# in arithmetic progressions

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##### 1: 27.11 Asymptotic Formulas: Partial Sums
where $\left(h,k\right)=1$, $k>0$. Letting $x\to\infty$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h\pmod{k}$ 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 $\left(h,k\right)=1$, then the number of primes $p\leq x$ with $p\equiv h\pmod{k}$ is asymptotic to $x/(\phi\left(k\right)\ln x)$ as $x\to\infty$.
##### 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. …
###### §1.2(iv) Means
The arithmetic mean of $n$ numbers $a_{1},a_{2},\dots,a_{n}$ 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.