Read e-book online Analytic Number Theory, Approximation Theory, and Special PDF

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

ISBN-10: 1493902571

ISBN-13: 9781493902576

ISBN-10: 149390258X

ISBN-13: 9781493902583

This ebook, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical learn in a number of difficulties. It comprises thirty-five articles, written by means of eminent scientists from the foreign mathematical group, together with either learn and survey works. topics coated comprise analytic quantity thought, combinatorics, specific sequences of numbers and polynomials, analytic inequalities and functions, approximation of capabilities and quadratures, orthogonality and distinct and intricate functions.

The mathematical effects and open difficulties mentioned during this publication are offered in an easy and self-contained demeanour. The booklet comprises an summary of outdated and new effects, tools, and theories towards the answer of longstanding difficulties in a large clinical box, in addition to new leads to speedily progressing components of study. The e-book could be worthy for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.

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Additional resources for Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava

Example text

1/. At a first glance the range for in (89) looks rather restrictive, but it will turn out that this is not the case. T /, but for a weighted integral. 1 2 The smooth structure and fast decay of the Gaussian weight function e 4 . T /. First we note that, similarly to 36 A. T; / WD 1 X p Äj Á 3=2 e T =2 ˛j Äj Hj3 . 12 / cos Äj log 4eT j D1 1 2 4 . D/ T; (93) assuming that (89) holds. Suppose henceforth that T " 6 put first T1 D T log T , T2 D 2T C log T . t; /dt D T1 Â 1 j 1 Z 2T > T . 21 C ix/j .

1 C i r/ i 2 F . 12 C i r; 12 C i rI 1 C 2i rI 1=y/ dy sinh. 3. aI bI cI z/. 1. With this function, after several simplifications, one is led to Motohashi’s explicit formula with a logarithmic error term. 2. T; /D p 2T 1 X ˛j Hj3 . 12 /Äj 1=2 sin Äj log j D1 Äj Á e 4eT 1 2 4 . log3DC9 T /; (90) where the O-constant depends only on D. 2 are more difficult than the proof of Atkinson’s formula and will not be given here. , [85]) plays an important rôle. T /, the error term in the asymptotic formula (82).

21 C i t/j4 log t 1 C t Clog2 t t log2 t ! j . 21 C i u/j4 e juj du : (119) The Mean Values of the Riemann Zeta-Function on the Critical Line 45 Suppose that j . r D 1; 2; : : : ; R 1/: Then (119) gives Z V4 tr Clog2 T log T log2 T tr j . r D 1; 2; : : : ; R/: If we consider separately points ftr g with even and odd indices and denote their number by R0 and R1 , respectively, then with a slight abuse of notation summation gives Rj V 4 log T Rj X j . trC1 (121) Finally we note that Z 2T j T . 21 C i t/j dt 6 12 R Z X rD1 tr C tr j .

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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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