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Math-4226 · mathematics · Number theory

What is the smallest Skewes's number

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a real $x\ge2$, let $\pi(x)=\#\{p\le x:p\text{ is a positive prime}\}$ and define the unshifted logarithmic integral by \[\operatorname{li}(x)=\lim_{\delta\downarrow0}\left(\int_0^{1-\delta}\frac{dt}{\log t}+\int_{1+\delta}^{x}\frac{dt}{\log t}\right).\] Thus $\operatorname{li}$ uses Cauchy principal value at $1$, rather than the offset integral starting at $2$. The least natural-number crossover is \[x_0=\min\{n\in\mathbb N:n\ge2,\ \pi(n)>\operatorname{li}(n)\}.\] Littlewood's sign-change theorem guarantees that the displayed set is nonempty.

Statement recorded in the supplied source.
What is the exact value of $x_0$, the smallest natural number at which $\pi(x)$ exceeds $\operatorname{li}(x)$? [S2]

Short English statement

Determine the very first integer where the prime-counting function exceeds the unshifted logarithmic integral.

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