Jeffrey C. Lagarias: Publications list

For papers that have already been published, the versions available here are those of the early preprints, and do not contain corrections made in the editing process, notes added in proof, etc. For earlier papers whose postscript versions don't work properly, please look at the pdf versions instead.
Recent eprints are also available at the arXiv in pdf and ps versions.
Most of them can be obtained at the front end for the arXiv at [http://front.math.ucdavis.edu]

[Unpublished preprints]

  • Succinct certificates for solutions to binary quadratic Diophantine equations
    J. C. Lagarias,
    [arxiv.org/math/0611209].

  • 3x+1 Problem Annotated Bibliography (1963-1999) ,
    Jeffrey C. Lagarias,
    unpublished,
    [arxiv.org/math/0309224]

  • 3x+1 Problem Annotated Bibliography II (2000-) ,
    Jeffrey C. Lagarias,
    unpublished-updated periodically,
    [arxiv.org/math/0608208]


    [2015]

  • A Skolem-Mahler-Lech theorem for iterated automorphisms of K-algebras,
    Jason Bell and Jeffrey C. Lagarias,
    Canad. J. Math. 67 (2015) , No. 2, 286--314.
    [arxiv:1312.5305].

  • Splitting behavior of S_n-polynomials,
    Jeffrey C. Lagarias and Benjamin L. Weiss,
    Research in Number Theory 1 (2015) , No. 7, 30 pages.
    [arxiv:1408.6251].


    [2014]

  • Ergodicity of the action of K^* on A_K,
    Jeffrey C. Lagarias,
    IMRN 2014 , No. 18, 5165--5186.
    [arxiv:1211.3256].

  • Path sets in one-sided symbolic dynamics,
    W. C. Abram and Jeffrey C. Lagarias,
    Adv. Applied Math. 56 (2014) , No. 1, 109--134.
    [arxiv:1207.5004].

  • p-adic path set fractals and arithmetic,
    W. C. Abram and Jeffrey C. Lagarias,
    J. Fractal Geometry 1 (2014) , No. 1, 45--81.
    [arxiv:1210.2478].

  • Intersections of multiplicative translates of 3-adic Cantor sets,
    W. C. Abram and Jeffrey C. Lagarias,
    J. Fractal Geometry 1 (2014) , No. 4, 349--390.
    [arxiv:1308.3133].


    [2013]

  • Euler's constant: Euler's work and modern developments,
    Jeffrey C. Lagarias,
    Bull. Amer. Math. Soc. 50 (2013) , No. 4, 527--628.
    [arxiv:1303.1856].

  • The maximal density of product-free sets in Z/nZ,
    P. Kurlberg, J. C. Lagarias and C. Pomerance,
    Int. Math. Res. Notices IMRN Vol. 2013, No. 4, 827-845.
    [arxiv:1111.2634].

  • On sets of integers which are both sum-free and product-free,
    P. Kurlberg, J. C. Lagarias and C. Pomerance,
    INTEGERS 12B (2012/2013),Paper #A4, 9pp.
    [arxiv:1201.1317].

  • Random Hill's Equation, Random Walks and Products of Random Matrices,
    F. Adams, A. Bloch and J. C. Lagarias
    Chap. 17 in: Recend Trends in Dynamical Systems, Springer Proc. Math. Stat. 35
    Springer: Basel 2013, 447--470.
    DOI 10.1007/978-3-0348-0451-6-17 .


    [2012]

  • Mysteries in packing regular tetrahedra,
    Jeffrey C. Lagarias and Chuanming Zong,
    Notices Amer. Math. Soc. 59 (2012) , No. 11, 1540-1549.
    [Notices AMS reprint].

  • Product-free sets of high density,
    P. Kurlberg, J. C. Lagarias and C. Pomerance,
    Acta Arithmetica 155 (2012), No. 2, 163-173.
    [arxiv:1107.5589].

  • The Lerch zeta function I. Zeta integrals,
    J. C. Lagarias and W-C. Winnie Li,
    Forum Mathematicum 24 (2012), No. 1, 1-48
    [arxiv:1005.4712].

  • The Lerch zeta function II. Analytic continuation,
    J. C. Lagarias and W-C. Winnie Li,
    Forum Mathematicum 24 (2012), No. 1, 49--84
    [arxiv:1005.4967].

  • Convergence of the restricted Nelder-Mead algorithm in two dimensions,
    J. C. Lagarias, B. Poonen and M. H. Wright
    SIAM J. Optimization 22 (2012), No. 2, 501-532
    [arxiv:1104.4350].

