000			02567cam a2200277 i 4500
001			137676
003			ISI Library, Kolkata
005			20170517115556.0
008			160525s2016 riu b 001 0 eng
020			_a9781470430450
040			_aISI Library
082	0	4	_a510MS _223 _bAm512
100	1		_aDiamond, Harold G., _eauthor
245	1	0	_aBeurling generalized numbers / _cHarold G. Diamond, Wen-Bin Zhang (Cheung Man Ping).
260			_aProvidence : _bAmerican Mathematical Society, _c©2016.
300			_axi, 244 pages ; _c26 cm.
490	0		_aMathematical surveys and monographs ; _vv 213.
504			_aIncludes bibliographical references and index.
505	0		_a1. Overview -- 2. Analytic machinery -- 3. dN as an exponential and Chebyshevs's identity -- 4. Upper and lower estimates of N (x) -- 5. Mertens' formulas and logarithmic density -- 6. O-Density of g-integers -- 7. Density of g-integers -- 8. Simple estimates of pie (x) -- 9. Chebyshev bounds-elementary theory -- 10. Wiener-Ikehara tauberian theorems -- 11. Chebyshev bounds-analytic methods -- 12. Optimality of a Chebyshev bound -- 13. Beurling's PNT -- 14. Equivalences to th PNT -- 15. Kahane's PNT -- 16. PNT with remainder -- 17. Optimality of the dIVP remainder term -- 18. The Dickman and Buchstab functions.
520			_aGeneralized numbers" is a multiplicative structure introduced by A. Beurling to study how independent prime number theory is from the additivity of the natural numbers. The results and techniques of this theory apply to other systems having the character of prime numbers and integers; for example, it is used in the study of the prime number theorem (PNT) for ideals of algebraic number fields. Using both analytic and elementary methods, this book presents many old and new theorems, including several of the authors' results, and many examples of extremal behavior of g-number systems. Also, the authors give detailed accounts of the $L^2$ PNT theorem of J. P. Kahane and of the example created with H. L. Montgomery, showing that additive structure is needed for proving the Riemann hypothesis. Other interesting topics discussed are propositions "equivalent" to the PNT, the role of multiplicative convolution and Chebyshev's prime number formula for g-numbers, and how Beurling theory provides an interpretation of the smooth number formulas of Dickman and de Bruijn.
650		0	_aPrime numbers.
650		0	_aReal numbers.
650		0	_aRiemann hypothesis.
700	1		_aZhang, Wen-Bin, _eauthor
942			_2ddc _cBK
999			_c423574 _d423574