- #1

eljose

- 492

- 0

Let be Pi(x) the prime number counting function,we can write the integral relating Zeta function and Pi(x) function as:

[tex]\lnH(s)=\int_{-\infty}^{\infty}dt\frac{\pi(e^{e^t})e^{t-s})}{e^{f(t-s)}-1} [/tex] with the function:

[tex]H(s)=\zeta(e^{-s}) [/tex] and [tex]f(t-s)=e^{t-s} [/tex]

so the strategy to obtain the formula would be obtain a L differential operator of the form:

[tex]L=a_{0}(x)+a_{1}(x)D+a_{2}(x)D^{2} [/tex] D means d/dx

so we would have that [tex]LG(x-s)=\delta(x-s) [/tex] [tex]G(x-s)=\frac{e^{t-s}}{e^{f(t-s)}-1} [/tex] so applying L to the Kernel of integral we would get the formula:

[tex]L[LnH(s)]=\pi(e^{e^{s}) [/tex] to obtain the operator L we would use the Green function theory

[tex]\lnH(s)=\int_{-\infty}^{\infty}dt\frac{\pi(e^{e^t})e^{t-s})}{e^{f(t-s)}-1} [/tex] with the function:

[tex]H(s)=\zeta(e^{-s}) [/tex] and [tex]f(t-s)=e^{t-s} [/tex]

so the strategy to obtain the formula would be obtain a L differential operator of the form:

[tex]L=a_{0}(x)+a_{1}(x)D+a_{2}(x)D^{2} [/tex] D means d/dx

so we would have that [tex]LG(x-s)=\delta(x-s) [/tex] [tex]G(x-s)=\frac{e^{t-s}}{e^{f(t-s)}-1} [/tex] so applying L to the Kernel of integral we would get the formula:

[tex]L[LnH(s)]=\pi(e^{e^{s}) [/tex] to obtain the operator L we would use the Green function theory

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