  • The Takagi function and its properties,
    Jeffrey C. Lagarias,
    in: Functions in Number Theory and their Probabilistic Aspects,
    (K. Matsumoto, Editor in Chief, S. Akiyama, H. Nakada, H. Sugita, A. Tamagawa, Eds.),
    RIMS K\^{o}ky\^{u}roku Bessatsu B34, Aug. 2012, pp. 153--189.
    [arxiv:1112.4205].

  • Counting smooth solutions to the equation A+B=C,
    J. C. Lagarias and K. Soundararajan,
    Proc. London Math. Soc. 104 (2012), No. 4, 770-798
    [arxiv:1102.4911].

  • Level sets of the Takagi function: Local level sets,
    J. C. Lagarias and Z. Maddock,
    Monatshefte fur Math.. 166 (2012), No. 2, 201-238.
    [arxiv.org/abs/1011.3183]


    [2011]

  • Level sets of the Takagi function: Generic level sets,
    J. C. Lagarias and Z. Maddock,
    Indiana Univ. Math. J. 60 (2011), 1857--1884. [arxiv.org/abs/1011.3183]

  • The integral of the Riemann xi function,
    J. C. Lagarias and D. Montague,
    Commentarii Math. Univ. Sancti Pauli, 60 (2011), No. 1 & 2, 143-169.
    [arxiv.org/abs/1106.4348]

  • Smooth solutions to the abc equation: the xyz conjecture,
    J. C. Lagarias and K. Soundararajan,
    J. Theorie Nombres Bordeaux, 23 (2011), No. 1, 209-234.
    [arxiv.org/abs/0911.4147]


    [2010]

  • Cyclic systems of simultaneous congruences,
    Jeffrey C. Lagarias,
    International Journal of Number Theory 6 (2010), No. 2, 219--245
    [arxiv.org/abs/0712.3954].

  • Stochastic Models for the 3x+1 and 5x+1 problems,
    Alex Kontorovich and Jeffrey C. Lagarias
    in The Ultimate Challenge: The 3x+1 Problem
    Amer. Math. Soc.: Providence 2010, 131-188.
    [arxiv:0910.1944].


    [2009]

  • Ternary expansions of powers of 2 ,
    Jeffrey C. Lagarias,
    J. London Math. Soc. 79 (2009), 562--588.
    [arxiv.org/math/0512006].

  • The Schrodinger operator with Morse potential on the right half line,
    Jeffrey C. Lagarias,
    Communications in Number Theory and Physics, 3 (2009), No. 2, 323--361
    [arxiv.org/abs/0712.3238].


    [2008]

  • The Kruskal count ,
    J. C. Lagarias, E. Rains and R. Vanderbei,
    pp. 371--391 in: The Mathematics of Preference, Choice and Order. Essays in Honor of Peter Fishburn.
    (S. Brams, W. V. Gehrlein, F. S. Roberts Eds.), Springer-Verlag: New York.
    [arxiv.org/math/0110143]



    [2007]

  • Li Coefficients for Automorphic L-Functions ,
    Jeffrey C. Lagarias,
    Annales Inst. Fourier 57 (2007), 1689--1740 .
    [arxiv.org/math/0404394] .

  • Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings ,
    Nicholas Eriksson and Jeffrey C. Lagarias,
    Ramanujan Journal 14 (2007), No.3, 437--469.
    [arxiv.org/math/0403296]


    [2006]

  • Benford's law for the 3x+1 function ,
    Jeffrey C. Lagarias and K. Soundararajan,
    J. London Math Soc., 74 (2006), 289--303.
    [arxiv.org/math/0509175].

  • The Riemann hypothesis for certain integrals of Eisenstein series ,
    Jeffrey C. Lagarias and Masatoshi Suzuki,
    J. Number Theory 118 (2006), 98--122.
    [arxiv.org/math/0412039].

  • Wild and Wooley numbers,
    Jeffrey C. Lagarias,
    Amer. Math. Monthly 113 (2006), 97--108.
    [arxiv.org/math/0411141].

  • The 3x+1 semigroup ,
    David Applegate and Jeffrey C. Lagarias,
    J. Number Theory, 117 (2006), 146--159 .
    [arxiv.org/math/0411140].

