27 Functions of Number TheoryComputation27.20 Methods of Computation: Other Number-Theoretic Functions27.22 Software

Lehmer (1914) lists all primes up to 100 06721. Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\mathrm{log}x$, and $\mathrm{li}\left(x\right)$. Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^{4}$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma \left(n\right)$ of these divisors. Table II lists all solutions $n$ of the equation $f\left(n\right)=m$ for all $m\le 2500$, where $f\left(n\right)$ is defined by (27.14.2). Table III lists all solutions $n\le {10}^{4}$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma \left(n\right)=m$ for all $m\le {10}^{4}$. Table 24.7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24.6 lists $\varphi \left(n\right),d\left(n\right)$, and $\sigma \left(n\right)$ for $n\le 1000$; Table 24.8 gives examples of primitive roots of all primes $\le 9973$; Table 24.9 lists all primes that are less than 1 00000.

The partition function $p\left(n\right)$ is tabulated in Gupta (1935, 1937), Watson (1937), and Gupta et al. (1958). Tables of the Ramanujan function $\tau \left(n\right)$ are published in Lehmer (1943) and Watson (1949). Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. Those published prior to 1918 are mentioned in Dickson (1919). The bibliography in Lehmer (1941) gives references to the places in Dickson’s History where the older tables are cited. Lehmer (1941) also has a section that supplies errata and corrections to all tables cited.