Goldbach%20conjecture
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1: 27.13 Functions
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§27.13(ii) Goldbach Conjecture
… ►Goldbach’s assertion is that for all even . This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. … ►The current status of Goldbach’s conjecture is described in the Wikipedia. … ►Hardy and Littlewood (1925) conjectures that when is not a power of 2, and that when is a power of 2, but the most that is known (in 2009) is for some constant . …2: 20 Theta Functions
Chapter 20 Theta Functions
…3: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Gauss and Legendre conjectured that is asymptotic to as :
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4: Bibliography K
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A proof of Askey’s conjectured
-analogue of Selberg’s integral and a conjecture of Morris.
SIAM J. Math. Anal. 19 (4), pp. 969–986.
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A proof of the -Macdonald-Morris conjecture for
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Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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5: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Proofs of two conjectures on the real zeros of the cylinder and Airy functions.
SIAM J. Math. Anal. 53 (4), pp. 4328–4349.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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6: 16.23 Mathematical Applications
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§16.23(iii) Conformal Mapping
►The Bieberbach conjecture states that if is a conformal map of the unit disk to any complex domain, then . In the proof of this conjecture de Branges (1985) uses the inequality …7: David M. Bressoud
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► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special
Relativity, published by Springer-Verlag in 1992, A Radical
Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S.
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8: 21.9 Integrable Equations
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►Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4).
The first part of this conjecture was established in Krichever (1976); the second part was proved in Shiota (1986).
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9: 8 Incomplete Gamma and Related
Functions
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10: 28 Mathieu Functions and Hill’s Equation
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