Don Zagier

This webpage contains a list of my publications, and also links to the PDF files of all papers (but not of books; you'll have to either go to the bookstore or else pirate your own copies). The list contains the bibliographical information (journal, page numbers, …) for papers that are already published, but the corresponding PDF files are often my own versions and may differ slightly from the published one, in particular in the formatting and numbering.

The Max Planck Institute for Mathematics, the principal institution at which I work, is listed below, again with links. The complete postal addresses and office telephone numbers are also given. To reach me by e-mail, please use the address dbz @ mpim-bonn . mpg . de for personal or mathematical correspondence and don . zagier @ mpim-bonn . mpg . de for professional correspondence (committees, conferences, refereeing, … ) or any matters connected with the MPIM or Bonn University.

Max Planck Institute for Mathematics
Vivatsgasse 7
53111 Bonn, Germany
Tel.: (+49) 228 402 233
Sec. (Ms. M. Mäkelä): (+49) 228 402 229


Research articles

  1. Expansion of an n -point function as a sum of commutators
    J. Math. Phys. 11 (1970) 3253-3257 (pdf)
  2. Relations among invariants of circle actions on three-manifolds
    Math. Annalen 195 (1972) 321-329 (pdf)
  3. The Pontrjagin class of an orbit space
    Topology 11 (1972) 253-264 (pdf)
  4. Higher dimensional Dedekind sums
    Math. Annalen 202 (1973) 149-172 (pdf)
  5. Formes modulaires à une et deux variables
    C.R. Acad. Sci. Paris (A) 279 (1974) 683-686 (pdf)
  6. A Kronecker limit formula for real quadratic fields
    Math. Annalen 213 (1975) 153-184 (pdf)
  7. Nombres de classes et formes modulaires de poids 3/2
    C.R. Acad. Sci. Paris (A) 281 (1975) 883-886 (pdf)
  8. Modular forms associated to real quadratic fields
    Invent. Math. 30 (1975) 1-46 (pdf)
  9. Nombres de classes et fractions continues
    Journées Arithmétiques de Bordeaux, Astérisque 24-25 (1975) 81-97 (pdf)
  10. On the values at negative integers of the zeta-function of a real quadratic field
    L'Enseignement Mathématique 22 (1976) 55-95 (pdf)
  11. (with F. Hirzebruch) Classification of Hilbert modular surfaces
    in Complex Analysis and Algebraic Geometry, Iwanami Shoten & Cambridge Univ. Press, Tokyo (1977) 43-77 (pdf)
  12. (with F. Hirzebruch) Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
    Invent. Math. 36 (1976) 57-113 (pdf)
  13. Traces des opérateurs de Hecke
    Séminaire Delange-Pisot-Poitou 1975-1976, Exposé No. 23, 12 pages (pdf)
    reprinted (in English translation) as: The Eichler-Selberg trace formula on SL_2(\Bbb Z) ,
    Appendix to S. Lang, Introduction to Modular Forms, Grundlehren d. math. Wiss. 222, Springer-Verlag, Berlin-Heidelberg-New York (1976) 44-54 (pdf)
    Correction in Modular Forms of One Variable VI, Lecture Notes in Math. 627, Springer-Verlag, Berlin-Heidelberg-New York (1977) 171-173 (pdf)
  14. Een ongelijkheid tegengesteld aan die van Cauchy
    Proc. Koninkl. Ned. Akad. v. Wetensch. (Indag. Math.) 80 (1977) 349-351 (pdf)
  15. Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs
    Journées Arithmétiques de Caen 1976, Astérisque 41-42 (1977) 135-151 (pdf)
  16. Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields
    in Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer-Verlag, Berlin-Heidelberg-New York (1977) 105-169 (pdf)
  17. (with G. van der Geer) The Hilbert modular group for the field \Bbb Q(\sqrt{13})
    Invent. math. 42 (1977) 93-133 (pdf)
  18. (with S. Suter) Dilatationen von abelschen Gruppen
    L'Enseignement Math. 24 (1978) 125-142 (pdf)
  19. (with H. Stark) A property of L -functions on the real line
    J. Number Theory 12 (1980) 49-52 (pdf)
  20. (with A. Libgober and J. Wood) Congruences modulo powers of 2 for the signature of complete intersections
    Quart. J. of Math. 31 (1980) 209-218 (pdf)
  21. (with W. Kohnen) Values of L -series of modular forms at the center of the critical strip
    Invent. Math. 64 (1980) 175-198 (pdf)
  22. Sur la conjecture de Saito-Kurokawa (d'après H. Maass)
    Séminaire de Théorie des Nombres, Paris 1979-1980 (Sém. Delange-Pisot-Poitou), Progr. in Math. 12, Birkhäuser-Verlag, Boston-Basel-Stuttgart (1980) 372-394 (pdf)
  23. (with B. Gross) On the critical values of Hecke L -series
    in Fonctions abéliennes et nombres transcendants, D. Bertrand and M. Waldschmidt (eds.), Mémoire de la Soc. Math. de France 2 (1980) 49-54 (pdf)
  24. Eisenstein series and the Riemann zeta function
    in Automorphic Forms, Representation Theory and Arithmetic, Springer-Verlag, Berlin-Heidelberg-New York (1981) 275-301 (pdf)
  25. Eisenstein series and the Selberg trace formula. I
    in Automorphic Forms, Representation Theory and Arithmetic, Springer-Verlag, Berlin-Heidelberg-New York (1981) 303-355 (pdf)
