New PDF release: An Introduction to the Theory of Numbers

New PDF release: An Introduction to the Theory of Numbers

By Leo Moser

ISBN-10: 1931705011

ISBN-13: 9781931705011

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G(n) = f (d). d|n We will obtain two expressions for x F (x) = g(n). n=1 28 Chapter 3. Distribution of Primes F (x) is the sum f (1) f (1) + f (2) f (1) + f (3) f (1) + f (2) + f (4) f (1) f (5) f (1) + f (2) + f (3) + + + + + Adding along vertical lines we have x x f (d); d d=1 if we add along the lines indicated we obtain x F n=1 Thus x x . x f (d) = n=1 d|n x n d=1 x F f (d) = d n=1 x n . The special case f = µ yields x 1= d=1 x x µ(d) M = d n=1 x n , which we previously considered. From x µ(d) d=1 x = 1, d we have, on removing brackets, allowing for error, and dividing by x, x d=1 Actually, it is known that ∞ d=1 µ(d) < 1.

The proof of Bertrand’s Postulate by this method is left as an exercise. Bertrand’s Postulate may be used to prove the following results. 1 1 1 (1) + + · · · + is never an integer. 2 3 n (2) Every integer > 7 can be written as the sum of distinct primes. (3) Every prime pn can be expressed in the form pn = ±2 ± 3 ± 5 ± · · · ± pn−1 with an error of at most 1 (Scherk). (4) The equation π(n) = ϕ(n) has exactly 8 solutions. About 1949 a sensation was created by the discovery by Erd˝ os and Selberg of an elementary proof of the Prime Number Theorem.

Using properties of integers only. One of the main tools for doing this is the following rational analogue of the logarithm function. Let 1 1 1 + +···+ and 2 3 n We prove in an quite elementary way that h(n) = 1 + | k (ab) − k (a) − k (b)| k (n) < 1 . k = h(kn) − h(k). 27 Chapter 3. Distribution of Primes The results we have established are useful in the investigation of the magnitude of the arithmetic functions σk (n), ϕk (n) and ωk (n). Since these depend not only on the magnitude of n but also strongly on the arithmetic structure of n we cannot expect to approximate them by the elementary functions of analysis.

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An Introduction to the Theory of Numbers by Leo Moser

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