The Riemann Zeta-function: Theory and Applications

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Courier Corporation, Jan 1, 2003 - Mathematics - 517 pages
"A thorough and easily accessible account." — MathSciNet, Mathematical Reviews on the Web, American Mathematical Society. This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estimates, the distribution of primes, the Dirichlet divisor problem and various other divisor problems, and Atkinson's formula for the mean square. End-of-chapter notes supply the history of each chapter's topic and allude to related results not covered by the book. 1985 edition.
 

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Contents

ELEMENTARY THEORY
1
EXPONENTIAL INTEGRALS AND EXPONENTIAL SUMS
55
THE VORONOI SUMMATION FORMULA
83
THE APPROXIMATE FUNCTIONAL EQUATIONS
97
THE FOURTH POWER MOMENT
129
THE ZEROFREE REGION
143
MEAN VALUE ESTIMATES OVER SHORT INTERVALS
171
HIGHER POWER MOMENTS
199
OMEGA RESULTS
231
ZEROS ON THE CRITICAL LINE
251
ZERODENSITY ESTIMATES
269
THE DISTRIBUTION OF PRIMES
297
THE DIRICHLET DIVISOR PROBLEM
351
VARIOUS OTHER DIVISOR PROBLEMS
385
ATKINSONS FORMULA FOR THE MEAN SQUARE
441
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Page 506 - Progress towards a conjecture on the mean value of Titchmarsh series, in "Recent Progress in Analytic Number Theory", symposium Durham 1979 (Vol. 1), Academic Press, London, 1981, 303-318.
Page 507 - A. Selberg. On the normal density of primes in short intervals and the difference between consecutive primes, Arch. Math. Naturvid. 47, 87-105 (1943).
Page 502 - A constructive approach to Kronecker approximations and its applications to the Mertens conlecture, J Reine Angew. Math. 2g6/2g7, 322-340(1976). M. Jutila. On a density theorem of H. L' Montgomery for /.-functions, Ann.
Page 508 - On the remainder term of the prime-number formula II. Acta Math. Acad. Sci. Hung. 1, 155-166 (1950).

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