12.5 $E_i(y)$, $L_i(x)=E_i(\log\,x)$, and $-Ei\left(\rho_1\log\,x\right)$

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This page illustrates a formula for $E_i(y)$ derived from evaluation of the following Mellin convolution using the Fourier series representation of $\delta(x-1)$.

(1) $\quad E_i(y)=\int_0^\infty\delta(x-1)\,E_i(y\,x)\,dx\,,\quad y<0$


The following plot illustrates the derived formula for $E_i(y)$ (blue curve) converges to the reference function provided by the Wolfram Language (dashed-gray curve) for $y<0$.

Derived Formula for $E_i(y)$ Evaluated from $-3<y<3$ (blue curve)

The following  plot illustrates the derived formula for $E_i(y)$ (red and blue curves) converges to the corresponding reference function provided by the Wolfram Language (dashed-gray curve) for $y<0$. The evaluation limit for the blue curve is a little over twice the evaluation limit for the red curve, and consequently the blue curve converges to the dashed-gray reference function $E_i(y)$ much better than the red curve.

Derived Formula for $E_i(y)$ Converges as Evaluation Limit is Increased from Red to Blue

The following plot illustrates the real part of the derived formula for $Li(x)=E_i(\log\,x)$ at successively increasing evaluation limits (green, orange, and blue curves) converges to the real part of the corresponding reference function provided by the Wolfram Language (dashed-gray curve) for $0<x<1$.

$Li(x)=E_i(\log\,x)$

The following plot illustrates the real part of the derived formula for $-Ei\left(\rho_1\log\,x\right)$ (orange curve) converges to the real part of the corresponding reference function provided by the Wolfram Language (blue curve) for $0<x<1$. Note the function illustrated below is the contribution of the first zeta zero $\rho_1$ in Riemann’s explicit formula for the prime-power counting function $J(x)$.

$-Ei\left(\rho_1\log\,x\right)$