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

This e-book, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical study in various difficulties. It includes thirty-five articles, written by means of eminent scientists from the overseas mathematical neighborhood, together with either examine and survey works. topics coated contain analytic quantity concept, combinatorics, detailed sequences of numbers and polynomials, analytic inequalities and purposes, approximation of services and quadratures, orthogonality and particular and complicated functions.

The mathematical effects and open difficulties mentioned during this e-book are awarded in an easy and self-contained demeanour. The ebook includes an outline of outdated and new effects, tools, and theories towards the answer of longstanding difficulties in a large clinical box, in addition to new ends up in quickly progressing parts of study. The booklet should be worthwhile for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.

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

**Example text**

21 C i t/j. Problem 5. Does there exist an analogue of Atkinson’s formula for the mean square of j . 21 C i t/j, also for the fourth moment of j . 21 C i t/j? No one has ever found such a formula, so the answer is probably not, but this has not been proved. To formulate Motohashi’s result on (83), we need some notation from the spectral theory of the non-Euclidean Laplacian. p/p s Cp 2s 33 / 1 . z/. s/ can be continued analytically to an entire function on C. "j cosh. Äj / cos. z D x C iy/. z/ is an odd function of x.

C i t/jk dt H: (133) T H Proof. Let 1 D and therefore C 2; s1 D Z 1 C i t; T T C 12 H T 1 2H j . 1 1 H 2 6 t 6 T C 12 H . s1 / C i t/jk dt Let now E be the rectangle with vertices C iT ˙ iH; and let X be a parameter which satisfies H: 2 C iT ˙ iH . 1 (134) 2 D C 3/ The Mean Values of the Riemann Zeta-Function on the Critical Line c T 51 6 X 6 Tc (135) for some constant c > 0. u; v 2 R/ we have ˇ ˇ j exp. e e 2 iw Áˇ ˇ / ˇ Ce Áˇ ˇ e / ˇ D exp. cos u cosh v/: iu v The above function exp. 1. log T /2 : Therefore the condition (135) ensures that, for suitable C; c1 > 0, Z k .

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). T / D j . T ! 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 .