## Analytic Number TheoryThis volume presents an authoritative, up-to-date review of analytic number theory. It contains outstanding contributions from leading international figures in this field. Core topics discussed include the theory of zeta functions, spectral theory of automorphic forms, classical problems in additive number theory such as the Goldbach conjecture, and diophantine approximations and equations. This will be a valuable book for graduates and researchers working in number theory. |

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### Contents

1 Subvarieties of Linear Tori and the Unit Equation A Survey | 1 |

2 Remarks on the Analytic Complexity of Zeta Functions | 21 |

3 Normal Distribution of Zeta Functions and Applications | 31 |

4 Goldbach Numbers and Uniform Distribution | 43 |

5 The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number ... | 53 |

6 The BrunTitchmarsh Theorem | 85 |

7 A Decomposition of Riemanns ZetaFunction | 95 |

8 Multiplicative Properties of Consecutive Integers | 103 |

14 The Goldbach Problem with Primes in Arithmetic Progressions | 227 |

15 On the Sum of Three Squares of Primes | 253 |

16 Trace Formula over the Hyperbolic Upper Half Space | 265 |

17 Modular Forms and the Chebotarev Density Theorem II | 287 |

18 Congruences between Modular Forms | 309 |

19 Regular Singularities in GFunction Theory | 321 |

20 Spectral Theory and Lfunctions | 337 |

21 Irrationality Criteria for Numbers of Mahlers Type | 343 |

9 On the Equation xm lx 1 yq with x Power | 119 |

10 Congruence Families of Exponential Sums | 127 |

11 On Some Results Concerning the Riemann Hypothesis | 139 |

12 Mean Values of Dirichlet Series via Laplace Transforms | 169 |

13 The Mean Square of the Error Term in a Genelarization of the Dirichlet Divisor Problem ... | 209 |

22 Hypergeometric Functions and Irrationality Measures | 353 |

23 Forms in Many Variables | 361 |

24 Remark on the Kuznetsov Trace Formula | 377 |

### Common terms and phrases

absolutely convergent algebraic numbers analogous Analytic Number Theory approximate functional equation approximations argument assume asymptotic Bombieri coefﬁcients completes the proof complex computable congruence conjecture conjugacy class consider convergence Corollary critical line cusp forms deduce deﬁned deﬁnition denote density Diophantine Dirichlet series divisor problem error term estimate exists exponential sums factor ﬁnd ﬁnite ﬁrst ﬁxed follows Fourier functional equation functions f Goldbach h-length Hence holds holomorphic hypergeometric hypothesis implies inequality inﬁnitely interval irrationality measure L-functions L-series Lemma linear lower bound main term Math mean value Mellin transform method minor arcs modular forms modulo Motohashi multiplicative non-negative integers notation number ﬁeld number of solutions obtain polynomial positive integers prime Proc proof of Theorem proved rational numbers result Riemann Hypothesis Riemann zeta function Riemann zeta-function satisﬁes satisfying Selberg sequence sieve spectral Suppose trace formula transform tuple upper bound variables zeta function