  • Apollonian Circle Packing: Geometry and Group Theory, II. Super-Apollonian group and Integral Packings ,
    R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. H. Yan,
    Discrete & Computational Geometry 35 (2006), 1--36.
    [arxiv.org/math/0010302]

  • Apollonian Circle Packing: Geometry and Group Theory, III. Higher dimensions ,
    R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. H. Yan,
    Discrete & Computational Geometry 35 ( 2006), 37--72.
    [arxiv.org/math/0010324]

  • Hilbert spaces of entire functions and Dirichlet L-functions,
    J. C. Lagarias,
    pp. 365--377 in: Frontiers in Number Theory, Physics and Geometry 1:
    On Random Matrices, Zeta Functions and Dynamical Systems
    (P. E. Cartier, B. Julia, P. Moussa and P. van Hove, Eds.), Springer-Verlag: Berlin 2006.
    [PostScript]  [Pdf]

  • The Riemann Hypothesis: Arithmetic and Geometry ,
    J. C. Lagarias,
    pp. 127--141 in: Surveys in Noncommutative Geometry,
    Clay Insititute Monographs, No. 6, CMI/AMS: Providence, RI 2006.
    [Pdf]



    [2005]

  • Apollonian Circle Packing: Geometry and Group Theory, I. The Apollonian group ,
    R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. H. Yan ,
    Discrete & Computational Geometry 34 (2005), 547--585 .
    [arxiv.org/math.MG/0010298]

  • A note on absolute derivations and zeta functions ,
    Jeffrey C. Lagarias,
    pp. 279--285 in: Number Fields and Function Fields--Two Parallel Worlds
    (G. van der Geer, B. Moonen, R. Schoof, Eds.), Birkhauser: Boston 2005 .
    [arxiv.org/math.NT/0404350] .

  • Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers ,
    J. C. Lagarias and E. Rains,
    J. Difference Eqns. Appl. 11 (2005), No. 12, 1089--1108.
    [arxiv.org/math.DS/0301294]

  • Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles ,
    J. C. Lagarias and E. Rains,
    J. Difference Eqns. Appl. 11 (2005), No. 13, 1137--1163.
    [arxiv.org/math.MG/0303007]

  • Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra ,
    J. C. Lagarias and E. Rains,
    J. Difference Eqns. Appl. 11 (2005), No. 14, 1205--1224.
    [arxiv.org/math.MG/0505103]

  • Zero spacing distributions for differenced L-functions ,
    J. C. Lagarias,
    Acta Arithmetica 120 (2005), No. 2, 159--184.
    [arxiv.org/math.NT/0601653]


    [2004]

  • Area inequalities for embedded disks spanning unknotted curves ,
    J. Hass, J. C. Lagarias and W. P. Thurston,
    J. Differential Geometry 66 (2004), 495--525.
    [arxiv.org/math.DG/0306313]

  • Approximate Squaring ,
    J. C. Lagarias and N. J. A. Sloane,
    Experimental Math. 13 (2004), No. 1, 113--128.
    [arxiv.org/math.NT/0309389]

    [2003]

  • Repetitive Delone sets and quasicrystals ,
    Jeffrey C. Lagarias and Peter A. B. Pleasants,
    Ergod. Th. Dyn. Sys. 23 (2003), 831--867 . [PostScript]  [Pdf]
    [arxiv.org/math.DS/9909033]

  • Lower bounds for the total stopping time of 3x+1 iterates ,
    D. Applegate and J. C. Lagarias,
    Math. Comp. 72 (2003), 1035--1049 . [PostScript]  [Pdf]
    [arxiv.org/math.NT/0103054]

  • Apollonian Packings: Number Theory ,
    R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks, and C. H. Yan ,
    J. Number Theory 100 (2003), 1--45. . [PostScript]  [Pdf]
    [arxiv.org/math.NT/0009113]

  • Substitution Delone Sets ,
    J. C. Lagarias and Yang Wang,
    Discrete & Computational Geometry 29} (2003), No. 2, 175--209 . [PostScript]  [Pdf]
    [arxiv.org/math.NT/0110222]

  • On a two-variable zeta function for number fields ,
    J. C. Lagarias and E. Rains,
    Annales Inst. Fourier 53 (2003), No. 1, 1--68 . [PostScript]  [Pdf]
    [arxiv.org/math.NT/0104176]

  • Bounds for the 3x+1 problem using difference inequalities ,
    Ilia Krasikov and J. C. Lagarias,
    Acta Arithmetica 109}, (2003), No. 3, 237--258 . [PostScript]  [Pdf]
    [arxiv.org/math.NT/0205002]

  • The minimal number of triangles needed to span a polygon embedded in R^d ,
    Joel Hass and J. C. Lagarias,
    in: Discrete and Computational Geometry: The Goodman-Pollack Festschrift
    (B. Aronov, S. Basu, J. Pach, and M. Sharir, Eds), Springer-Verlag: New York 2003, pp. 509--526 . [PostScript]  [Pdf]
    [arxiv.org/math.GT/0211364]

    [2002]

  • An elementary problem equivalent to the Riemann hypothesis ,
    J. C. Lagarias,
    Amer. Math. Monthly 109 (2002), 534--543. [PostScript]  [Pdf]
    [arxiv.org/math.NT/0008177]

