# §27.21 Tables

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

The partition function $\mathop{p\/}\nolimits\!\left(n\right)$ is tabulated in Gupta (1935, 1937), Watson (1937), and Gupta et al. (1958). Tables of the Ramanujan function $\mathop{\tau\/}\nolimits\!\left(n\right)$ 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.

No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944).