Soldner's Constant

Consider the following formulation of the Prime Number Theorem,

$$\pi(x)=\sum{\mu(m)\over m} \int_c^x {dt\over \ln t},$$

where $\mu(m)$ is the Möbius Function and $c$ (sometimes also denoted $\mu$) is Soldner's constant. Ramanujan found $c=1.45136380\ldots$ (Hardy 1969, Le Lionnais 1983, Berndt 1994). Soldner (cited in Nielsen 1965) derived the correct value of $c$ as 1.4513692346..., where $c$ is the root of

$$L(x)=\lim_{\epsilon\to 0}\int_0^{1-\epsilon} {dt\over \ln t}+\int_{1+\epsilon}^\infty {dt\over\ln t}$$

(Le Lionnais 1983).

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 123-124, 1994.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 45, 1959.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983.

Nielsen, N. Théorie des Integrallogarithms. New York: Chelsea, p. 88, 1965.