  26. (with G. Roland and N. Yui) A parametric family of quintic polynomials with Galois group D_5
    J. Number Theory 15 (1982) 137-142 (pdf)
  27. The Rankin-Selberg method for automorphic functions which are not of rapid decay
    J. Fac. Sci. Tokyo 28 (1982) 415-438 (pdf)
  28. (with M. Eichler) On the zeros of the Weierstrass \wp -function
    Math. Annalen 258 (1982) 399-407 (pdf)
  29. On the number of Markoff numbers below a given bound
    Math. Comp. 39 (1982) 709-733 (pdf)
  30. On the decomposability of the Gini coefficient and other indices of inequality
    J. Math. Econ. 12 (1983) 104-118 (pdf)
  31. (with W. Kohnen) Modular forms with rational periods
    in Modular Forms, R.A. Rankin (ed.), Ellis Horwood, Chichechester (1984) 197-249 (pdf)
  32. Asymptotics and congruence properties of the a_n
    Appendix to M. Newman and D. Shanks, "On a sequence arising for series for \pi "
    Math. Comp. 42 (1984) 199-217 (appendix: pp. 212-216) (pdf)
  33. (with J. Buhler and B. Gross) On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
    Math. Comp. 44 (1985) 473-481 (pdf)
  34. (with W. Neumann) A note on an invariant of Fintushel and Stern
    in Geometry and Topology (Proceedings, Univ. of Maryland 1983-1984), Lecture Notes in Math. 1167, Springer-Verlag, Berlin-Heidelberg-New York (1985) 241-244 (pdf)
  35. Modular points, modular curves, modular surfaces and modular forms
    in Arbeitstagung Bonn 1984, Lecture Notes 1111, Springer-Verlag, Berlin-Heidelberg-New York (1985) 225-248 (pdf)
  36. Hyperbolic manifolds and special values of Dedekind zeta functions
    Invent. Math. 83 (1986) 285-302 (pdf)
  37. (with B. Gross) Singular moduli
    J. reine Angew. Math. 355 (1985) 191-220 (pdf)
  38. (with J. Lagarias and A. Odlyzko) On the capacity of disjointly shared networks
    Computer Networks 10 (1985) 275-285 (pdf)
  39. Modular parametrizations of elliptic curves
    Can. Math. Bull. 28 (1984) 372-384 (pdf)
  40. (with W. Neumann) Volumes of hyperbolic three-manifolds
    Topology 24 (1985) 307-332 (pdf)
  41. Volume of R_3
    Appendix to M. Jankins and W. Neumann, "Rotation numbers of products of circle homeomorphisms"
    Math. Ann. 271 (1985) 381-400 (appendix: pp. 398-400) (pdf)
  42. (with B. Gross) Heegner points and derivative of L -series
    Invent. Math. 85 (1986) 225-320 (pdf)
  43. (with J. Harer) The Euler characteristic of the moduli space of curves
    Invent. Math. 85 (1986) 457-485 (pdf)
  44. (with H. Jacquet) Eisenstein series and the Selberg trace formula. II
    Trans. AMS 300 (1987) 1-48 (pdf)
  45. Large integral points on elliptic curves
    Math. Comp. 48 (1987) 425-436 (pdf)
    Addendum in Math. Comp. 51 (1988) 375 (pdf)
  46. (with B. Gross and W. Kohnen) Heegner points and derivatives of L -series. II
    Math. Annalen 278 (1987) 497-562 (pdf)
  47. Note on the Landweber-Stong elliptic genus
    in Elliptic Curves and Modular Forms in Algebraic Topology, Proceedings, Princeton 1986, Lecture Notes 1326, Springer-Verlag, Berlin-Heidelberg-New York (1988) 216-224 (pdf)
  48. (with G. Kramarz) Numerical investigations related to the L -series of certain elliptic curves
    J. Ind. Math. Soc. 52 (1987) 51-69 (pdf)
  49. (with N. Skoruppa) A trace formula for Jacobi forms
    J. reine Angew. Math. 393 (1989) 168-198 (pdf)
  50. (with N. Skoruppa) Jacobi forms and a certain space of modular forms
    Invent. Math. 94 (1988) 113-146 (pdf)
  51. On an approximate identity of Ramanujan
    Proc. Ind. Acad. Sci. 97 (1987) 313-324 (pdf)
  52. Proof of the identity (28)
    Appendix to H. Gillet and C. Soulé, "Analytic torsion and the arithmetic Todd genus"
    Topology 30 (1991) 21-54 (appendix: pp. 48-52) (pdf)
  53. The Bloch-Wigner-Ramakrishnan polylogarithm function
    Math. Annalen 286 (1990) 613-624 (pdf)
  54. The Birch-Swinnerton-Dyer conjecture from a naive point of view
    in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 377-389 (pdf)
  55. Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields
    in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 391-430 (pdf)
  56. A one-sentence proof that every prime p\equiv1\pmod4 is a sum of two squares
    Amer. Math. Monthly 97 (1990) 144 (pdf)
  57. Periods of modular forms and Jacobi theta functions
    Invent. math. 104 (1991) 449-465 (pdf)
  58. Hecke operators and periods of modular forms
    Israel Math. Conf. Proc. 3 (1990) 321-336 (pdf)
  59. (with J. Alexander) The entropy of a certain infinitely convolved Bernoulli measure
    J. Lond. Math. Soc. 44 (1991) 121-134 (pdf)
  60. (with YJ. Choie) Rational period functions for PSL(2,\Bbb Z)
    in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics, 143, AMS, Providence (1993) 89-107 (pdf)
  61. Some strange 3-adic identities (Solution to Problem 6625)
    Amer. Math. Monthly 99 (1992) 66-69 (pdf)
  62. (with S. Gupta) On the coefficients of the minimal polynomials of Gaussian periods