  • Bounds for local density of sphere packings and the Kepler conjecture ,
    J. C. Lagarias,
    Discrete & Computational Geometry 27 (2002), 165--193. [PostScript]  [Pdf]
    [arxiv.org/math.MG/0008151]

  • Pinwheel scheduling: achievable densities ,
    P. C. Fishburn and J. C. Lagarias,
    Algorithmica 34 (2002), No. 1, 14--38. [PostScript]  [Pdf]

  • Beyond the Descartes circle theorem ,
    J. C. Lagarias, C. L. Mallows and A. Wilks,
    Amer. Math. Monthly 109 (2002), 338--361. [PostScript]  [Pdf]
    [arxiv.org/math.MG/0101066]

  • Counting dyadic equipartitions of the unit square ,
    J. C. Lagarias, J. H. Spencer and J. P. Vinson,
    Discrete Math. 257 (2002), No. 2-3, 481--499 . [PostScript]  [Pdf]

  • Local Complexity of Delone Sets and Crystallinity,
    J. C. Lagarias and P. A. B. Pleasants,
    Bull. Canadian Math. Soc. 45 (2002), No. 4, 634--652. [PostScript]  [Pdf]
    [arxiv.org/math.MG/0105088]

  • The EKG sequence ,
    J. C. Lagarias, E. Rains and N. J. A. Sloane,
    Experimental Math. 11 (2002), No. 3, 437--446 . [PostScript]  [Pdf]
    [arxiv.org/math.MG/0204011]

    [2001]

  • The number of Reidemeister moves needed for unknotting ,
    Joel Hass and Jeffrey C. Lagarias,
    J. Amer. Math. Soc. 14 (2001), 399--428. . [PostScript]  [Pdf]
    [arxiv.org/math.GT/9807012]

  • Universal spectra and Tijdeman's conjecture on factorization of cyclic groups ,
    J. C. Lagarias and Sandor Szabo,
    J. Fourier Anal. Appl. 7 (2001), 63--70 . [PostScript]  [Pdf]
    [arxiv.org/math.FA/0008132]

  • Robustness of single loop sigma-delta modulation with constant inputs ,
    S. Gunturk, J. C. Lagarias and V. Vaishampayan,
    IEEE Trans. Information Theory , 47 (2001), No. 5, 1735--1745 . [PostScript]  [Pdf]

  • On the normality of arithmetical constants ,
    J. C. Lagarias,
    Experimental Math. , 10 (2001), 355--368 . [PostScript]  [Pdf]
    [arxiv.org/math/0101055]

  • Cutting sequences for geodesic flow on the modular surface and continued fractions ,
    D. Grabiner and J. C. Lagarias,
    Monatshefte Math. 133, No. 4 (2001), 295--339 . [PostScript]  [Pdf]
    [arxiv/math:9707215]

    [2000]

  • Orthogonal bases of exponentials for the n-cube ,
    Jeffrey C. Lagarias, James A. Reeds and Yang Wang,
    Duke Math. J. 103 (2000), 25--37 . [PostScript]  [Pdf]

  • Well-spaced labellings of points in rectangular grids ,
    J. C. Lagarias,
    SIAM J. Discrete Math. 13 (2000), 521-534. [PostScript]  [Pdf]

  • Orthogonality criteria for compactly refinable functions and refinable function vectors ,
    J. C. Lagarias and Yang Wang,
    J. Fourier. Anal. Appl. 6 (2000), 153--170 . [PostScript]  [Pdf]

  • Mathematical quasicrystals and the problem of diffraction ,
    J. C. Lagarias,
    in: Directions in Mathematical Quasicrystals, M. Baake and R. V. Moody, Eds.),.
    CRM Monograph Series, Vol. 13, Amer. Math. Soc.: Providence, RI, 2000, pp. 61--93 . [PostScript]  [Pdf]

  • The impact of aperiodic order on mathematics ,
    J. C. Lagarias,
    Materials Science & Engineering A 294--296 (2000), 186--191. [PostScript]  [Pdf]

  • The lap counting function and zeta function of the tent map ,
    Leopold Flatto and Jeffrey C. Lagarias,
    Nonlinearity 13 (2000), 61--93 [PostScript]  [Pdf]

    [1999]

  • Complements to Li's criterion for the Riemann hypothesis ,
    Enrico Bombieri and Jeffrey C. Lagarias,
    J. Number Theory 77 (1999), 274--287 . [PostScript]  [Pdf]

  • On a positivity property of the Riemann xi function ,
    Jeffrey C. Lagarias,
    Acta Arithmetica 89 (1999), No. 3, 217--234 . [PostScript]  [Pdf]

