Number Theory

P-adic analysis and mathematical physics by V.S. Vladimirov, I.V. Volovich, and E.I. Zelenov.

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By V.S. Vladimirov, I.V. Volovich, and E.I. Zelenov.

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Example text

Is equal to 1J 1 + n" P. + -P = M(TJ). 7 Here [a\ is the entire part of a number a. 6) we get i»i \ "-"o M[n) = , n-no-njp n - n - f » i p - . . - n,p* ^ +•••+ P 1-p"1-p- — 1-p1 - p -1 = n n ; rt p-1 p-1 p-1 ' p-1 0 r 2 1 1 : 0 l n - s„ 7i —p-'p - l p - - i1 1 + (no + n l P + ... 8) follows. 1) from which i t follows ( e ) ' = ef. 8) for the radius r ( e ) (see Sec. 9) r - < r ( e ) < 1. 9) r(e ) = 2 . 1) on the circle \x\ = 2 . Let x = 2 and k = 2". 3) - 1 2 1 = 2-*2*- = 2 - 1 7 4 0, fc-oo. 1) diverges, and by Lemma 3 of Sec.

Necessity of the condition is obvious: K I p = l-S* - S -i\ k — ' 0, k - » co. p To prove the sufficiency we use the Cauchy criterion. As K | —• 0, k —* oo then for any e > 0 there exists a N = N such that for all it > N the inequality K | < E is fulfilled. 1) converges. n max b L < " n<* R.

The space A is a Banach algebra. • Prove completeness of A. Let a sequence {/",« —* oo}, / " £ A be fundamental. As fn " -n\=™jft-fjr\p then the sequences {/£, n —. oo) are fundamental for every fc = 0 , 1 , . . , thus they converge to some f G Q uniformly with respect to k (see Sec. 3) and hence k lim f -co k p = lim l i m f? oo = 0. cc 0l0*-jl m m ; < max | / j | max | P p f f t | = l l / l l l|ff| P R e m a r k .

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