    Math. Comp. 60 (1993) 385-398 (pdf)
  63. Realizability of a model in infinite statistics
    Commun. Math. Phys. 147 (1992) 199-210 (pdf)
  64. Special values and functional equations of polylogarithms
    in The Structural Properties of Polylogarithms, ed. L. Lewin, Mathematical Surveys and Monographs 37, AMS, Providence (1991) 377-400 (pdf)
  65. (with H. Cohen and L. Lewin) A sixteenth-order polylogarithm ladder
    Experimental Math. 1 (1992) 25-34 (pdf)
  66. (with F. Rodriguez Villegas) Square roots of central values of Hecke L-series
    in Advances in Number Theory (Proceedings of the Third Conference of the Canadian Number Theory Association), eds. F.Q. Gouvea and N. Yui, Oxford University Press (1993) 81-99 (pdf)
  67. Algebraic numbers close to both 0 and 1
    Math. Comp. 61 (1993) 485-491 (pdf)
  68. Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula
    Israel Mathematical Conference Proceedings 9 (1996) 445-462 (pdf)
  69. Modular forms and differential operators
    Proc. Ind. Acad. Sciences 104 (1994) 57-75 (pdf)
  70. Values of zeta functions and their applications
    in First European Congress of Mathematics, Volume II, Progress in Math. 120, Birkhäuser-Verlag, Basel, (1994) 497-512 (pdf)
  71. (with N. Yui) On the singular values of Weber modular functions
    Math. Comp. 66 (1997) 1645-1662 (pdf)
  72. (with J. Szmidt and J. Urbanowicz) Congruences among generalized Bernoulli numbers
    Acta Arithmetica 71 (1995) 273-278 (pdf)
  73. (with R. Hall and J. Wilson) Reciprocity formulae for general Dedekind-Rademacher sums
    Acta Arithmetica 73 (1995) 389-396 (pdf)
  74. On the distribution of the number of cycles of elements in symmetric groups
    Nieuw Archief voor Wiskunde 13 (1995) 489-495 (pdf)
  75. A converse to Cauchy's inequality
    American Mathematical Monthly, 102 (1995) 919-920 (pdf)
  76. (with F. Rodriguez Villegas) Which primes are sums of two cubes?
    in Number Theory (Proceedings of the Third Conference of the Canadian Number Theory Association), ed. K. Dilcher, CMS Conference Proceedings 15 (1995) 295-306 (pdf)
  77. Cohomology of moduli spaces of stable rank two bundles over a curve
    in The Moduli Spaces of Curves (R. Dijkgraaf, C. Faber, G. v.d. Geer, eds.), Prog. in Math. 129, Birkhäuser, Boston (1995) 533-563 (pdf)
  78. (with M. Kaneko) A generalized Jacobi theta function and quasimodular forms
    in The Moduli Spaces of Curves (R. Dijkgraaf, C. Faber, G. v.d. Geer, eds.), Prog. in Math. 129, Birkhäuser, Boston (1995) 165-172 (pdf)
  79. (with P. Cohen and Y. Manin) Automorphic pseudodifferential operators
    in Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, Prog. in Nonlinear Differential Equations and their Applications 26, Birkhäuser, Boston (1997) 17-47 (pdf)
  80. (with R. Kaufmann and Yu. Manin) Higher Weil-Petersson volumes of moduli spaces of stable n -pointed curves
    Commun. Math. Phys. 181 (1996) 763-787 (pdf)
  81. A note on Reyes's theorem about triangles
    Nieuw Archief voor Wiskunde 14 (1996) 373-374 (pdf)
  82. (with F. Beukers) Lower bounds of heights of points on hypersurfaces
    Acta Arith. 79 (1997) 103-111 (pdf)
  83. (with J. Lewis) Period functions and the Selberg zeta function for the modular group
    in The Mathematical Beauty of Physics, A Memorial Volume for Claude Itzykson, (J.M. Drouffe and J.B. Zuber, eds.), Adv. Series in Mathematical Physics 24, World Scientific, Singapore (1997) 83-97 (pdf)
  84. (with M. Kaneko) Supersingular j -invariants, hypergeometric series, and Atkin's orthogonal polynomials
    in Proceedings of the Conference on Computational Aspects of Number Theory, AMS/IP Studies in Advanced Math. 7, International Press, Cambridge (1997) 97-126 (pdf)
  85. (with L. Goettsche) Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b_+=1
    Selecta Mathematica 4 (1998) 69-115 (pdf)
  86. A modified Bernoulli number
    Nieuw Archief voor Wiskunde 16 (1998) 63-72 (pdf)
  87. A modular identity arising from mirror symmetry
    in Integrable Systems and Algebraic Geometry (M.-H. Saito, Y. Shimizu, K. Ueno, eds.), Proceedings of the Taniguchi Symposium 1997, Word Scientific, Singapore (1998) 477-480 (pdf)
  88. From quadratic functions to modular functions
    in Number Theory in Progress. Vol 2 (K. Győry, H. Iwaniec and J. Urbanowicz, eds.), Proceedings of Internat. Conference on Number Theory, Zakopane 1997, de Gruyter, Berlin (1999) 1147-1178 (pdf)
  89. (with H. Cohen and F. Rodriguez Villegas) Convergence acceleration of alternating series
    Experimental Math. 9 (2000) 3-12 (pdf)
  90. (with R. Lawrence) Modular forms and quantum invariants of 3-manifolds
    Asian J. of Math. 3 (1999) 93-108 (pdf)
  91. Vassiliev invariants and a strange identity related to the Dedekind eta-function
    Topology 40 (2001) 945-960 (pdf)
  92. (with M. Kontsevich) Periods.
