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The prime number theorem is obtained there in an equivalent form that the Cesàro sum of the values of the Liouville function is zero. The Liouville function is ( − 1 ) ω ( n ) {\displaystyle (-1)^{\omega (n)}} where ω ( n ) {\displaystyle \omega (n)} is the number of prime factors, with multiplicity, of the integer n {\displaystyle n} .
Here "many zeros" may mean many distinct zeros, or as few as one zero but with a high multiplicity, or even many zeros all with high multiplicity. Charles Hermite used auxiliary functions that approximated the functions e k x {\displaystyle e^{kx}} for each natural number k {\displaystyle k} in order to prove the transcendence of e ...
A large part of analytic number theory deals with multiplicative problems, and so most of its texts contain sections on multiplicative number theory. These are some well-known texts that deal specifically with multiplicative problems: Davenport, Harold (2000). Multiplicative Number Theory (3rd ed.). Berlin: Springer. ISBN 978-0-387-95097-6.
In prime factorization, the multiplicity of a prime factor is its -adic valuation.For example, the prime factorization of the integer 60 is . 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1.
The multiplicity of a prime factor p of n is the largest exponent m for which p m divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p 1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p ⁄ q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n.
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The other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977) .