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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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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\

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