    In Mathematics Unlimited--2001 and Beyond (B. Engquist and W. Schmid, eds.), Springer, Berlin-Heidelberg-New York (2001), 771-808 (pdf)
    Japanese translation: In Suugaku no saisentan--21 seiki e no chousen, Vol. 1, Springer, Tokyo (2002), 74-125
  93. Traces of singular moduli
    In "Motives, Polylogarithms and Hodge Theory" (Eds. F. Bogomolov, L. Katzarkov), Lecture Series 3, International Press, Somerville (2002), 209-244 (pdf)
  94. (with H. Gangl) Classical and elliptic polylogarithms and special values of L -series
    In The Arithmetic and Geometry of Algebraic Cycles, Proceedings, 1998 CRM Summer School, Nato Science Series C, Vol. 548, Kluwer, Dordrecht-Boston-London (2000) 561-615 (pdf)
  95. (with J. Lewis) Period functions for Maass wave forms. I
    Annals of Mathematics 153 (2001) 191-258 (pdf)
  96. (with V. Fridkin and Y. Stroganov) Ground state of the quantum symmetric finite size XXZ spin chain with anisotropy parameter \Delta=\frac12 .
    J. Phys. A 33 (2000) L121-L125 (pdf)
  97. Polynomials arising from the tautological ring
    Appendix to C. Faber and R. Pandharipande, "Logarithmic series and Hodge integrals in the tautological ring"
    Mich. Math. J. 48 (2000) 215-252 (appendix: pp. 240-252) (pdf)
  98. (with F. Rodriguez Villegas and J. Voloch) Constructions of plane curves with many points
    Acta Arithmetica 99 (2001) 85-96 (pdf)
  99. A proof of the Kac-Wakimoto affine denominator formula for the strange series
    Mathematical Research Letters 7 (2000) 597-604 (pdf)
  100. (with E. Wirsing) Multiplicative functions with \Delta f(n)\to0
    Acta Arithmetica C.1 (2001) 75-78 (pdf)
  101. (with T. Sasamoto) On the existence of a phase transition for an exclusion process on a ring
    J. Phys. A 34 (2001) 5033-5039 (pdf)
  102. (with V. Fridkin and Y. Stroganov) Finite size XXZ spin chain with anisotropy parameter \Delta=\frac12 . Proceedings of the Baxter Revolution in Mathematical Physics (Canberra, 2000)
    J. Stat. Phys. 102 (2001) 781-794 (pdf)
  103. (with Y. Ohno) Multiple zeta values of fixed weight, depth, and height
    Indagationes mathematicae 12 (2001) 483-487 (pdf)
  104. (with P. Kleban) Crossing probabilities and modular forms
    J. Stat. Phys. 113 (2003), 431-454 (pdf)
  105. Applications of the representation theory of finite groups
    Appendix to S. Lando and A. Zvonkin, Graphs on Surfaces and Their Applications
    Encycl. of Math. Sciences 141, Springer-Verlag, Berlin Heidelberg (2004), 399-427 (pdf)
  106. (with K. Ihara and M. Kaneko) Derivation and double shuffle relations for multiple zeta values
    Compositio Math. 142 (2006), 307-338 (pdf)
  107. (with T. Dokchitser and R. de Jeu) Numerical verification of Beilinson's conjecture for K_2 of hyperelliptic curves
    Compositio Math. 142 (2006), 339-373 (pdf)
  108. (with H. Gangl and M. Kaneko) Double zeta values and modular forms
    In Automorphic Forms and Zeta Functions, Proceedings of the Conference in Memory of Tsuneo Arakawa, S. Böcherer, T. Ibukiyama, M. Kaneko, F. Sato (eds.), World Scientific, New Jersey (2006), 71-106 (pdf)
  109. The Mellin transform and other useful analytic techniques
    Appendix to E. Zeidler, Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists
    Springer-Verlag, Berlin-Heidelberg-New York (2006), 305-323 (pdf)
  110. The dilogarithm function
    In Frontiers in Number Theory, Physics and Geometry II, P. Cartier, B. Julia, P. Moussa, P. Vanhove (eds.), Springer-Verlag, Berlin-Heidelberg-New York (2006), 3-65 (pdf)
  111. (with A. Connes) A geometric property of parallelograms inscribed in ellipses.
    Amer. Math. Monthly 114 (2007), 909-914 (pdf)
  112. The equivalence of the two sets of relations
    Appendix to A. Kouvidakis and G. van der Geer, "Cycle relations on Jacobian varieties"
    Compositio Math. 143 (2007), 900-908 (appendix: pp. 905-907) (pdf)
  113. A proof of (2.40)
    Appendix to T. Eguchi, Y. Sugawara and A. Taormina"Liouville field, modular forms and elliptic genera"
    J. of High Energy Physics 119 (2007), 1-20 (appendix: pp. 17-19) (pdf)
  114. Evaluation of S(m,n)
    Appendix to M. Green, J. Russo and P. Vanhove, "Low energy expansion of the four-particle genus-one amplitude in type II superstring theory" J. of High Energy Physics (2008) 020, 33-34 (pdf)
  115. (with K. Thas) Finite projective planes, Fermat curves and Gaussian periods
    J. Eur. Math. Soc. 10 (2008) 173-190 (pdf)
  116. (with L. Weng) Deligne products of line bundles over moduli spaces of curves.
