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- W2076143374 abstract "with M(a) =l.u.b._.<t<,,, If(a+it) . Proximate linear orders forf(s) have been defined by Sunyer i Balaguer [7, p. 28] by extending in the natural way the notion of Lindel6f's proximate order [6, p. 326]. Since log M(a) is an increasing convex [4, pp. 74-75 ] and therefore continuous [8, p. 172] function of a, the arguments of Shah [5] and this author [2, pp. 20-25] can be applied, with trivial modifications, to the present case. The following propositions of existence of linear proximate orders and linear lower proximate orders T(a), are thus easily obtained: (A) If O<p< oo, then for any given number a (O<a< oo), there exists a positive continuous function R(a) such that: (i) the derivatives R' (a) and R(af) exist everywhere but for isolated points where R'(a ?0) and R (a?+0) exist. (ii) limao0 oR'(a) = a aR(-) = 0. (iii) lim,-o. R(a) = p. (iv) lim sup,.. log M(a)/exp [o-R(o-) ] =a. (B) If 0<r< oo, then for any given number b (O<b< oo), there exists a continuous positive function T(a) satisfying conditions (i) and (ii) of part (A) and such that: (iii') lim?=. T(a) = r. (iv') lim inf,., log M(a-)/exp [a T(a)] = b. In what follows we will study the upper and lower limits for a-* oo of the quotient log m(a)/exp [aP(a) ] where m (a) is defined as" @default.
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- W2076143374 date "1962-03-01" @default.
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- W2076143374 title "On the proximate linear orders of entire Dirichlet series" @default.
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- W2076143374 doi "https://doi.org/10.1090/s0002-9947-1962-0139725-5" @default.
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