Riemann Zeta-Function: Theory and Applications黎曼Zeta函数：理论及应用

"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 ed.

NOTATION
ERRATA
1.　ELEMENTARY THEORY
　1.1.　Definition of (s) and Elementary Properties,
　1.2.　The Functional Equation,
　1.3.　The Hadamard Product Formula,
　1.4.　The Riemann-von Mangoldt Formula,
　1.5.　An Approximate Functional Equation,
　1.6.　Mean Value Theorems,
　1.7.　Various Dirichlet Series Connected with (s),
　1.8.　Other Zeta-Functions,
　1.9.　Unproved Hypotheses,
2.　EXPONENTIAL INTEGRALS AND EXPONENTIAL SUMS
　2.1.　Exponential Integrals,
　2.2.　Exponential Sums,
　2.3.　The Theory of Exponent Pairs,
　2.4.　Two-Dimensional Exponent Pairs,
3.　THE VORONOI SUMMA TION FORMULA
　3.1.　Introduction,
　3.2.　The Truncated Voronoi Formula,
　3.3.　The Weighted Voronoi Formula, :
　3.4.　Other Formulas of the Voronoi Type,
4.　THE APPROXIMATE FUNCTIONAL EQUA"I IONS
　4.1.　The Approximate Functional Equation for (s),
　4.2.　The Approximate Functional Equation for 2(s),
　4.3.　The Approximate Functional Equation for Higher Powers,
　4.4.　The Reflection Principle,
5.　THE FOURTH POWER MOMENT
　5.1.　Introduction, 129
　5.2.　The Mean Value Theorem for Dirichlet Polynomials,
　5.3.　Proof of the Fourth Power Moment Estimate,
6.　THE ZERO-FREE REGION
　6.1.　A Survey of Results,
　6.2.　The Method of Vinogradov-Korobov,
　6.3.　Estimation of the Zeta Sum,
　6.4.　The Order Estimate of (s) Near a = 1,
　6.5.　The Deduction of the Zero-Free Region,
7.　MEAN　VALUE ESTIMATES OVER SHORT INTERVALS
　7.1.　Introduction,
　7.2.　An Auxiliary Estimate,
　7.3.　The Mean Square When o Is in the Critical Strip,
　7.4.　The Mean Square When o =1/2
　7.5.　The Order of f(s) in the Critical Strip,
　7.6.　Third and Fourth Power Moments in Short Intervals,
8.　HIGHER POWER MOMENTS
　8.1.　Introduction,
　8.2.　Some Convexity Estimates,
　8.3.　Power Moments for o = 1/2,
　8.4.　Power Moments for 1/2 < a < 1,
　8.5.　Asymptotic Formulas for Power Moments When1/29.　OMEGA RESULTS
10.　ZEROS ON THE CRITICAL LINE
11.　ZERO-DEMSOTU ESTOMATES
12.　THE DISTRIBUTION OF PRIMES
13.　THE DIRICHLET DIVEISOR RROBLEM
14.　VARIOUS OTHER DIVISOR PROBLEMS
15.　ATKINSON'S FORMULA FOR THE MEAN SQUARE
APPENDIX
REFERENCES
AUTHOR INDEX
SUBJECT INDEX