    Commun. Math. Physics 281 (2008), no. 3, 793-803 (pdf)
  117. Exact and asymptotic formulas for v_n
    Appendix to D. Grünberg and P. Moree, "Sequences of enumerative geometry: congruences and asymptotics. Experim. Math. 17 (2008), 409-426 (Appendix: pp. 423-425) (pdf)
  118. (with A. Zinger) Some properties of hypergeometric series associated with mirror symmetry
    In Modular Forms and String Duality, Fields Institute Communications Vol. 54 (2008), 163-177 (pdf)
  119. Ramanujan's mock theta functions and their applications [d'après Zwegers and Bringmann-Ono],
    Séminaire Bourbaki, 60ème année, 2007-2008, n ^o 986, Astérisque 326 (2009), Soc. Math. de France, 143-164 (pdf)
  120. Elliptic modular forms and their applications
    In The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway (= [10] of "Books"; ed. K. Ranestad), Universitext, Springer-Verlag, Berlin-Heidelberg-New York (2008), pp. 1-103 (pdf)
  121. Integral solutions of Apéry-like recurrence equations
    In Groups and Symmetries: From the Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes, Vol. 47 (2009), Amer. Math. Society, 349-366 (pdf)
  122. (with T. Dimofte, S. Gukov and J. Lenells) Exact results for perturbative Chern-Simons theory with complex gauge group
    Communications in Number Theory and Physics 3 (2009), 363-443 (pdf)
  123. On Harder's \text{SL}(2,\Bbb R) - \text{SL}(3,\Bbb R) identity
    Appendix to G. Harder, "Arithmetic aspects of rank one Eisenstein cohomology"
    In Cycles, Motives and Shimura Varieties, V. Srinivas (ed.), Tata Institute (2010), 131-190 (appendix: pp. 191-195) (pdf)
  124. (with S. Wimmer-Zagier) Some questions and observations around the mathematics of Seki Takakazu
    In Seki, Founder of Modern Mathematics in Japan--A Commemoration on His Tercentenary, E. Knobloch; H. Komatsu; Dun Liu (Eds.), Springer Proceedings in Mathematics and Statistics 39 (2013), 275-297 (pdf)
  125. Section 4.2: Proof of the Harvey-Moore identity
    Appendix to A. Klemm, D. Maulik, R. Pandharipande, E. Scheidegger, Noether-Lefschetz theory and the Yau-Zaslow conjecture, J. Amer. Math. Soc. 23 (2010), 1013-1040 (appendix: pp. 1037-1038) (pdf)
  126. Quantum modular forms
    In Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675 (pdf)
  127. (with Z.-W. Sun) On a curious property of Bell numbers
    Bull. Aust. Math. Soc. 84 (2011), 153-­158 (pdf)
  128. A Family of Combinatorial Identities
    Appendix to Jonathan M. Borwein; Armin Straub; James Wan; Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64 (2012), 961-990 (appendix: pp. 987-989) (pdf)
  129. (with H. Cohen) Vanishing and non-vanishing theta values
    Annales mathématiques du Québec 37 (2013), 45-61 (pdf)
  130. (with M. Vlasenko) Higher Kronecker limit formulas for real quadratic fields
    J. reine angew. Math. 679 (2013), 23-64 (pdf)
  131. Evaluation of the multiple zeta values \zeta(2,\dots,2,3,2,\dots,2)
    Annals of Math. 175 (2012), 977-1000 (pdf)
  132. Appendix A: Asymptotics of the regular cube
    In Stavros Garoufalidis; Roland van der Veen, Asymptotics of classical spin networks, Geom. Topol. 17 (2013), 1-37 (appendix: pp. 28-33) (pdf)
  133. (with R. Bruggeman and J. Lewis) Function theory related to the group PSL(2, \Bbb R ).
    in From Fourier Analysis and Number Theory to Radon Transforms and Geometry. In memory of Leon Ehrenpreis, Farkas, H.M.; Gunning, R.C.; Knopp, M.I.; Taylor, B.A. (Eds.), Springer, Developments in Mathematics, Vol. 28 (2013), 107-201 (pdf)
  134. (with A. Malter and D. Schleicher) New looks at old number theory
    Amer. Math. Monthly 120 (2013), 243-264 (pdf)
  135. Appendix. Curious and exotic identities for Bernoulli numbers
    In T. Arakawa, M. Kaneko and T. Ibukiyama, Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics (2014), Springer Japan, xi + 274 pages (appendix: pp. 239-262) (pdf)
  136. Lines on the Dwork quintic pencil and its higher degree analogues
    J. Differential Geometry 97 (2014), 177-189 (pdf)
  137. Partitions, quasimodular forms, and the Bloch-Okounkov theorem
    Ramanujan Journal 41 (2016), 345-368 (pdf)
  138. (with M. Möller) Modular embeddings of Teichmüller curves
    Compositio Mathematica 152 (2016), 2269-2349 (pdf)
  139. (with V. Golyshev) Proof of the Gamma Conjecture for Fano 3-folds of Picard rank one
    in Russian: Izvest. Russ. Acad. Sci., Math. Series 80 (2016), 27-54
    in English: Izvest. Russ. Acad. Sci., Math. Series 80 (2016), 24-49 (pdf)
  140. (with A. Popa) A combinatorial refinement of the Kronecker-Hurwitz class number relation
    Proc. AMS 145 (2017), 1003-1008 (pdf)
  141. Eigenvalues of the coprimality matrix
    Appendix to G. Mussardo, G. Giudici and J. Viti, "The coprime quantum chain", J. Stat. Mech. (2017) 1-68 (appendix: pp. 51-66) (pdf)
  142. (with A. Pixton) Combinatorial identity
    Appendix to R. P. Thomas and R. Kool, "Stable pairs with descendents on local surfaces I: the vertical component", Pure and Appl. Math. Quart. 13 (2017) 581-638 (appendix: pp. 635-636) (pdf)
  143. (with B. Dubrovin and D. Yang) Classical Hurwitz numbers and related combinatorics
    Moscow Mathematical Journal 17 (2017) 601-633 (pdf)
  144. (with J. Lewis) Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis.
