By Gradimir V. Milovanović, Michael Th. Rassias (eds.)

ISBN-10: 1493902571

ISBN-13: 9781493902576

ISBN-10: 149390258X

ISBN-13: 9781493902583

This e-book, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical learn in numerous difficulties. It comprises thirty-five articles, written via eminent scientists from the foreign mathematical group, together with either study and survey works. topics lined contain analytic quantity idea, combinatorics, distinct sequences of numbers and polynomials, analytic inequalities and functions, approximation of capabilities and quadratures, orthogonality and specific and complicated functions.

The mathematical effects and open difficulties mentioned during this ebook are offered in an easy and self-contained demeanour. The booklet comprises an summary of previous and new effects, tools, and theories towards the answer of longstanding difficulties in a large medical box, in addition to new ends up in swiftly progressing parts of study. The publication might be important for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.

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21 C c C i t/jT c Since s D 0 is a simple pole of j . s/ 1C log2 T log2 T j . c C jvj/ 1 dv: (109) 40 A. Ivi´c Now set s 0 D s C c C iv D Z 1 1Ci 1 k 2 i C c C i t C iv. n/e n s0 n 1: nD1 In the last integral we shift the line of integration to Re w D c and use again the residue theorem and Stirling’s formula. 1/. s / 1C j . 2 C i t/ 1C log2 T e juj Z 1 log2 T e jvj 1 1C j . c C juj/ 1 du c 1 D log T: T In the remaining integral we make the substitution v D x of integration. 2 C i t/ 1 log T C ÂZ u and invert the order 1 1 j .

The transformations of In lead ultimately to the expressions in (39)–(42). Atkinson’s formula lay forgotten for almost 30 years until Heath-Brown [30, 31] used it to obtain important results. 2. 4. 5. T 1=4 /: (63) The omega result (63) is far from the upper bound (44). Problem 1. T /? T 1=4C" /; but this is certainly beyond reach by present methods. 2. T 1C" /: nD1 The non-diagonal terms m ¤ n give rise to an expression of the type Z 2T X 1 . , Chap. 2 of [52]). 1. x/j 6 G in Œa; b. x/e dx ˇ ˇ ˇ a ˇ G=m: (66) 26 A.

T / " T 1=2C" : (99) The conjectural bound in (99) is supported by two mean value results, proved by Motohashi and the author [60, 61]. 4. t/dt T 2 logC T: (100) The value C D 22 in (100) is worked out by Motohashi in [85]. t/dt T 2; (101) so that (100) and (101) determine, up to a logarithmic factor, the true order of the integral in question. Problem 7. What is the true order of magnitude of the integral in (100)? T ! T 3=2 ı / hold (for any given ı; " > 0) if (102) holds. T / is very strong.

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Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava by Gradimir V. Milovanović, Michael Th. Rassias (eds.)

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