  • The Computational Complexity of Knot and Link Problems
    Joel Hass, Jeffrey C. Lagarias and Nick Pippenger,
    J. A. C. M. 46 (1999), 185--211 . [PostScript]  [Pdf]
    [arxiv.org/math.GT/9807016]

  • Number theory zeta functions and dynamical zeta functions ,
    J. C. Lagarias,
    in: Spectral Problems in Geometry and Arithmetic (T. Branson, Ed.),
    Contemporary Math.,Vol. 237, AMS: Providence, R. I. 1999, pp. 45--86 [PostScript]  [Pdf]

  • Processor Shadowing: Maximizing Expected Throughput in Fault-Tolerant Systems ,
    J. L. Bruno, E. G. Coffman, Jr., J. C. Lagarias, T. J. Richardson and P. W. Shor,
    Mathematics of Operations Research 24 (1999), 362--382.

  • Bounds for the Number of DC Operating Points of Transistor Circuits ,
    Jeffrey C. Lagarias and Ljiljana Trajkovic,
    IEEE Transactions in Circuits and Systems 46 (1999) , No. 10, 1216--1221. [Postscript]  [Pdf]

  • Geometric Models for Quasicrystals I. Delone Sets of Finite Type ,
    Jeffrey C. Lagarias,
    Discrete & Computational Geometry 2 (1999), 161--191. [PostScript]  [Pdf]

  • Geometric Models for Quasicrystals II. Local Rules Under Isometries ,
    Jeffrey C. Lagarias,
    Discrete & Computational Geometry 21 (1999), 345--372. [PostScript]  [Pdf]

  • Beurling generalized integers with the Delone property ,
    Jeffrey C. Lagarias,
    Forum Math. 11 (1999), 295--312. [PostScript]   [Pdf]

  • Who wins misere Hex?
    J. Lagarias and D. Sleator
    in: The Mathemagician and Pied Puzzler (E. Berlekamp and T. Rogers, Eds.),
    A.K. Peters: Natick Ma. 1999, pp. 253--266. [Pdf]

    [1998]

  • Multiregular Point Systems ,
    Nikolai P. Dolbilin, Jeffrey C. Lagarias, and Marjorie Senechal,
    Discrete & Computational Geometry , 20 (1998), 477--498. [PostScript]  [Pdf]

  • Convergence Properties of the Nelder-Mead Simplex Algorithm in Low Dimensions
    Jeffrey C. Lagarias, James A. Reeds, Margaret H. Wright and Paul E. Wright,
    SIAM J. Optimization , 9 (1998), No. 1, 112--147.
    Also: Computing Sciences Research Center, Bell Laboratories, Technical Report 96-4-07, [PostScript]   [Pdf]

  • Threading Homotopies and DC Operating Points of Nonlinear Circuits ,
    Ross Geoghegan, Jeffrey C. Lagarias and Robert C. Melville,
    SIAM J. Optimization 9 (1998), No. 1, 159--178. [PostScript]  [Pdf]

  • Interference-Minimizing Colorings of Distance-Regular Graphs ,
    P. C. Fishburn, J. H. Kim, J. C. Lagarias and P. E. Wright,
    SIAM J. Discrete Math. 11 (1998), pp. 15-40. [PostScript]  [pdf]

  • Counting d-Step Paths in Extremal Dantzig Figures ,
    J. C. Lagarias and N. Prabhu,
    Discrete & Computational Geometry 19 (1998), pp. 19-31. [PostScript]  [Pdf]

    [1997]

  • The Computational Complexity of Knot and Link Problems. (Preliminary Version)
    Joel Hass, Jeffrey C. Lagarias and Nicholas Pippenger,
    Proc. 38th Annual Symposium on Foundations of Computer Science ,
    IEEE Press 1997, pp. 172-181. [PostScript]  [Pdf]

  • Saddle point and Hamiltonian structure in excitatory-inhibitory networks ,
    H. S. Seung, T. J. Richardson, J. C. Lagarias and J. J. Hopfield,
    Proc. Neural Information Processing Systems '97 , [PostScript]  [Pdf]

  • Linear Fractional Transformations of Continued Fractions with Bounded Partial Quotients ,
    J. C. Lagarias and J. O. Shallit,
    Journal de Theorie des Nombres de Bordeaux 9 (1997), pp. 267-279. [Postscript]  [Pdf]

  • Spectral Sets and Factorizations of Finite Abelian Groups ,
    Jeffrey C. Lagarias and Yang Wang,
    J. Functional Analysis, 143 (1997), pp. 73-98. [PostScript]  [Pdf]

  • Multivariate Descartes Rule of Signs and Sturmfels's Challenge Problem ,
    Jeffrey C. Lagarias and Thomas J. Richardson,
    The Mathematical Intelligencer 19, No. 3 (1997), pp. 9-15. [PostScript]   [Pdf]

