e s + s e α µζ d 7 µ Γ d = d e. 8 3

4 The firs erm on he rhs??, for α >, follows from + s = e s e α d = e s = { } ep e s ν α d wih compleely monoone funcion e π 3 e α d d 9 ν α = e α erfc = Ee Z α where Z α has densiy /π + for >. Hence evaluaing a is where s > and given he ideniies, e is e / e is d = π 3 and e / π 3 d = ogeher wih yields, for s >, e is d = is 3 π 3 e is + is = e is e / d. 4 π 3 This implies ha, for s > and any µd, e is α µd + is e = e is ν α d 5 where ν α is he compleely monoone funcion ν α = e z sin dz z/e α µd. 6 π πz Using??,??,?? ogeher wih??, gives ξα ξα + is = ep α ξ is + e is νξ α d 7 4

5 as required. Here ν ξ α = ν α + ν Γ α + ν ζ α and ν Γ α = π ν ζ α = π = π n p prime e / e e α+ d 8 Λn n α e log n/ 9 log p r p αr e r log p/ 3 να = e α erfc. 3 Now we provide an equivalen resul for he case α =. Theorem. There eiss a GGC disribuion H ξ, such ha ξ = E[ep s s H ξ ] 3 ξ Proof. As he Laplace ransform of GGC is analyic on he cu plane C \, and he RHS of?? is he produc of such funcions, we can evaluae a i + s o yield ξα ξα s = ep α := E[ep ξ s Hα ξ s + e s ν αd 33 ] 34 where α/ξ/ < and Hα ξ is a GGC. Then evaluaing a s + s, wih c α = ξ /ξα, gives ξ ξ s = c αe [ ep + ] s H α = c α E [ ep s s H α ] := E[ep s H ξ ]. 37 As GGC is closed under eponenial iling and Eep bsh = Eep s T bh where T is a sable subordinaor, which is GGC, H ξ a GGC disribuion. This provides an alernaive derivaion of Polson 7, Theorem. This is known as Thorin s condiion for he Riemann Hypohesis Bondesson, 99, p.8 where using symmery ξ s = ξ + s and ξ ξ + s = Eep shξ. 38 5

6 The Laplace ransform, Eep sh ξ, of a GGC disribuion, is analyic on he cu plane, namely C \,, and, in paricular, i canno have any singulariies here. By analyic coninuaion, he same is rue of ξ /ξ + s. The denominaor, ξ + s canno have any zeros in he cu plane, and ξ + s has no zeros for Res >. Then ξs has no zeroes for Res > and, as ξs = ξ s, no zeroes for Res < eiher. Hence all non-rivial zeros of he ζ-funcion lie on he criical line + is. 3 References Bondesson, L. 99. Generalised Gamma Convoluions and Relaed Classes of Disribuions and Densiies. Springer-Verlag, New York. Ken, J. T. 98. The Specral Decomposiion of a Diffusion Hiing Time. Annals of Probabiliy,, 7-9. Polson, N. G. 7. On Hilber s 8h problem. arxiv Riemann, B Über die Anzahl der Primzahlen uner einer gegebenen Grösse. Monasberiche der Berliner Akademie. Tichmarsh, E.C The Theory of he Riemann Zea-funcion. Oford Universiy Press. 6