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Sequel to my Proof for Riemann Hypothesis.

Riemann conjured that the function

[Tex] \xi = \int \frac {1}{\ln(x)} [/Tex]

has a root at [Tex] \frac{1}{2} [/Tex] when s=2

Let

[Tex] f ^-1 (x)=\frac{1}{\ln (x)} [/Tex]

[Tex] f(x)= e^ \frac {1}{ln (x)} [/Tex]

Taking log on both sides, \frac {1}{\ln(x)} = \ln e^\frac{1}{1/ln (x)}

Integrating both sides,

[Tex] \int \frac {1}{ln(x)}= \int \ln e^{1}{ln x} [/Tex]

[Tex] \int \ln(e^\\frac {1}{ln (x)} = e^ \ln (e^\frac{1}{ln (x)} [/Tex]

Value of \xi function is got by taking the integral between 2 and that number.

Taking limits of r.h.s from \frac{1}{2} to 1 and from 1 to 2 and adding them up for finding the value of the \xi function at \frac {1}{2} due to Cauchy’s principal numbers method.

We get,

[Tex] e^\ln e^ \infty – e^ \ln e^\frac{1){0.5} + e^ \ln e^0.5 – e^ \ln e^\infty [/Tex]

[Tex] (\infty - 0.2363) + (4.2321 - \infty) [/Tex]

Here we see that small numbers are added to uncountable infinity which preserve infinity value and so the equation becomes,

[Tes]\infty - \infty = 0 [/Tex]

thus proving Riemann hypothesis that \xi has a root at \frac {1}{2}

Mathew Cherian

[\Tex]

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# Sequel for my proof for Riemann's hypothesis

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