  • Convexity and the Average Curvature of Plane Curves ,
    Jeffrey C. Lagarias and Thomas J. Richardson,
    Geometriae Dedicata 67 (1997), pp. 1-30. [Postscript]   [Pdf]

  • The d-Step Conjecture and Gaussian Elimination ,
    J. C. Lagarias, N. Prabhu, and J. A. Reeds,
    Discrete & Computational Geometry 18 (1997), pp. 53-82. [PostScript]   [Pdf]

  • Integral Self-Affine Tiles in R^n II. Lattice Tilings ,
    Jeffrey C. Lagarias and Yang Wang,
    J. Fourier Anal. Appl. 3 (1997), pp. 83-102. [PostScript]   [Pdf]

  • The lap counting function for linear mod one transformations
    III. the period of a Markov chain     ,
    Leopold Flatto and Jeffrey C. Lagarias,
    Ergodic Th. & Dynam. Sys. 17 (1997), pp. 369-403.

  • The lap counting function for linear mod one transformations
    II.the Markov chain for generalized lap numbers     ,
    Leopold Flatto and Jeffrey C. Lagarias,
    Ergodic Th. & Dynam. Sys. 17 (1997), pp. 123-146.

    [1996]

  • Tiling the line with translates of one tile ,
    Jeffrey C. Lagarias and Yang Wang,
    Inventiones math. 124 (1996), pp. 341-365.

  • Structure of tilings of the line by a function ,
    Mihail N. Kolountzakis and Jeffrey C. Lagarias,
    Duke Math. J. 82 (1996), pp. 653-678. [PostScript]  [Pdf]

  • The 3x+1 Conjugacy Map ,
    Daniel J. Bernstein and Jeffrey C. Lagarias,
    Canadian J. Math. 48 (1996), pp. 1154-1169. [PostScript]  [Pdf]

  • Meyer's Concept of Quasicrystal and Quasiregular Sets ,
    J. C. Lagarias,
    Comm. Math. Physics 179 (1996), pp. 365-376.

  • The lap counting function for linear mod one transformations
    I. explicit formulas and renormalizability     ,
    Leopold Flatto and Jeffrey C. Lagarias,
    Ergodic Th. & Dynam. Sys. 16 (1996), pp. 451-492.

  • Haar Bases for L^2(R^n) and Algebraic Number Theory,
    Jeffrey C. Lagarias and Yang Wang,
    J. Number Theory 57 (1996), pp. 181-197.

  • Integral Self-Affine Tiles in R^n I. Standard and Nonstandard Digit Sets ,
    Jeffrey C. Lagarias and Yang Wang,
    J. London Math. Soc. 54 (1996), pp. 161-179.

  • Self-Affine Tiles in R^n ,
    Jeffrey C. Lagarias and Yang Wang,
    Advances in Math. 121 (1996), pp. 21-49.

  • Non-negative Radix Expansions for the Orthant ,
    Jeffrey C. Lagarias and Yang Wang,
    Trans. Amer. Math. Soc. 348 (1996), pp. 99-117. [PostScript]  [Pdf]

    [1995]

  • Point Lattices ,
    Jeffrey C. Lagarias,
    Chapter 19 in: Handbook of Combinatorics, Volume I
    (R. L. Graham, M. Gro"tschel and L. Lovasz, eds.), Elsevier Science B. V.: Amsterdam 1995, pp. 919-966. [1995]

  • Point Lattices ,
    Jeffrey C. Lagarias,
    Chapter 19 in: Handbook of Combinatorics, Volume I
    (R. L. Graham, M. Gro"tschel and L. Lovasz, eds.), Elsevier Science B. V.: Amsterdam 1995, pp. 919-966.

  • On the range of fractional parts {/xi (p/q)^n} ,
    Leopold Flatto, Jeffrey C. Lagarias and Andrew D. Pollington,
    Acta Arithmetica 70 (1995), pp. 125-147.

  • The Distribution of 3x+1 Trees ,
    David Applegate and Jeffrey C. Lagarias,
    Experimental Math. 4 (1995), pp. 193-209. [Postscript]   [Pdf]

  • Haar-type Orthonormal Wavelet Bases in R^2 ,
    Jeffrey C. Lagarias and Yang Wang,
    J. Fourier Anal. Appl. 2 (1995) 1-14.

  • Polytopes that fill R^n and scissors congruence ,
    J. C. Lagarias and D. Moews,
    Discrete & Computational Geometry 13 (1995), pp. 573-584.
    [ Acknowledgement of Priority , ibid,14 (1995), pp. 359-360.]

