Analytic Number Theory, Approximation Theory, and Special by Gradimir V. Milovanović, Michael Th. Rassias (eds.)

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

This ebook, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical examine in a number of difficulties. It includes thirty-five articles, written by way of eminent scientists from the foreign mathematical group, together with either examine and survey works. topics coated comprise analytic quantity concept, combinatorics, specified sequences of numbers and polynomials, analytic inequalities and purposes, approximation of services and quadratures, orthogonality and distinctive and intricate functions.

The mathematical effects and open difficulties mentioned during this publication are provided in an easy and self-contained demeanour. The publication includes an outline of outdated and new effects, equipment, and theories towards the answer of longstanding difficulties in a large clinical box, in addition to new leads to speedily progressing components of analysis. The e-book should be necessary for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.

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T / is very strong. T 3=2C" / holds. 3. T 3=2C" / the (hitherto unproved) bound . 2. Let k 2 N be fixed and T =2 6 t 6 2T . Then Z j . 21 C i t/jk log T 1C log2 T log2 T ! j . 21 C i t C iT /jk e jvj dv : (106) Proof. Let D be the rectangle with vertices ˙c ˙ i log2 T , where c D 1= log T . z/dz: (107) D As T ! 1/. By the functional equation (6) and (15) j . 21 c C i t/j j . 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.

Let k 2 N be fixed and T =2 6 t 6 2T . Then Z j . 21 C i t/jk log T 1C log2 T log2 T ! j . 21 C i t C iT /jk e jvj dv : (106) Proof. Let D be the rectangle with vertices ˙c ˙ i log2 T , where c D 1= log T . z/dz: (107) D As T ! 1/. By the functional equation (6) and (15) j . 21 c C i t/j j . 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.

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. mn/ 3=4 jn1=2 m1=2 j : " T m¤n6T The contribution of P the remaining terms in (65) is estimated similarly. 2 //1=2 T; where N D T depends on t. T / log2 T . 2 follows if we replace T by T 2 and sum the resulting expressions. T log4 T /: (67) 2 The bound in (67) was obtained independently by Preissman [86] and the author [42] in Chap.

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