§25.1 Special Notation

Keywords:: Jonquière’s function, Riemann zeta function, zeta function
Permalink:: http://dlmf.nist.gov/25.1

(For other notation see Notation for the Special Functions.)

$k,m,n$	nonnegative integers.
$p$	prime number.
$x$	real variable.
$a$	real or complex parameter.
$s=\sigma+it$	complex variable.
$z=x+iy$	complex variable.
$\EulerConstant$	Euler’s constant (§5.2(ii)).
$\mathop{\psi\/}\nolimits\!\left(x\right)$	digamma function ${\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ except in §25.16. See §5.2(i).
$\mathop{B_{{n}}\/}\nolimits,\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$	Bernoulli number and polynomial (§24.2(i)).
$\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$	periodic Bernoulli function $\mathop{B_{{n}}\/}\nolimits\!\left(x-\left\lfloor x\right\rfloor\right)$ .
$m\divides n$	$m$ divides $n$ .
primes	on function symbols: derivatives with respect to argument.

The main function treated in this chapter is the Riemann zeta function $\mathop{\zeta\/}\nolimits\!\left(s\right)$ . This notation was introduced in Riemann (1859).

The main related functions are the Hurwitz zeta function $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ , the dilogarithm $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)$ , the polylogarithm $\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(z\right)$ (also known as Jonquière’s function $\mathop{\phi\/}\nolimits\!\left(z,s\right)$ ), Lerch’s transcendent $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ , and the Dirichlet $\mathop{L\/}\nolimits$ -functions $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ .