By Gradimir V. Milovanović, Michael Th. Rassias (eds.)
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 neighborhood, together with either learn and survey works. matters coated contain analytic quantity concept, combinatorics, distinctive sequences of numbers and polynomials, analytic inequalities and purposes, approximation of services and quadratures, orthogonality and specific and intricate functions.
The mathematical effects and open difficulties mentioned during this publication are provided 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 The booklet comprises an summary of outdated and new effects, tools, and theories towards the answer of longstanding difficulties in a large medical box, in addition to new ends up in swiftly progressing components of analysis. The booklet can be valuable 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
C i t/j log t . D 0:15609 : : :/, due to Huxley . The function . / is continuous, convex downwards and satisfies . / D 0 for > 1 and . s/. 1=2/ D 0 or, equivalently, that . / D 0 for > 1=2. T / k;" T 1C" for each k. For several other equivalent statements to the Lindelöf hypothesis, see Chap. 13 of Titchmarsh’s book . The Lindelöf hypothesis is a consequence of the Riemann hypothesis (op. ), since the latter implies the bound . C > 0/: (116) It is not known whether the Lindelöf hypothesis implies the Riemann hypothesis, but this does not seem to be likely.
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.
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. We have Hj . 12 / > 0 and X ˛j Hj3 . 12 / T 2 log8 T: (84) Äj 6T Motohashi’s general formula for weighted integral of the fourth moment of j . 21 C i t/j will involve not only Maass wave forms (non-holomorphic cusp forms) but holomorphic cusp forms as well. n/, of the Petersson unitary space of holomorphic cusp forms of weight 2k for the full modular group. n/n s 34 A.
Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava by Gradimir V. Milovanović, Michael Th. Rassias (eds.)