By Gradimir V. Milovanović, Michael Th. Rassias (eds.)
This ebook, in honor of Hari M. Srivastava, discusses crucial advancements in mathematical study in quite a few difficulties. It comprises thirty-five articles, written by way of eminent scientists from the foreign mathematical neighborhood, together with either examine and survey works. topics lined contain analytic quantity concept, combinatorics, detailed sequences of numbers and polynomials, analytic inequalities and purposes, approximation of capabilities and quadratures, orthogonality and specific and complicated functions.
The mathematical effects and open difficulties mentioned during this ebook are offered in an easy and self-contained demeanour. The booklet includes an summary of outdated and new effects, equipment, and theories towards the answer of longstanding difficulties in a large medical box, in addition to new ends up in speedily progressing components of study. The publication should be important 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
T/ WD . 12 C i t/ 1=2 . 1 ; s/; (124) commonly called Hardy’s function. s/. t/ 2 R when t 2 R and that j . 12 C i t/j D 1, hence j . t/j when t 2 R. ˛1 ; : : : ; ˛2k / WD 0 T Z. 12 C t C ˛1 / : : : Z. 12 C t C ˛2k /dt; where the ˛j are distinct complex numbers with Re ˛j > 1=4, so that the integrand becomes j . 12 C i t/j2k when ˛1 D D ˛2k D 0. 2 /. ˛1 ; : : : ; ˛2k /. Then one proceeds heuristically as follows: 1. t =2 / is ignored everywhere, and the product of Z-values is expanded, producing 22k terms.
T / C x 2 log T: (70) 28 A. T /. 3. T / T 1=3 log8=3 T; which is a non-trivial result, but the sharpest exponent of Watt  requires much more effort. One can improve the omega result (63). 3Clog 4/=4 exp. log log log T / 5 8 : Soundararajan’s method does not yield an ˝C or ˝ result, but just an ! result. However, it improves either (75) or (76), but one cannot tell which one. A quantitative omega result was obtained by the author . It was shown p there that there exist constants B; C > 0 such that every interval ŒT; T C C T , for T > T0 , contains numbers 1 ; 2 such that E.
T / D ˝˙ . T /; (98) but there is still a large gap between (98) and the upper bound in (96). Problem 6. T /? T / " T 1=2C" : (99) The conjectural bound in (99) is supported by two mean value results, proved by Motohashi and the author [60, 61]. 4. t/dt T 2 logC T: (100) The value C D 22 in (100) is worked out by Motohashi in . t/dt T 2; (101) so that (100) and (101) determine, up to a logarithmic factor, the true order of the integral in question. Problem 7. What is the true order of magnitude of the integral in (100)?