  • Density Bounds for the 3x+1 Problem I. Tree-Search Method ,
    David Applegate and Jeffrey C. Lagarias,
    Math. Comp. 64 (1995), pp. 411-426. [Pdf]

  • Density Bounds for the 3x+1 Problem II. Krasikov Inequalities .
    David Applegate and Jeffrey C. Lagarias,
    Math. Comp. 64 (1995), pp. 427-438. [Pdf]

  • A Walk Along the Branches of the Extended Farey Tree ,
    Jeffrey C. Lagarias and Charles Tresser,
    IBM J. Res. Devel. 39 (1995), pp. 283-294.

  • The continuous Diophantine approximation mapping of Szekeres ,
    Jeffrey C. Lagarias and Andrew D. Pollington,
    J. Australian Math. Soc., Series A, 59 (1995), pp. 148-172.

  • The Finiteness Conjecture for the Generalized Spectral Radius of a Set of Matrices ,
    Jeffrey C. Lagarias and Yang Wang,
    Lin. Alg. Appl. 214 (1995), pp. 17-42.

    [1994]

  • Geodesic Multidimensional Continued Fractions ,
    J. C. Lagarias,
    Proc. London Math. Soc. 69 (1994), pp. 464-488.

  • On the distribution of multiplicative translates of sets of residues (mod p),
    J. Hastad, J. C. Lagarias, and A. M. Odlyzko,
    J. Number Theory, 46 (1994), pp. 108-122. [Pdf- Andrew Odlyzko web page]  [Pdf]

  • The Zeta Function of the Beta Transformation ,
    Leopold Flatto, Jeffrey C. Lagarias and Bjorn Poonen
    Ergodic Th. & Dynam. Sys. 14 (1994),pp. 237-266.

  • On the thermodynamic formalism for multifractal functions ,
    Ingrid Daubechies and Jeffrey C. Lagarias,
    Rev. Math. Physics 6 (1994), pp. 1033-1070.
    [Also in: , The State of Matter, A volume dedicated to E. H. Lieb ,
    (M. Aizenmann and H. Araki, Eds.), World Scientific: Singapore 1994, pp. 213-264.]

  • Cube Tilings in R^n and Nonlinear Codes ,
    J. C. Lagarias and P. W. Shor,
    Discrete & Computational Geometry 11 (1994), pp. 359-391.

  • Asymmetric Tent Maps II. Purely Periodic Points ,
    J. C. Lagarias, H. A. Porta and K. B. Stolarsky,
    Illinois J. Math. 38 (1994), pp. 574-588.

    [1993]

  • The Quality of the Diophantine Approximations found by the Jacobi-Perron Algorithm and Related Algorithms ,
    J. C. Lagarias,
    Monatshefte fu"r Math. 115 (1993), pp. 299-328.

  • A polyomino tiling problem of Thurston and its configurational entropy ,
    J. C. Lagarias and D. S. Romano,
    J. Combinatorial Theory, Series A 63 (1993), pp. 338-358.

  • A collinear scaling interpretation of Karmarkar's linear programming algorithm ,
    Jeffrey C. Lagarias,
    SIAM J. Optimization 3 (1993), pp. 630-636.

  • Asymmetric Tent Maps I. Eventually Periodic Points ,
    J. C. Lagarias, H. A. Porta and K. B. Stolarsky,
    J. London Math. Soc. 47 (1993), pp. 542-556.

  • Probabilistic Algorithms for Speedup ,
    Joan Feigenbaum and Jeffrey C. Lagarias,
    Statistical Science 8 (1993), pp. 20-25.

  • Pseudorandom Numbers ,
    Jeffrey C. Lagarias,
    Statistical Science 8 (1993), pp. 31-39.

    [1992]

  • Keller's Cube Tiling Conjecture is False in High Dimensions ,
    Jeffrey C. Lagarias and Peter W. Shor,
    Bull. Amer. Math. Soc. 27 (1992), pp. 279-283.
    [arxiv.org/math/9210222]

  • A Lie bracket decomposition and its application to flows on symmetric matrices ,
    Jeffrey C. Lagarias ,
    Lin. Alg. Appl. 175 (1992), pp. 239-257.

  • The 3x+1 Problem: Two Stochastic Models ,
    Jeffrey C. Lagarias and Alan Weiss,
    Annals of Applied Probability 2 (1992), pp. 229-261.

  • Sets of matrices all infinite products of which converge ,
    Ingrid Daubechies and Jeffrey C. Lagarias,
    Lin. Alg. Appl. 161 (1992), pp. 227-263.
    Corrigendum/Addendum ,
    Lin. Alg. Appl. 327 (2001), pp. 69--83. [PostScript]  [Pdf]

  • Two-scale difference equations II. Local regularity of solutions and fractals ,
    Ingrid Daubechies and Jeffrey C. Lagarias,
    SIAM J. Math. Anal. 23 (1992), pp. 1031-1079.

