Riemann%20hypothesis
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1: 21.7 Riemann Surfaces
§21.7 Riemann Surfaces
►§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
►In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … ►is a Riemann matrix and it is used to define the corresponding Riemann theta function. … ►§21.7(iii) Frobenius’ Identity
…2: 25.1 Special Notation
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►The main function treated in this chapter is the Riemann zeta function .
This notation was introduced in Riemann (1859).
►The main related functions are the Hurwitz zeta function , the dilogarithm , the polylogarithm (also known as Jonquière’s function ), Lerch’s transcendent , and the Dirichlet -functions .
nonnegative integers. | |
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3: 21.2 Definitions
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§21.2(i) Riemann Theta Functions
… ►For numerical purposes we use the scaled Riemann theta function , defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … ►§21.2(ii) Riemann Theta Functions with Characteristics
… ►It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …4: 25.17 Physical Applications
§25.17 Physical Applications
►Analogies exist between the distribution of the zeros of on the critical line and of semiclassical quantum eigenvalues. This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …5: 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. …6: 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 . …7: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).