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

Here,
is defined as the solution of the equation

(7)

which satisfies the conditions

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 .

See also Critical Strip ,

Goursat Problem ,

Logarithmic Integral ,

Mangoldt
Function ,

Riemann Method ,

Prime
Number Theorem ,

Riemann Prime Counting
Function ,

Riemann Zeta Function
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References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 144-145, 1996. Hazewinkel,
M. (Managing Ed.). Encyclopaedia
of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical
Encyclopaedia." Dordrecht, Netherlands: Reidel, Vol. 4, p. 289
and Vol. 8, p. 125, 1988. Knuth, D. E. The
Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, 1998. Riemann, B. "Über die Darstellbarkeit
einer Function durch eine trigonometrische Reihe." Reprinted in Gesammelte
math. Abhandlungen. New York: Dover, pp. 227-264, 1957. Referenced
on Wolfram|Alpha Riemann Function
Cite this as:
Weisstein, Eric W. "Riemann Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannFunction.html

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