a note on the real part of the riemann zeta-function

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1、ANOTEONTHEREALPARTOFTHERIEMANNZETA-FUNCTIONJUANARIASDEREYNA,RICHARDP.BRENT,ANDJANVANDELUNEDedicatedtoHermanJ.J.teRieleontheoccasionofhisretirementfromtheCWIinJanuary2012Abstract.WeconsidertherealpartReζ(s)oftheRiemannzeta-functionζ(s)inthehalf-planeRe(s

2、)≥1.Weshowhowtocom-puteaccuratelytheconstantσ0≈1.19whichisdefinedtobethesupremumofσsuchthatReζ(σ+it)canbenegative(orzero)forsomerealt.WealsoconsiderintervalswhereReζ(1+it)≤0andshowthattheyarerare.Thefirstoccursfort≈682112.9,andhaslength≈0.05.Welistthefirst

3、50suchintervals.1.IntroductionInthisnoteweconsidertherealpartoftheRiemannzeta-functionζ(s)inthehalf-planeH={s∈C

4、Re(s)≥1}.Asusual,wewrites=σ+it,soRe(s)=σ≥1.WearemainlyinterestedintheregionswhereReζ(s)≤0.Sincelimσ↑∞ζ(σ+it)=1(uniformlyint),Reζ(σ+it)cannotb

5、ezeroforarbitrarilylargeσ>1.Wedefineσ0:=sup{σ∈R

6、(∃t∈R)Reζ(σ+it)=0}.Thus,Reζ(s)>0ifσ>σ0.InvandeLune[9]itwasshownthatσ0isthe(unique)positiverealrootoftheequationX1πarcsin=,pσ2pwhereprunsthroughtheprimes(weadoptthisconventionthrough-arXiv:1112.4910v1[math

7、.NT]21Dec2011out).In[9]itwasalsoshownthatσ0>1.192andthatReζ(σ0+it)nevervanishes.Themainaimofthisnoteistoshowhowσ0canbecomputedtoarbitrarilyhighprecisionbyanefficientalgorithm.WealsomentionsomeresultsonthebehaviourofReζ(σ+it)for1≤σ<σ0,andinparticularonthel

8、ineσ=1.2.Accuratecomputationoftheconstantσ0Inthissectionweassumethatσ≥σ1>1,whereσ1isasuitableconstant(e.g.1.1).Weshowhowtheconstantσ0canbecomputedwithinagivenerrorbound.Therearethreemainsteps.1ONTHEREALPARTOFTHERIEMANNZETA-FUNCTION2(1)Giveanalgorithmtoe

9、valuatetheprimezeta-function[5]X−σP(σ)=p,pforrealσ>1.(2)Usingstep1,giveanalgorithmtoevaluatethefunctionf(σ)definedbyX1πf(σ)=arcsin−.pσ2p(3)Useasuitablezero-findingalgorithmtolocateazerooff(σ)ina(sufficientlysmall)intervalwheref(σ)changessign,forexample[1.

10、1,1.2].Step1iseasy.FromtheEulerproductforζ(σ)andM¨obiusinversion,wehaveaformulaessentiallyknowntoEuler[4,1748]:X∞µ(r)(1)P(σ)=logζ(rσ),rr=1whichisvalidforσ>1(seeTitchmarsh[13,eqn.(1.6.1)]).Theseriesconvergesrapidlyinviewofthefollo