Riemann%20%CE%BE-function
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11: 25.5 Integral Representations
§25.5 Integral Representations
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25.5.1
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25.5.5
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25.5.19
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§25.5(iii) Contour Integrals
…12: 21.9 Integrable Equations
§21.9 Integrable Equations
… ►Typical examples of such equations are the Korteweg–de Vries equation … ► … ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …13: 21.6 Products
§21.6 Products
►§21.6(i) Riemann Identity
… ►Then …This is the Riemann identity. On using theta functions with characteristics, it becomes …14: 8.22 Mathematical Applications
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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
… ►See Paris and Cang (1997). ►If denotes the incomplete Riemann zeta function defined by …so that , then …For further information on , including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …15: 25.4 Reflection Formulas
§25.4 Reflection Formulas
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25.4.1
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25.4.3
►where is Riemann’s -function, defined by:
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25.4.4
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16: 25.18 Methods of Computation
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§25.18(i) Function Values and Derivatives
►The principal tools for computing are the expansion (25.2.9) for general values of , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for . …Calculations relating to derivatives of and/or can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). … ►§25.18(ii) Zeros
►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of lie on the critical line . …17: 21.4 Graphics
§21.4 Graphics
►Figure 21.4.1 provides surfaces of the scaled Riemann theta function , with …This Riemann matrix originates from the Riemann surface represented by the algebraic curve ; compare §21.7(i). … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ►18: 25.10 Zeros
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§25.10(i) Distribution
… ►The Riemann hypothesis states that all nontrivial zeros lie on this line. … ►§25.10(ii) Riemann–Siegel Formula
… ►Riemann also developed a technique for determining further terms. …19: 27.4 Euler Products and Dirichlet Series
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►The completely multiplicative function
gives the Euler product representation of the Riemann zeta function
(§25.2(i)):
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27.4.3
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►The Riemann zeta function is the prototype of series of the form
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27.4.5
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►In (27.4.12) and (27.4.13) is the derivative of .