# Download An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery PDF

By Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery

The 5th version of 1 of the traditional works on quantity idea, written via internationally-recognized mathematicians. Chapters are particularly self-contained for larger flexibility. New gains contain extended remedy of the binomial theorem, ideas of numerical calculation and a piece on public key cryptography. includes a superb set of difficulties.

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Extra info for An Introduction to the Theory of Numbers

Sample text

We set c(n) : A(r), C (u) = \$(u), f (") = n-' in (1'14) and use the estimate of À(r). For Res ) 1 we get and _ ('(r) ((s) =s-ll. J Since / dn=pn+r-pn)c(logp") I side ) | +e1 for an/ 61 ) 0, and, consequently, according to Weierstrass'theorem, the right-hand side is a regular function in the half-plane Res I l+ et. It follows that all the singularities of the left-hand side, including the complex zeros of ((s), lie in the half-plane Res ( |, and, since they are symmetrical with respect to the straight line Re s = i, they all lie in Res = |.

Log;-; m (,(") f lln)n-" = n=2 I tt(n)n-"n-;', ((") = n=2 -o" | I 4+f 4 4*r,+ 1+f, -logzt'*5log 4 we have and, consequently, t'(2+it) _ €'(-1+ir) _€'(2+it) _€'(2-it) - tQ+tù c Res = o t'1- -1 Let us now calculate n(r,r = *ror#- o(rog"). ) and the terms themselves are equal to O(1). 25) and the last relation we obtain dt 2+lt- Riemann's Zeta-Function ") gives C'(o) \\o ) - 1 o-7 . 2e) 30 Ch. 1. Complex Integration Method where c1 ) 0 is an absolute constant. 17) that (r+ (('Ù -p" ((s) ( c2 rog(lrl + z) - *" \s P.

And the terms themselves are equal to O(1). 25) and the last relation we obtain dt 2+lt- Riemann's Zeta-Function ") gives C'(o) \\o ) - 1 o-7 . 2e) 30 Ch. 1. Complex Integration Method where c1 ) 0 is an absolute constant. 17) that (r+ (('Ù -p" ((s) ( c2 rog(lrl + z) - *" \s P. * +), Pn/' n=1, where c2 ) 0 is an absolute constant. it follows that 1 o-Ar >0, R"s-Pt' =Re o - 0x*i(t-101 @ - 0o), I(t-tt), 1n. R"^ = n#->0. 30) (o-0")r+(t-t"),' This inequality yields the following inequality: ((o * i2t\