    Res. Math. Sci. 87 (2018) 406-429 (pdf)
  145. (with D. Chen and M. Möller) Quasimodularity and large genus limits of Siegel-Veech constants
    J. Amer. Math. Soc. 31 (2018) 1059-1163 (pdf)
  146. (with C. Dupont) An approach via series
    Appendix to C. Dupont, "Odd zeta motive and linear forms in odd zeta values", Compositio Math. 154 (2018) 342-379 (appendix: pp. 373-378) (pdf)
  147. (with A. Popa) An elementary proof of the Eichler-Selberg trace formula
    J. reine angew. Math. 762 (2020) 105-122 (pdf)
  148. (with YJ. Choie and Y-K. Park) Periods of modular forms on \Gamma_0(N) and products of Jacobi theta functions
    Journal of the European Mathematical Society 21 (2019), 1379-1410 (pdf)
  149. (with M. Griffin, K. Ono, and L. Rolen) Jensen polynomials for the Riemann zeta function and other sequences.
    Proc. Nat. Acad. Sci. 23 (2019), 11103-11110 (pdf)
  150. (with V. Golyshev and D. van Straten), Dimensional interpolation and the Selberg integral.
    Journal of Geometry and Physics 145 (2019) 103455, 7 pages (pdf)
  151. (with S. Garoufalidis) Asymptotics of Nahm sums at roots of unity
    The Ramanujan Journal 55 (2021) 219-238 (pdf)
  152. (with L. Weng) Higher-rank zeta functions for elliptic curves
    Proc. Nat. Acad. Sciences 117 (2020), 4546-4558 (pdf)
  153. (with L. Weng) Higher-rank zeta functions and SL_n -zeta functions for curves
    Proc. Nat. Acad. Sciences 117 (2020), 6398-6408 (pdf)
  154. (with B. Dubrovin and D. Yang) Gromov-Witten invariants of the Riemann sphere
    Pure and Applied Mathematics Quarterly 16 (2020), 153-190 (pdf)
  155. (with D. Chen, M. Möller, and A. Sauvaget) Masur-Veech volumes and intersection theory on the moduli space of Abelian differentials.
    Invent. Math. 222 (2020), 283-373 (pdf)
  156. (with F. Zerbini), Genus-zero and genus-one string amplitudes and special multiple zeta values.
    Communications in Number Theory and Physics 14 (2020), 413-452 (pdf)
  157. (with D. Yang and Y. Zhang) Masur-Veech volumes of quadratic differentials and their asymptotics
    J. Geom. Phys. 158 (2020), 103870, 12 pp. (pdf)
  158. (with V. Golyshev), Interpolated Apéry numbers, quasiperiods of modular forms, and motivic gamma functions
    In Integrability, Quantization, and Geometry, Vol. II. Quantum theories and algebraic geometry, I. Krichever, S. Novikov, O. Ogievetsky and S. Shlosman (Eds.), Proceedings Symp. Pure Math. 103.2, Amer. Math. Soc., Providence, RI, (2021), 281-301 (pdf)
  159. (with B. Dubrovin and D. Yang) On tau-functions for the KdV hierarchy
    Selecta Math. (N.S.) 27 (2021), no. 1, Paper No. 12, 47 pages (pdf)
  160. (with L. Touzo and M. Marsili) Information thermodynamics of financial markets: the Glosten-Milgrom model
    J. Stat. Mech. Theory Exp. 2021, no. 3, Paper No. 033407, 21 pp. (pdf)
  161. Power partitions and a generalized eta transformation property.
    Hardy-Ramanujan Journal 44 (2021), 1-18 (pdf)
  162. (with F. Gonçalves) Strichartz estimates with broken symmetries
    Rev. Mat. Iberoam. 38 (2022), no. 5, 1709-1722 (pdf)
  163. (with E. Garcia-Failde) A curious identity that implies Faber's conjecture
    Bulletin of the London Mathematical Society 54 (2022), 1839-1845. (pdf)
  164. (with F. Calegari and S. Garoufalidis) Bloch groups, algebraic K -theory, units, and Nahm's conjecture
    Annales Scientifiques de l'École Normale Supérieure, 4th series, 56 (2023), no. 2, 383-426
    (arXiv:1712.04887 and arXiv:1812.07690) (pdf)
  165. (with D. Radchenko) Arithmetic properties of the Herglotz function.
    J. reine angew. Math. 2023 797 (2023), 229-253 (arXiv:2012.15805) (pdf)
  166. (with S. Garoufalidis) Hyperbolic 3-manifolds, the Bloch group, and the work of Walter Neumann.
    Celebratio Mathematica 2023, 19 pages (arXiv:2304.12545) (pdf)
  167. (with N. Skoruppa and V. Gritsenko) Theta blocks.
    Journal of the European Mathematical Society, to appear, 60 pages. (arXiv:1907:00188) (pdf)
  168. (with B. Dubrovin and D. Yang) Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality.
