There are several branch of Number Theory and we focus in algebraic and analytic here. As the title indicated, algebraic number theory employs algebra result to number theory, analytic one applies analysis theory. It is recommended you have a solid background of algebra before you begin to study algebraic number theory. Riemann hypothesis is one of the Millennium Prize Problems. Fermat's last theorem and the Poincaré conjecture have been solved, this is probably the biggest challenge to date.
- Baker A. A Concise Introduction to the Theory of Numbers
- Cohen H. Number Theory: Volume I: Tools and Diophantine Equations
Cohen H. Number Theory: Volume II: Analytic and Modern Tools - See his web for errata list.
- Crandall R. and Pomerance C. Prime Numbers: A Computational Perspective
- Guy R.K. Unsolved Problems in Number Theory
- Hardy G.H. and Wright E.M. An Introductory to the Theory of Numbers - A must have for this field.
- Hasse H. Number Theory - 'Every number-theorist should have this book on his or her shelf,' Irving Kaplansky states in Bulletin of the American Mathematical Society.
- Ireland K. and Rosen M. A Classical Introduction to Modern Number Theory
- Manin Y.I. and Panchishkin A.A. Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories
- Rademacher H. Lectures on Elementary Number Theory
- Serre J.-P. A Course in Arithmetic - English translation of Cours d'arithmétique.
Algebraic Number Theory:
- Achter J. and Chai C.L. Algebraic Number Theory (FREE!)
- Alaca S. and Williams K.S. Introductory Algebraic Number Theory
- Ash R.B. A Course In Algebraic Number Theory (FREE!)
- Chai C.-L. Algebraic Number Theory and Introduction to Automorphic L-functions (FREE!)
- Chapman R. Algebraic Number Theory [.pdf] (FREE!)
- Chapman R. Notes on Algebraic Numbers [.pdf] (FREE!)
- Cohen H. A Course in Computational Algebraic Number Theory - See his web for errata list.
- Cohn H. Advanced Number Theory
- Fesenko I. Introduction to Algebraic Number Theory [.pdf] (FREE!)
- Ghorpade S.R. Lectures on Topics in Algebraic Number Theory [.pdf] (FREE!)
- Gross D. Algebraic Number Theory Notes or here [.ps] or here [.pdf] (FREE!)
- Lang S. Algebraic Number Theory
- Milne J.S. Algebraic Number Theory (FREE!)
- Mollin R.A. Algebraic Number Theory
- Murty R.M. and Esmonde J. Problems in Algebraic Number Theory
- Neukirch J. Algebraic Number Theory
- Pollard H. and Diamond H.G. The Theory of Algebraic Numbers
- Reid L.W. The Elements of the Theory of Algebraic Numbers
- Stein W. A Brief Introduction To Classical and Adelic Algebraic Number Theory (FREE!)
- Stewart I. and Tall D. Algebraic Number Theory and Fermat's Last Theorem
- Swinnerton-Dyer H. P. F. A Brief Guide to Algebraic Number Theory
- Weyl H. Algebraic Theory of Numbers
Class Field Theory:
- Artin E. and Tate J. Class Field Theory
- Fesenko I.B. and Vostokov S.V. Local Fields and Their Extensions (here for FREE!)
- Gras G. Class Field Theory: From Theory to Practice
- Milne J.S. Class Field Theory (FREE!)
- Neukirch J. Class Field Theory
- Serre J.-P. Algebraic Groups and Class Fields
- Serre J.-P. Local Fields
- Weil A. Basic Number Theory
Analytic Number Theory:
- Apostol T.M. Introduction to Analytic Number Theory - Another nice book by Apostol.
- Bateman P.T. and Diamond H.G. Analytic Number Theory: An Introductory Course
- Chandrasekharan K. Introduction to Analytic Number Theory
- Chen W.W.L. Distribution of Prime Numbers (FREE!)
- Davenport H. and Montgomery H.K. Multiplicative Number Theory
- Elkies N.D. An Introduction to Analytic Number Theory (FREE!)
- Everest G. Analytic Number Theory [.pdf] (FREE!)
- Hildebrand A.J. Introduction to Analytic Number Theory (FREE!)
- Kedlaya K.S. Analytic Number Theory (FREE!)
- Knopfmacher J. Abstract Analytic Number Theory - He developed this from analytic number theory and applied it to other areas.
- Martin G. and Leung D. Analytic Number Theory [.pdf] (FREE!)
- Murty M.R. Problems in Analytic Number Theory
- Newman D.J. Analytic Number Theory
- Siegel C.L. Advanced Analytic Number Theory
- Stopple J. A Primer of Analytic Number Theory: From Pythagoras to Riemann
- Tijdeman R. Combinatorial and Analytic Number Theory [.pdf] (FREE!)
Riemann Zeta Function and the Hypothesis:
- Borwein P., Choi S., Rooney B. and Weirathmueller A. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike
- Edwards H.M. Riemann's Zeta Function
- Koblitz N. p-adic Numbers, p-adic Analysis, and Zeta-Functions
- Lapidus M.L. and van Frankenhuijsen M. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings - More advanced work.
- Mazur B. and Stein W. What is Riemann's Hypothesis? [.pdf] (FREE!) - See Stein's page for extra material, like Riemann's paper.
- Patterson S.J. An Introduction to the Theory of the Riemann Zeta-Function
- Rockmore D. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers
- Titchmarsh E.C. The Theory of the Riemann Zeta-Function
Further readings (including Modular Forms, Elliptic Curves, etc.):
See Franz Lemmermeyer's page for further study.
- Apostol T.M. Modular Functions and Dirichlet Series in Number Theory
- Armitage J.V. and Eberlein W.F. Elliptic Functions
- Baker A. Transcendental Number Theory
- Borwein P. Computational Excursions in Analysis and Number Theory
- Borwein J.M. and Borwein P.B. PI and the AGM: A Study in Analytic Number Theory and Computational Complexity
- Cassels J.W.S. and Frohlich A. Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (here is a scanned version for FREE!)
- Diamon F. and Shurman J. A First Course in Modular Forms
- Fesenko I. Lectures on Complete Discrete Valuation Fields [.pdf] (FREE!)
- Greaves G. Sieves in Number Theory
- Husemöller D. Elliptic Curves
- Knapp A.W. Elliptic Curves (here is a scanned version for FREE!)
- Knopp M.I. Modular Functions in Analytic Number Theory
- Koblitz N.I. Introduction to Elliptic Curves and Modular Forms
- Lang S. Elliptic Functions
- Lang S. Introduction to Modular Forms (here is a scanned version for FREE!)
- Lang S. (ed.) Number Theory III: Diophantine Geometry (here is a scanned version for FREE!)
- Miyake T. Modular Forms (here is a scanned version for FREE!)
- Sándor J., Mitrinovic D.S. and Crstici B. Handbook of Number Theory I, II
- Shimura G. Abelian Varieties with Complex Multiplication and Modular Functions
- Shimura G. Elementary Dirichlet Series and Modular Forms - Has a supplementary nature to his Introduction to Arithmetic Theory of Automorphic Functions, to some extend, as he says.
- Shimura G. Introduction to Arithmetic Theory of Automorphic Functions (here is a scanned version for FREE!)
- Silverman J.H. Advanced Topics in the Arithmetic of Elliptic Curves
- Silverman J.H. The Arithmetic of Elliptic Curves
- Silverman J.H. and Tate J. Rational Points on Elliptic Curves