By Gradimir V. Milovanović, Michael Th. Rassias (eds.)
This booklet, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical learn in a number of difficulties. It includes thirty-five articles, written through eminent scientists from the overseas mathematical neighborhood, together with either learn and survey works. topics coated contain analytic quantity concept, combinatorics, particular sequences of numbers and polynomials, analytic inequalities and purposes, approximation of capabilities and quadratures, orthogonality and particular and intricate functions.
The mathematical effects and open difficulties mentioned during this publication are awarded in an easy and self-contained demeanour. The booklet includes an summary of previous and new effects, equipment, and theories towards the answer of longstanding difficulties in a large clinical box, in addition to new ends up in quickly progressing components of analysis. The booklet can be invaluable for researchers and graduate scholars within the fields of arithmetic, physics and different computational and utilized sciences.
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Extra resources for Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava
1 / > B 1=4 1 ; E. t/ (see the review paper of Tsang ). The author  proved (see also Chap. 35AC38C"/=108 if 0 6 A 6 35=4; if A > 35=4: (78) The proof depended on the use of the best-known exponent for the order of j . 21 C i t/j, which has been since improved to 32=205C" D 0:15609 : : :C", due to Huxley . This enabled Heath-Brown  to extend the first bound in (78) to A 6 28=3, which one expects to be the true order of magnitude of the integral in question for all k > 0. t/ such that ( .
1. The range 0 < applications. 2. t = / inserted in the integrand facilitates convergence problems and analytic continuation essential to Motohashi’s method of proof. 1 below). On the other hand, it is clear that the resulting expression(s) for the fourth moment will not be the analogue(s) of Atkinson’s formula for the mean square of j . 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.
By Ramachandra in [89, 90]. T /, it was proved (op. log T /k 2 (137) for any fixed integer k > 1. The lower bound furnished by (137) is of the same order of magnitude as the conjectural formula (122). Recently Radziwill and Soundararajan  showed that (unconditionally) Z T 0 j . log T /k 2 holds for any real k > 1 and T > T0 . This bound not only holds for any k > 1, but 4 also it is explicit and at the same time continuous in k, although the constant e 30k is certainly not the best one possible.