## The Riemann Zeta-function: Theory and Applications"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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### Common terms and phrases

absolute constant Ak(x analogous analytic continuation application approximate functional equation asymptotic formula Atkinson's formula Cauchy-Schwarz inequality Chapter choose circle problem complex conjecture consider convexity Corollary critical line D. R. Heath-Brown defined denote Dirichlet polynomials Dirichlet series discussed E. C. Titchmarsh E. C. Titchmarsh 1951 elementary error term exponential integrals exponential sums factor fixed integer follows G. H. Hardy gives hence holds implies inversion formula Iwaniec J. E. Littlewood Jutila Kolesnik Lindelof hypothesis line of integration lower bound main term Mangoldt notation number of zeros number theory obtain omega results partial summation pole prime number theorem Proof of Lemma proof of Theorem real numbers replaced residue theorem Riemann hypothesis right-hand side satisfy Section seen shows suitable summation formula suppose Theorem 7.2 theory of exponent trivially uniformly upper bounds von Mangoldt function Voronoi formula write zero-density estimates zero-free region zeta-function theory

### Popular passages

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).