Mathscinet   (Providence site;   Rice mirror site)  (Note: The list of journal links below contains links not found in Mathscinet, and contains free links in several cases where Mathscinet contains non-free links.)
Jahrbuch über die Fortschritte der Mathematik (reviews of math papers 1868–1942)
Zentralblatt MATH (reviews of math papers 1931–present; a sample search is au:zieve & ti:polynomial* ^an:1382.11090

Math arXiv  (papers;   e-mail alert;   paper submission)

Springer Graduate Texts in Mathematics
Springer Lecture Notes in Mathematics
Springer Lecture Notes in Computer Science
(Many other Springer-published books and conference proceedings are also available via SpringerLink; click on the item, then you can usually download either a chapter or the entire book. Also for many books there's a link at the top right of the webpage which lets you buy a softcover edition for $24.99.)

7FAST (Free service includes both interlibrary loan and the scanning of any article or book chapter of an item in our library)
Mirlyn (Search UMich library)

Number Theory Web  
Conferences in   number theory   or   arithmetic geometry   (and  rumored conferences)

Electronic journals with number theory content:  (more e-journals listed here)
(Unless otherwise noted, UMich has full electronic runs of each journal, and has hard copy of all issues not available electronically)
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
Acta Arithmetica
Acta Mathematica (latest issue) (older issues)
Acta Mathematica Hungarica
Acta Mathematica et Informatica Universitatis Ostraviensis (now called Communications in Mathematics)
Acta Mathematica Sinica (New (English) series)
Acta Mathematica Universitatis Ostraviensis (now called Communications in Mathematics)
Acta Scientiarum Mathematicarum (Szegediensis)
Advances in Mathematics
Aequationes Mathematicae
Algebra i Analiz (English translation; since 2004)
Algebra & Number Theory
Algebra Colloquium (since 2005) (vol.7 numbers 1–3, vol. 10 number 1)
Algorithmica
American Journal of Mathematics (volumes 118–present) (volumes 1–117)
American Mathematical Monthly
Annales Academiae Scientiarum Fennicae. Mathematica
Annales de l'Institut Fourier (Grenoble)
Annales Polonici Mathematici
Annales scientifiques de l'École Normale Supérieure
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2014–present) (1871–2013)
Annali di Matematica Pura ed Applicata (1923–present) (1898–1922) (1867–1897) (1858–1865)
Annals of Combinatorics
Annals of Mathematical Stastistics
Annals of Mathematics (last ten years) (older issues)
Applicable Algebra in Engineering, Communication and Computing
Archiv der Mathematik
Arkiv för Matematik (latest issue) (earlier issues)
Asian Journal of Algebra
Asian Journal of Mathematics
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni (volumes 17–present) (volumes 1–16)
Australasian Journal of Combinatorics
Banach Center Publications
Bulletin de la Société Mathematique de France
Bulletin des sciences Mathématiques (volumes 122–present)
Bulletin of the American Mathematical Society
Bulletin of the Australian Mathematical Society
Bulletin of the Belgian Mathematical Society–Simon Stevin (continues Bull.Soc.math.Belgique Ser.A,B)
Bulletin of the Korean Mathematical Society
Bulletin of the London Mathematical Society
Canadian Journal of Mathematics
Canadian Mathematical Bulletin
Colloquium Mathematicum (volumes 87–present)
Combinatorica
Commentarii Mathematici Helvetici (since 1997) (older issues)
Communications in Algebra
Communications in Mathematics
Communications on Pure and Applied Mathematics
Complex Variables, Theory and Application
Complex Variables and Elliptic Equations
Compositio Mathematica (volumes 105–present) (volumes 1–104)
Comptes Rendus Mathematique (volumes 334–present)
Comptes Rendus de l'Académie des Sciences. Série I, Mathématique (volumes 324–333) (volumes 298–323) (volumes 293–296)
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Série A et B: Sciences Mathématiques. Sciences Physiques (volumes 284–291) (volumes 266–275) (volumes 1–261)
Computational Complexity
Designs, Codes, and Cryptography
Discrete Mathematics
Doklady Mathematics (volumes 73–present)
Duke Mathematical Journal
Electronic Journal of Combinatorics
L'Enseignement Mathématique (last five years) (older issues)
European Journal of Combinatorics
Experimental Mathematics
xctExpositiones Mathematicae (volumes 19–present)
Finite Fields and their Applications
Forum Mathematicum (volumes 10–present) (volumes 1–9)
Functional Analysis and its Applications
Functiones et Approximatio Commentarii Mathematici (volumes 35–present)
Fundamenta Mathematicae
Geometric and Functional Analysis
Glasgow Mathematical Journal
IEEE Transactions on Information Theory (volumes 9–present) (volumes 1–8)
Illinois Journal of Mathematics
