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1: 27.11 Asymptotic Formulas: Partial Sums
where ( h , k ) = 1 , k > 0 . Letting x in (27.11.9) or in (27.11.11) we see that there are infinitely many primes p h ( mod k ) if h , k are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if ( h , k ) = 1 , then the number of primes p x with p h ( mod k ) is asymptotic to x / ( ϕ ( k ) ln x ) as x .
2: 25.15 Dirichlet L -functions
This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions27.11). Related results are: …
3: 1.2 Elementary Algebra
In (1.2.2), (1.2.4), and (1.2.5) n is a positive integer. …
Arithmetic Progression
Geometric Progression
§1.2(iv) Means
The arithmetic mean of n numbers a 1 , a 2 , , a n is …
4: Bibliography P
  • A. R. Paterson (1983) A First Course in Fluid Dynamics. Cambridge University Press, Cambridge.
  • M. S. Petković and L. D. Petković (1998) Complex Interval Arithmetic and its Applications. Mathematical Research, Vol. 105, Wiley-VCH Verlag Berlin GmbH, Berlin.
  • G. Pólya, R. E. Tarjan, and D. R. Woods (1983) Notes on Introductory Combinatorics. Progress in Computer Science, Vol. 4, Birkhäuser Boston Inc., Boston, MA.
  • G. Pólya (1949) Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.
  • M. H. Protter and C. B. Morrey (1991) A First Course in Real Analysis. 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  • 5: Bibliography K
  • P. L. Kapitsa (1951a) Heat conduction and diffusion in a fluid medium with a periodic flow. I. Determination of the wave transfer coefficient in a tube, slot, and canal. Akad. Nauk SSSR. Žurnal Eksper. Teoret. Fiz. 21, pp. 964–978.
  • R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.
  • C. Kormanyos (2011) Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.
  • Y. A. Kravtsov (1968) Two new asymptotic methods in the theory of wave propagation in inhomogeneous media. Sov. Phys. Acoust. 14, pp. 1–17.
  • Y. A. Kravtsov (1988) Rays and caustics as physical objects. In Progress in Optics, E. Wolf (Ed.), Vol. 26, pp. 227–348.