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- W1787089592 abstract "i=1 (xji − y i ) = 0 (1 ≤ j ≤ k) with xi, yi ∈ [1, P ] ∩ Z are of great utility. This is perhaps best illustrated by the seminal works of Vinogradov from the first half of this century (see, for example, [1, 6]). Despite modern developments, such estimates remain the primary tool in establishing the best known results concerning the zerofree region of the Riemann zeta function, and the smallest number G(k) of variables for which the asymptotic formula holds in Waring’s problem. When s k + 1 the latter asymptotic formula seems far beyond the grasp of current technology. Our primary purpose in this memoir is to establish in a rather sharp form the desired asymptotic formula in the case s = k + 1. When s is a natural number, let Ts(P ) denote the number of s-tuples x and y in which 1 ≤ xi, yi ≤ P (1 ≤ i ≤ s), and the xi are a permutation of the yj , so that in particular, Ts(P ) = s!P s + Os(P s−1). In Section 2 we establish the strong form below of the asymptotic formula Jk+1,k(P ) ∼ Tk+1(P ), and in connection with this we define (1.2) αn = min 1≤r≤n r∈N (r + n/r)." @default.
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- W1787089592 date "1997-01-01" @default.
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- W1787089592 title "A special case of Vinogradov's mean value theorem" @default.
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