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number-theoretic functions

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11: 27.2 Functions
Other examples of number-theoretic functions treated in this chapter are as follows. …
27.2.9 d ( n ) = d | n 1
12: 27.11 Asymptotic Formulas: Partial Sums
The behavior of a number-theoretic function f ( n ) for large n is often difficult to determine because the function values can fluctuate considerably as n increases. …
27.11.2 n x d ( n ) = x ln x + ( 2 γ - 1 ) x + O ( x ) ,
13: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
§6.16(ii) Number-Theoretic Significance of li ( x )
If we assume Riemann’s hypothesis that all nonreal zeros of ζ ( s ) have real part of 1 2 25.10(i)), then …where π ( x ) is the number of primes less than or equal to x . …
14: 24.19 Methods of Computation
§24.19(i) Bernoulli and Euler Numbers and Polynomials
A similar method can be used for the Euler numbers based on (4.19.5). …
§24.19(ii) Values of B n Modulo p
For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p - 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . …
  • Buhler et al. (1992) uses the expansion

    24.19.3 t 2 cosh t - 1 = - 2 n = 0 ( 2 n - 1 ) B 2 n t 2 n ( 2 n ) ! ,

    and computes inverses modulo p of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

  • 15: Bibliography M
  • Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
  • P. M. Morse and H. Feshbach (1953a) Methods of Theoretical Physics. Vol. 1, McGraw-Hill Book Co., New York.
  • P. M. Morse and H. Feshbach (1953b) Methods of Theoretical Physics. Vol. 2, McGraw-Hill Book Co., New York.
  • H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.
  • H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.