New PDF release: 250 problems in elementary number theory

By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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Additional resources for 250 problems in elementary number theory

Example text

As concerns these problems, the most essential result has been achieved by Erdòs and Rankin who stated the following theorem: there exists an infinite sequence of natural numbers n such that the estimate R(x)r-t-'dr. converges absolutely and uniformly in the half-plane Re s p(n) = 23 @ À(r) = O(rà+'), the improper integral on the right-hand M(r)= t Riemann's Zeta-Function a ) 0 is arbitrarv, (log log p" )(loglog log log p" ) (log log log p")' ) where c ) 0 is an absolute constant, holds true (see [41, 187]).

37 Moser's parers [156-162] were a new appreciable step in the development of this theme. Here are two most prominent results obtained by Moser: the real numbers a(u) arc defined by the relation -==i"(l), r/((s) 7=, t't' Zeros on the Critical Line = "-i "-,r,'l à o61n-,, , , uln\-i" a(n)a(m)l-l =e \L \m/ n,m1M n-,,1' at *f -,trr-r, / s-(ìr)'-ttr"s*dt -J* trans- Ch. 3B = 2. 1. The Theory of Riemann's Zeta-Function e\/iH | "1n1î@(*)-'' u'stog;)'. e) ) M1*', then it follows from lo(")l'. This explains why the quantity 11 is bounded from below in the theorems of Hardy-Littlewood-Selberg.

It should also be pointed out that the proof of Riemann's hypothesis will not solve all the problems of the theory of prime numbers. In particular, the problems of the upper and lower bounds of the quantities dn : pn+r - p,, where pp is the kth prime number, the problems of large values of d, and others will remain open. As concerns these problems, the most essential result has been achieved by Erdòs and Rankin who stated the following theorem: there exists an infinite sequence of natural numbers n such that the estimate R(x)r-t-'dr.

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250 problems in elementary number theory by Waclaw Sierpinski


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