    Adv. Math. 433 (2023), Paper No. 109311, 73 pp. (arXiv:2101.10924) (pdf)
  169. (with S. Garoufalidis) Knots and their related q -series,
    SIGMA 19 (2023), Paper No. 082, 39 pp. (arXiv:2304.09377) (pdf)
  170. (with K. Nagatomo and Y. Sakai) Modular linear differential equations and generalized Rankin-Cohen brackets
    Trans. AMS, to appear, 34 pages, arXiv:2210.10686 (pdf)
  171. (with K. Bönisch A. Klemm, and E. Scheidegger) D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms and periods
    Communications in Mathematical Physics 405 (2024), Paper No. 134, 71 pages, arXiv:2203.09426 (pdf)
  172. (with S. Garoufalidis) Knots, perturbative series, and quantum modularity
    SIGMA, to appear, 97 pages, arXiv:2111.06645 (pdf)

Other articles (announcements, expository, recreational)

  1. Une formule-limite de Kronecker pour les corps quadratiques réels
    Sém. de Théorie de Nombres, Bordeaux 1973-74, Exposé No. 21, 5 pages (pdf)
  2. The first 50 million prime numbers [=English translation of Monograph 5 below]
    New Mathematical Intelligencer 0 (1977) 1-19 (pdf)
    Russian translation: Pervie 50 millionov prostikh chisel, Uspekhi Mat. Nauk 39 (1984) 175-190 (pdf)
    Japanese translation: Saisho no gosemman no sosuu, Suugaku Seminaru 24 (1985) 38-48 (pdf)
    Spanish translation: Los Primeros 50 Milliones de Números Primos, ?? (pdf)
  3. (with Joe Harris) Problem
    Math. Intell. 2(2), 107 (1979) (pdf)
  4. A simple proof of the prime number theorem [in Chinese]
    Shue Xue Yi Lin 1 (1982) 259-262 (pdf)
  5. What are modular forms? [in Chinese]
    Shue Xue Yi Lin 1 (1982) 329-335 (pdf)
  6. Lösungen von Gleichungen in ganzen Zahlen
    in Miscellanea mathematica, Springer-Verlag (1991) 311-326 (pdf)
  7. (with B. Gross) Points de Heegner et dérivées des fonctions
    C.R. Acad. Sci. Paris (I) 297 (1983) 85-87 (pdf)
  8. L -series of elliptic curves, the Birch-Swinnerton-Dyer conjecture, and the class number problem of Gauss
    Notices AMS 31 (1984) 739-743 (pdf)
  9. L -series and the Green's functions of modular curves
    Proc. Internat. Congress of Math. 1986, A.M.S. (1987) 689-698 (pdf)
  10. The remarkable dilogarithm
    J. Math. and Phys. Sciences 22 (1988) 131-145
    reprinted as: The dilogarithm function in geometry and number theory
    in Number Theory and Related Topics, papers presented at the Ramanujan Colloquium, Bombay 1988, Tata and Oxford (1989) 231-249 (pdf)
  11. Primzahlen: Theorie und Anwendung
    Rheinisch-Westfälische Akad. Wiss. Vorträge No. 370, West-deutscher Verlag (1989) Opladen, 35-47 (pdf)
  12. Elliptische Kurven: Fortschritte und Anwendungen
    Jahreber. d. Dt. Math.-Verein. 92 (1990) 58-76 (pdf)
  13. How often should you beat your kids?
    Math. Magazine 63 (1990) 89-92 (pdf)
  14. English-Japanese dictionary for mathematical terminology
    Japan Intelligencer, Springer-Verlag (1990) 74-77 (pdf)
  15. Introduction to modular forms
    in From Number Theory to Physics, eds. M. Waldschmidt et al, Springer-Verlag, Heidelberg (1992) 238-291 (pdf)
    Previous version: Modular Forms of One Variable
    Notes based on a course given in Utrecht, Spring 1991 (pdf)
  16. (with J.-F. Mestre, R. Schoof and L. Washington) Quotients homophones des groupes libres/Homophonic quotients of free groups
    Experimental Math. 2 (1993) 153-155 (pdf)
  17. Periods of modular forms, traces of Hecke operators, and multiple zeta values
    in Hokei-keishiki to L-kansuu no kenkyuu (= Research on Automorphic Forms and L-Functions), RIMS Kokyuroku 843 (1993) 162-170 (pdf)
  18. Two results in number theory with elementary aspects
    in Daisuuteki suuron--Saikin no shinten to sono haikei (= Algebraic Number Theory: Recent Developments and Their Backgrounds), RIMS Kokyuroku 844 (1993) 84-86 (pdf)
  19. Oplossingen van vergelijkingen in rationale getallen
    in De Laatste Stelling van Fermat, Universiteit Utrecht (1993) 8 pages (pdf)
    and in Speeltuin van de Wiskunde, B. de Smit and J. Top (eds.), Veen Magazines/Natuurwetenschap en Techniek, Diemen (2003), 105-112
  20. Quelques conséquences surprenantes de la cohomologie de SL(2,\Bbb Z)
    in Leçons de Mathématiques d'aujourd'hui, Cassini, Paris (2000), 99-123 (pdf)
  21. Improved zeta values and unproved formulas
    in Proceedings, Workshop on Polylogarithms, Essen (1997) 24-25 (pdf)
  22. Some problems related to polylogarithms
    in Proceedings, Workshop on Polylogarithms, Essen (1997) 51-54 (pdf)
  23. Newman's short proof of the prime number theorem
    Amer. Math. Monthly 104 (1997) 705-708 (pdf)
  24. (with H. Cohen) Number theory
    Accepted by the Enciclopedia del Novecento (1997?), 25 pages; never appeared
  25. Leçon inaugurale, Chaire de Théorie des Nombres
    Collège de France, Publication No. 161 (2002), 1-35 (pdf)
  26. New points of view on the Selberg zeta function
    Proceedings of the Japanese-German Seminar "Explicit Structures of Modular Forms and Zeta Functions", Ryushi-do (2002), 1-10 (pdf)
  27. Summen von Quadratzahlen, Summen von Kubikzahlen
    Jahrbuch 2001 der Deutschen Akademie der Naturforscher Leopoldina, Leopoldina 47 (2002) 407-413 (pdf)
  28. (with G. Harder) La conjecture de Birch et Swinnerton-Dyer: nombre magique pour problème elliptique
    Les Dossiers de La Recherche, 20, (2005), 48-53 (pdf)
  29. Ramanujan à Hardy: de la première à la dernière lettre
    Hors-série No. 25, Grands mathématiciens modernes, Tangente (2006), 16-19.
