…
►Tabulated for
,
to 10D by
Fettis and Caslin (1964).
►Tabulated for
,
to 7S by
Beli͡akov et al. (1962).
…
►Tabulated for
,
to 10D by
Fettis and Caslin (1964).
►Tabulated for
,
to 6D by
Byrd and Friedman (1971), for
,
and
to 8D by
Abramowitz and Stegun (1964, Chapter 17), and for
,
to 9D by
Zhang and Jin (1996, pp. 674–675).
…
►Tabulated (with different notation) for
,
,
to 5D by
Abramowitz and Stegun (1964, Chapter 17), and for
,
,
to 7D by
Zhang and Jin (1996, pp. 676–677).
…
…
►This reference gives
,
, and their logarithmic
-derivatives to 4D for
,
, where
is the modular angle given by
►
20.15.1
►Spenceley and Spenceley (1947) tabulates
,
,
,
to 12D for
,
, where
and
is defined by (
20.15.1), together with the corresponding values of
and
.
►Lawden (1989, pp. 270–279) tabulates
,
, to 5D for
,
, and also
to 5D for
.
►Tables of Neville’s theta functions
,
,
,
(see §
20.1) and their logarithmic
-derivatives are given in
Abramowitz and Stegun (1964, pp. 582–585) to 9D for
, where (in radian measure)
, and
is defined by (
20.15.1).
…
…
►
5.16.1
►
5.16.2
…
►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see
Andrews et al. (1999, Chapter 1) and
Terras (1999, pp. 90, 149).
…
►Functions in this section derive their properties from the
fundamental
theorem of arithmetic, which states that every integer
can be represented uniquely as a product of prime powers,
…where
are the distinct prime factors of
, each exponent
is positive, and
is the number of distinct primes dividing
.
…
►An equivalent form states that the
th prime
(when the primes are listed in increasing order) is asymptotic to
as
:
…
►It is the special case
of the function
that counts the number of ways of expressing
as the product of
factors, with the order of factors taken into account.
…Note that
.
…
…
►Spenceley and Spenceley (1947) tabulates
,
,
,
,
for
and
to 12D, or 12 decimals of a radian in the case of
.
…
►Lawden (1989, pp. 280–284 and 293–297) tabulates
,
,
,
,
to 5D for
,
, where
ranges from 1.
…