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11: 20 Theta Functions
Chapter 20 Theta Functions
…12: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
►§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ► …13: Tom M. Apostol
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►Apostol was born on August 20, 1923.
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►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project
MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects.
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►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem).
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14: 6.16 Mathematical Applications
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§6.16(ii) Number-Theoretic Significance of
►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then ►
6.16.5
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►where is the number of primes less than or equal to .
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15: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
… ► denotes the number of partitions of into at most parts. See Table 26.9.1. … ►It follows that also equals the number of partitions of into parts that are less than or equal to . …16: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the number of ways of placing distinct objects into labeled boxes so that there are objects in the th box.
It is also the number of -dimensional lattice paths from to .
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is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
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26.4.7
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is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
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17: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into distinct parts.
denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
… denotes the number of partitions of into odd parts.
denotes the number of partitions of into parts taken from the set .
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18: 26.2 Basic Definitions
19: 27.18 Methods of Computation: Primes
§27.18 Methods of Computation: Primes
►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers, , and Fermat numbers, . …20: 26.11 Integer Partitions: Compositions
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denotes the number of compositions of , and is the number of compositions into exactly
parts.
is the number of compositions of with no 1’s, where again .
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26.11.1
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►The Fibonacci numbers are determined recursively by
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►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).