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 via eminent scientists from the foreign mathematical group, together with either study and survey works. matters lined contain analytic quantity idea, combinatorics, certain sequences of numbers and polynomials, analytic inequalities and functions, approximation of features and quadratures, orthogonality and unique and complicated functions.

The mathematical effects and open difficulties mentioned during this booklet are offered 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 The publication includes an outline of outdated and new effects, equipment, and theories towards the answer of longstanding difficulties in a large clinical box, in addition to new leads to speedily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 parts of analysis. The e-book could be priceless for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.

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

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Log T /k j . 2 will be deduced from a large values estimate for log j . 21 C i t/j. 3. T; H; V / denote the measure of points t from ŒT H; T C H  such that log j . T; H; V / we have H exp. 2, first note that the contribution of t satisfying log j . log T /k=2 (144) Likewise the bound (139) holds, by (141) and (142), for the contribution of t satisfying log j . 21 C i t/j > 10k log2 T . Thus we can consider only the range V C where 1 6 j j 1 j 6 log j . 21 C i t/j 6 V C ; log3 T log3 T log3 T; V D 2` 1 2 k log3 T; 1 6 ` 6 3 2 C log 10 log 2 .

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. By j D Äj2 C 1 4 [ f0g we denote the eigenvalues (discrete spectrum) of the hyperbolic Laplacian Â Ã2 Â Ã2 ! 2; Z/). s/ is attached.

Dt (136) 52 A. Ivi´c Let now Z T CH j . C iv/jk dv; I WD T H and choose X D H " . Then (136) gives I H 1 2" , showing that I cannot be too 1=3 1=3 small. Then we choose X D H I , so that (since . 21 C i t/ jtj1=6 ) trivially T k=18 X H T; and (135) is satisfied. With this choice of X , (136) reduces to H H 2=3 I 1=3 , and (133) follows. Slightly sharper results than (133), involving powers of log log T , are known. , by Ramachandra in [89, 90]. T /, it was proved (op. log T /k 2 (137) for any fixed integer k > 1.