Indagationes Mathematicae (since 1990) (1954–1989) (note: look up articles in this journal by year and page, rather than volume number, since many issues have two quite different volume numbers)
INTEGERS: Electronic Journal of Combinatorial Number Theory
International Journal of Algebra and Computation
International Journal of Mathematics
International Journal of Mathematics and Mathematical Sciences (the link is to issues from 2014; change the URL to see other years)
International Mathematics Research Notices
Inventiones Mathematicae
Israel Journal of Mathematics
Izvestiya: Mathematics
Journal de Mathématiques Pures et Appliquées (volumes 76–present) (some earlier volumes)
Journal de Théorie des Nombres de Bordeaux (formerly Seminaire de Théorie des Nombres de Bordeaux)
Journal für die reine und angewandte Mathematik (Crelle's journal)
Journal of Algebra
Journal of Algebraic Combinatorics
Journal of Algebraic Geometry (volumes 11–present)
Journal of Algorithms
Journal of Combinatorial Theory. Series A
Journal of Combinatorial Theory
Journal of Geometry
Journal of Group Theory
Journal of K-Theory
Journal of Knot Theory and its Ramifications
Journal of Mathematical Sciences (formerly Journal of Soviet Mathematics)
Journal of Number Theory
Journal of Pure and Applied Algebra
Journal of Soviet Mathematics (now Journal of Mathematical Sciences)
Journal of Symbolic Computation
Journal of the American Mathematical Society
Journal of the Australian Mathematical Society
Journal of the European Mathematical Society
Journal of the Institute of Mathematics of Jussieu
Journal of the London Mathematical Society
Journal of the Ramanujan Mathematical Society (2013–present)
Journal of Topology
K-Theory
Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae: listed as Indagationes Mathematicae
Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Mathematical Sciences: listed as Indagationes Mathematicae
Kyoto Journal of Mathematics
L'Enseignement Mathématique (last five years) (older issues)
Linear & Multilinear Algebra
Linear Algebra and its Applications
Manuscripta Mathematica
Mathematica Scandinavica (volumes 1–81)
Mathematical Intelligencer
Mathematical Journal of Okayama University (volumes 46–present) (volumes 1–45)
Mathematical Notes
Mathematical Proceedings of the Cambridge Philosophical Society
Mathematical Research Letters
Mathematics of Computation
Mathematics of the USSR. Izvestija
Mathematics of the USSR. Sbornik
Mathematika
Mathematische Annalen
Mathematische Nachrichten
Mathematische Zeitschrift
Memoirs of the American Mathematical Society
Memoirs of the College of Science, University of Kyoto (volumes 26–present) (now called the Kyoto Journal of Mathematics)
Messenger of Mathematics
Michigan Mathematical Journal
Monatshefte für Mathematik
Moscow Mathematical Journal
Nagoya Mathematical Journal
New York Journal of Mathematics
New Zealand Journal of Mathematics (volumes 36–present)
Nonlinearity
Notices of the American Mathematical Society
Oberwolfach reports
Pacific Journal of Mathematics (alternate site for volumes 1–176)
Periodica Mathematica Hungarica
Proceedings of the American Mathematical Society
Proceedings of the Japan Academy. Series A: Mathematical Sciences
Proceedings of the London Mathematical Society
Proceedings of the National Academy of Sciences
Proceedings of the Steklov Institute of Mathematics (volumes 252–present)
Publicationes Mathematicae Debrecen (volumes 39–present)
Publications of the Research Institute for Mathematical Sciences (PRIMS)
Quarterly Journal of Mathematics
Ramanujan Journal
Rendiconti del Seminario Matematico della Università di Padova
Rendiconti Lincei. Matematica e Applicazioni. Serie 9 (volumes 17–present) (volumes 1–16; Ser. 8, vol. 49–83; Ser. 5, vol. 1–33) (formerly Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali)
Rocky Mountain Journal of Mathematics
Russian Mathematical Surveys (Uspekhi Matematischeskikh Nauk) (volumes 15–present)
Russian Mathematics (Iz. VUZ) (volumes 51–present)
Sbornik: Mathematics
Science China Mathematics (since 1997)
Seminaire de Théorie des Nombres de Bordeaux (now called Journal de Théorie des Nombres de Bordeaux, which was originally called series 2 of this journal)
Séminaire Delange–Pisot–Poitou (volumes 1–20)
St. Petersburg Mathematical Journal (since 2004) (older issues) (formerly Leningrad Math. J.; also called Algebra i Analiz, or Algebra and Analysis)
Tôhoku Mathematical Journal
Topological Methods in Nonlinear Analysis
Topology
Transactions of the American Mathematical Society
(Unless otherwise noted, UMich has full electronic runs of each journal listed above, and has hard copy of all issues not available electronically)

Please email zieve[at]umich.edu if you spot any mistakes, or know of other useful journals!