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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 F(x) obtained by Riemann in studying Fourier series, the function R(x,y;xi,eta) appearing in the application of the Riemann method for solving the Goursat problem, the Riemann prime counting function f(x), and the related the function R(n) obtained by replacing f(x) with li(x^(1/n)) in the Möbius inversion formula.

The Riemann function F(x) for a Fourier series

1/2a_0+sum_(n=1)^infty[a_ncos(nx)+b_nsin(nx)]

(1)

is obtained by integrating twice term by term to obtain

F(x)=1/4a_0x^2-sum_(n=1)^infty1/(n^2)[a_ncos(nx)+b_nsin(nx)]+Cx+D,

(2)

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

The Riemann function R(x,y;xi,eta) arises in the solution of the linear case of the Goursat problem of solving the hyperbolic partial differential equation

L^~u=u_(xy)+au_x+bu_y+cu=f

(3)

with boundary conditions

Here, R(x,y;xi,eta) is defined as the solution of the equation

R_(xy)-(aR)_x-(bR)_y+cR=0

(7)

which satisfies the conditions

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

u(x,y)=int_0^xdxiint_1^yR(xi,eta;x,y)f(xi,eta)deta.

(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.

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Riemann Function

Cite this as:

Weisstein, Eric W. "Riemann Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannFunction.html

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