All of zeros of Riemann's Zeta-Function are on σ=1/2

2015/08/07 by Feng, Nianrong, Wang, Yongzheng
#11H05 #11M06 #11M26 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.1508.02932

Abstract

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on σ=1/2 based on the functional equation π^-(s)/(2)Γ( (s)/(2) ) ζ(s)amp;=(1)/(s(s-1)) + ∫₁^∞ ψ(x) ( x^{(s)/(2) - 1} + x^-(1+s)/(2) ) dx, s=σ+it, which is in Riemann's ``Über die Anzahl der Primzahlen unter einer gegebenen Grosse". According to the geometric meaning of the functional equation and the argument principle, we obtain the number of zeros N₀(T) of the Riemann zeta function on the critical segment σ=1/2,0≤t≤T and the number of zeros N(T) of the Riemann zeta function in the rectangular region -1≤σ≤2,0≤t≤T, respectively. The result is N(T)amp;=N₀(T)=\fracarg[π^-(s)/(2)Γ((s)/(2) )ζ(s)]π+1
amp;=(T)/(2π)log(T)/(2π)-(T)/(2π)+O(logT), s=1/2+iT.

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