Riemann Function

There are a number of functions in various branches of mathematics known as Riemann functions. Examples include the Riemann P-series, Riemann-Siegel functions, Riemann theta function, Riemann zeta function, xi-function, the function obtained by Riemann in studying Fourier series, the function appearing in the application of the Riemann method for solving the Goursat problem, the Riemann prime counting function , and the related the function obtained by replacing with in the Möbius inversion formula.

The Riemann function for a Fourier series

(1)

is obtained by integrating twice term by term to obtain

(2)

where and are constants (Riemann 1957; Hazewinkel 1988, vol. 8, p. 118).

The Riemann function arises in the solution of the linear case of the Goursat problem of solving the hyperbolic partial differential equation

(3)

with boundary conditions

			(4)
			(5)
			(6)

Here, is defined as the solution of the equation

(7)

which satisfies the conditions

			(8)
			(9)

on the characteristics and , where is a point on the domain on which (8) is defined (Hazewinkel 1988). The solution is then given by the Riemann formula

(10)

This method of solution is called the Riemann method.