By Anatoly A. Karatsuba

ISBN-10: 0849328667

ISBN-13: 9780849328664

This ebook examines the applying of advanced research how to the speculation of top numbers. In a simple to appreciate demeanour, a connection is validated among mathematics difficulties and people of 0 distribution for designated capabilities. major achievements during this box of arithmetic are defined. Indicated is a connection among the recognized Riemann zeta-function and the constitution of the universe, details concept, and quantum mechanics. the idea of Riemann zeta-function and, particularly, distribution of its zeros are offered in a concise and complete method. the entire proofs of a few sleek theorems are given. major equipment of the research also are tested as utilized to basic difficulties of quantity thought.

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2. The Theory of Riemann's Zeta-Function 2. 13. The zeros of ((s) in the neighborhood of the critical line Littlewood [148] proved in 1924 that for I estirnate is ualid uniformly with respect to o: . 12) N(o,T) is the number of complex zeros p of the function ((s) subject to the condition Re p ) o, 0 < Im p < T. 13). Theorem D is formulated as follows ([200, p. 14) is ualid fo, i < o { 7 uniformly with respect to o. I2) and of Littlewood's theorem mentioned above, namely, almost all compler zeros of ((s) lie in the domain 9-g), < l" -:l | 2l logú' iD(ú)--+ -\-/ *oo as ú -+ {oo.

21) f the contour of the rectangle with vertices 2 t iT1, 2- iTr, -1* iTr, -I - iTt. 22) -T +6 ( arg s ( a'-ó -' I 2tri Ir Ir(t(o J \ ((o -1 _,;Tr) - ir1) (,(o 1 e\" ,- r. -/ s,l^Éro I iT1- pn + o(log"). 26) Assume now that 11 is the contour of the rectangle with vertices -l + iTt,--2 + iTl, 2 + i(Tr - 2), s - -1 + i(Tt -2). ), we obtain tl r \ / (,,,-ì,. ,;;)ds = olro*7;' f1 We can also write the left-hand side of the last relation as - s + losJz,t + | #, 0 t( r-r)a,=T,-tSr,lr,+rf t t:= "- t'-^rrr s - Pn, I \'r, Pn 26 o{ Ch.

This method belongs to Landau [125, pp. 78-85;2la]. gf (i + * ú + foo. /,(r) : IIt V@)10", J2(t) = '-i 'at o'l' where A is a positive number, a parameter of the problem. 2, we must set A equal to a sufficiently large positive constant. We denote by E a subset of the interval (7, T * 11) on which these integrals are equal. It is clear that if we obtain an estimate of the measure p(E) of the form p(E) < e(A)H, 32 Ch. 2. The Theorv of Riemann's Zeta-Function 1. ) 0 is a constant (see [6a, 65]). , T + H), H =To}', with the value of c as small as possible.

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Complex Analysis in Number Theory by Anatoly A. Karatsuba

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