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A COROLLARY OF RIEMANN HYPOTHESIS

This paper use the results of the value distribution theory , got a significant conclusion by Riemann hypothesis

N(r,f)=

r

n(t,f) n(0,f)

dt+n(0,f)logr

t

De nition.T(r,f)=m(r,f)+N(r,f).

T(r,f)iscalledthecharacteristicfunctionoff(z).

LEMMA1.

Iff(z)isananalyticalfunctionintheregion|z|<R(0<

ρ+r

T(ρ,f)(0<r<ρ<R)ρ r

R≤∞),then

T(r,f)≤log+M(r,f)≤

whereM(r,f)=max|z|=r|f(z)|

Theproofofthelemmaseethepage57ofthereferences[1].

Setf(z)isameromorphicfunctionintheregion|z|<R(0<

R≤∞),notidenticaltozero.Set|z|<ρ(0<ρ<R)isacircle,aλ(λ=1,2,...,h)andbµ(µ=1,2,...,k)respectivelyisthezerosandthepolesoff(z)inthecircle,appearednumberofeveryzerooreverypoleanditsorderthesame,andthatz=0isnotthezeroorthepoleoffunctionf(z),theninthecircle|z|<ρ,Wehavethefollowingformula

LEMMA2.

1

log|f(0)|=

hk ρρlogloglog f(ρei ) d +

|aλ||bµ|µ=1λ=1

thisformulaiscalledJensenformula.

Theproofofthelemmaseethepage48ofthereferences[1].

LEMMA3.

Setfunctionf(z)isthemeromorphicfunctionin|z|≤R,

f (0)=0

and

f(0)=0,∞,1,

thenwhen0<r<R,have

11

)T(r,f)<2N(R,)+N(R,f)+N(R,

ff 1

2

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