prime number theorem
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21: 25.2 Definition and Expansions
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25.2.8
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25.2.10
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►For see §24.2(i), and for see §24.2(iii).
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25.2.11
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►product over all primes
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22: 1.5 Calculus of Two or More Variables
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Implicit Function Theorem
►If is continuously differentiable, , and at , then in a neighborhood of , that is, an open disk centered at , the equation defines a continuously differentiable function such that , , and . … ►§1.5(iii) Taylor’s Theorem; Maxima and Minima
… ►§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
… ►Suppose also that converges and converges uniformly on , that is, given any positive number , however small, we can find a number that is independent of and is such that …23: 21.7 Riemann Surfaces
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►Since a Riemann surface is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus
(the number of handles in the surface).
On this surface, we choose
cycles (that is, closed oriented curves, each with at most a finite number of singular points) , , , such that their intersection indices satisfy
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►Then the prime form on the corresponding compact Riemann surface is defined by
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►Fay derives (21.7.10) as a special case of a more general class of addition theorems for Riemann theta functions on Riemann surfaces.
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►Next, define an isomorphism which maps every subset of with an even number of elements to a -dimensional vector with elements either or .
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24: 25.6 Integer Arguments
25: 1.9 Calculus of a Complex Variable
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DeMoivre’s Theorem
… ►Jordan Curve Theorem
… ►Cauchy’s Theorem
… ►Liouville’s Theorem
… ►Dominated Convergence Theorem
…26: Bibliography C
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The Staudt-Clausen theorem.
Math. Mag. 34, pp. 131–146.
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New proof of the addition theorem for Gegenbauer polynomials.
SIAM J. Math. Anal. 2, pp. 347–351.
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Short proofs of three theorems on elliptic integrals.
SIAM J. Math. Anal. 9 (3), pp. 524–528.
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The universal von Staudt theorems.
Trans. Amer. Math. Soc. 315 (2), pp. 591–603.
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Prime Numbers: A Computational Perspective.
2nd edition, Springer-Verlag, New York.
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27: Bibliography D
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Note on the addition theorem of parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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Bernoulli Numbers and Confluent Hypergeometric Functions.
In Number Theory for the Millennium, I (Urbana, IL, 2000),
pp. 343–363.
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Ramanujan’s master theorem for symmetric cones.
Pacific J. Math. 175 (2), pp. 447–490.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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28: 24.4 Basic Properties
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24.4.11
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§24.4(iv) Finite Expansions
… ►Raabe’s Theorem
… ►§24.4(ix) Relations to Other Functions
►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).29: 19.11 Addition Theorems
30: 3.8 Nonlinear Equations
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►If and , then is a simple zero of .
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►As in the case of Table 3.8.1 the quadratic nature of convergence is clearly evident: as the zero is approached, the number of correct decimal places doubles at each iteration.
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►We have and .
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