    Reprinted in Bibliothèque Tangente, Grands mathématiciens modernes, Éditions Pole, Paris (2006), 60-65. (pdf)
    Short popularized version of 36 below
  30. Ramanujan an Hardy: vom ersten bis zum letzten Brief
    Jahresbericht of the Mathematisches Forschungsinstitut Oberwolfach (2008), 27-42 [German translation of the preceding, together with the French original] (pdf)
    Reprinted in Mitteilungen der Deutschen Mathematiker-Vereinigung 18 (2010), 21-28 (pdf)
  31. Une passion pour les mathématiques
    in Mathématiques, un dépaysement soudain, Ed. Fondation Cartier pour l'art contemporain, Paris (2011), 90-97 (pdf)
    English translation: A passion for mathematics
    in Mathematics--A Beautiful Elsewhere, Ed. Fondation Cartier pour l'art contemporain, Paris (2011), 90-97 (pdf)
  32. (with M.F. Atiyah) Friedrich Hirzebruch (1927-2012)
    Notices Amer. Math. Soc. 61 (2014), 706-727 (pdf)
  33. A refinement of a theorem of J. E. Littlewood
    Amer. Math. Monthly 121 (2014), 618 (pdf)
  34. Nachruf auf Friedrich Hirzebruch
    Jahrbuch der Nordrhein-Westfälischen Akademie der Wissenschaften und der Künste (2014), 116-121 (pdf)
  35. Life and Work of Friedrich Hirzebruch
    Jahresbericht der DMV 117 (2015), pp. 93-132 (pdf)
  36. The arithmetic and topology of differential equations
    in Proceedings of the European Congress of Mathematics, Berlin, 18-22 July, 2016, Mehrmann, V.; Skutella, M. (Eds.), European Mathematical Society (2018), 717-776 (pdf)
  37. (with A. Pohl) Dynamics of geodesics and Maass cusp forms
    Enseign. Math. 66 3-4 (2020), 305-340 (pdf)


  1. Equivariant Pontrjagin Classes and Applications to Orbit Spaces
    Lecture Notes 290, Springer-Verlag, Berlin-Heidelberg-New York (1972), 130+x pages (pdf)
  2. (with F. Hirzebruch) The Atiyah-Singer Theorem and Elementary Number Theory
    Math. Lecture Series 1, Publish or Perish, Boston (1974), 262+xii pages (pdf)
  3. (co-editor with J-P. Serre) Modular functions of One Variable V
    Lecture Notes 601, Springer-Verlag, Berlin-Heidelberg-New York (1977), 294+vi pages (pdf)
  4. (co-editor with J-P. Serre) Modular functions of One Variable VI
    Lecture Notes 627, Springer-Verlag, Berlin-Heidelberg-New York (1977), 339+vi pages (pdf)
  5. Die ersten 50 Millionen Primzahlen
    Elemente der Mathematik, Beiheft 15, Birkhäuser-Verlag, Basel (1977), 24 pages (pdf)
  6. (with W. Borho, J. Rohlfs, H. Kraft and J.-C. Jantzen) Lebendige Zahlen
    Mathematische Miniaturen 1, Birkhäuser-Verlag, Boston-Basel (1981), 135 pages [my contribution, an updated version of the preceding, is on pp. 39-73] (pdf)
    Russian translation: Zhiviye Chisla, Mir, Moscow (1985), 128 pages [my contribution: pp. 42-72] (pdf)
    Bulgarian translation: Zhivy Chisla, Nauka i Izkustvo, Sofia (1987), 127 pages [my contribution: pp. 44-72]
  7. Zetafunktionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie
    Hochschultext, Springer-Verlag, Berlin-Heidelberg-New York (1981), 149+vi pages (pdf)
    Japanese translation: Suuron Nyuumon--zeta-kansuu to nijitai, Iwanami Shoten, Tokyo (1990), 174 pages
  8. (with M. Eichler) The Theory of Jacobi Forms
    Progress in Math. 55, Birkhäuser-Verlag, Basel-Boston (1985), 148 pages (pdf)
  9. Hokei-keishiki-ron no wadai kara (=Topics in the Theory of Automorphic Forms)
    [lecture notes, in Japanese, by M. Kaneko]
    Mathematical lecture note series, Kyushu University (1992), 80 pages (pdf)
  10. (with J. Bruinier, G. Harder and G. van der Geer) The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway (ed. K. Ranestad)
    Universitext, Springer-Verlag, Berlin-Heidelberg-New York (2008), 280 pages [my contribution: pp. 1-103] (pdf)
  11. (with R. Bruggeman and J. Lewis) Period functions for Maass forms and cohomology
    Memoirs of the AMS, Vol. 237, No. 1118 (2015), xii + 132 pages (pdf)
  12. (co-editor with J. Schwermer and S. Wimmer-Zagier) Friedrich Hirzebruch Gesammelte Abhandlungen/Collected Papers, Band III 1988-2012
    Springer Nature Switzerland (2019), xvi + 659 pages
  13. (with Atish Dabholkar and Sameer Murthy) Quantum Black Holes, Wall Crossing, and Mock Modular Forms
    arXiv:1208.4074, to appear in Cambridge Monographs in Mathematical Physics, 314 pages (pdf)


  1. Proof of Proposition 1
    Appendix to C. Joyner and U. Smilansky, "Dyson's Brownian-motion model for random matrix theory - revisited", arXiv:1503.06417, (appendix: pp. 12-14) (pdf)
  2. On the values of the function F_{k,\varepsilon}(s)
    Appendix to S. Goette and J. Nordström, "Nu-invariants of extra-twisted connected sums"
    68 pages (appendix: pp. 63-67), arXiv:2010.16367 (pdf)
  3. (with O. Lechtenfeld) Computing the potential
    Appendix to O. Lechtenfeld, "A hyperbolic Kac-Moody Calogero model", 26 pages (appendix: pp. 20-24), arXiv:2203.06519, (pdf)
  4. (with T. Ibukiyama) Higher spherical polynomials
    Under revision, current length 221 pages
  5. (with F. Cléry, E. Ferapontov, and A. Odesskii) Integrable Lagrangians and Picard modular forms
    43 pages (pdf)