  • Self-packing of Centrally Symmetric Convex Bodies in R^2 ,
    P. G. Doyle, J. C. Lagarias and D. S. Randall,
    Discrete & Computational Geometry 8 (1992), pp. 171-189.

  • Number Theory and Dynamical Systems ,
    Jeffrey C. Lagarias,
    in: The Unreasonable Effectiveness of Number Theory ,
    (S. A. Burr, Ed.), Proc. Symp. Applied Math. No. 46, AMS: Providence 1992, pp. 35-72.

    [1991]

  • Two-scale difference equations I. Global regularity of solutions ,
    Ingrid Daubechies and Jeffrey C. Lagarias,
    SIAM J. Math. Anal. 22 (1991), pp. 1388-1410.

  • Bounds for lattice polytopes containing a fixed number of lattice points ,
    Jeffrey C. Lagarias and Gu"nter W. Ziegler,
    Canad. J. Math. 43 (1991), pp. 1022-1035.

  • Karmarkar's Algorithm and Newton's Method ,
    D. A. Bayer and J. C. Lagarias,
    Math. Programming 50 (1991), pp. 291-320.

  • Monotonicity inequalities for the Toda flow, QR-flow and subspace iteration ,
    J. C. Lagarias,
    SIAM J. Matrix Anal. Appl. 12 (1991), pp. 449-462.

  • Sets Uniquely Determined by Projections II. Discrete Case ,
    P. C. Fishburn, J. C. Lagarias, J. A. Reeds and L. A. Shepp,
    Discrete Math. 91 (1991), pp. 141-151.

  • Singularities of minimal surfaces and networks and related extremal problems in Minkowski space ,
    Z. Furedi, J. C. Lagarias and F. Morgan,
    in: DIMACS Geometry Year (R. Pollack, ed.), DIMACS Series Vol. 6, AMS: Providence 1991, pp. 95-109.

  • Effects of misspecification of lag structure in certain two-variable distributed lag models ,
    J. C. Lagarias,
    Computers & Math. with Applications 22, No. 10 (1991), pp. 3-23.

    [1990]

  • The non-linear geometry of linear programming
    III. Projective Legendre transform coordinates and Hilbert geometry     ,
    J. C. Lagarias,
    Trans. Amer. Math. Soc. 320 (1990), pp. 193-225.

  • Bounds for Korkin-Zolotarev reduced bases and successive minima of a lattice and its reciprocal lattice ,
    J. C. Lagarias, H. W. Lenstra, Jr. and C. P. Schnorr,
    Combinatorica 10 (1990), pp. 343-358.

  • Simultaneously good bases of a lattice and its reciprocal lattice ,
    J. Hastad and J. C. Lagarias,
    Math. Annalen 2287 (1990), pp. 167-174.

  • Tiling with polyominoes and combinatorial group theory ,
    J. H. Conway and J. C. Lagarias,
    J. Combinatorial Theory, Series A 53 (1990), pp. 183-208.

  • The set of rational cycles for the 3x+1 problem ,
    J. C. Lagarias,
    Acta Arithmetica 56 (1990), pp. 33-53.

  • Sets Uniquely Determined by Projection I. Continuous Case,
    P. C. Fishburn, J. C. Lagarias, J. A. Reeds and L. A. Shepp,
    SIAM J. Applied Math. 50 (1990), pp. 288-306.

  • I. I. Dikin's convergence result for the affine scaling algorithm ,
    R. Vanderbei and J. C. Lagarias,
    in: Mathematical Developments Arising From Linear Programming ,
    (J. C. Lagarias and M. J. Todd, eds.), Comtemporary Math. vol. 114, AMS: Providence 1990, pp. 109-119.

  • Pseudorandom number generators in number theory and cryptography ,
    J. C. Lagarias,
    in: Cryptology and Computational Number Theory ,
    (C. Pomerance, ed.), Proc. Symp. Applied Math. No. 42, AMS: Providence 1990, pp. 114-144.

    [1989]

  • The non-linear geometry of linear programming
    I. Affine and projective scaling trajectories     ,
    D. A. Bayer and J. C. Lagarias,
    Trans. Amer. Math. Soc. 314 (1989), pp. 499-526. [PostScript-nonreadable]  [Pdf]

  • The non-linear geometry of linear programming
    II. Legendre transform coordinates and central trajectories     ,
    D. A. Bayer and J. C. Lagarias,
    Trans. Amer. Math. Soc. 314 (1989), pp. 527-581.

  • Polynomial time algorithms for finding integer relations among real numbers ,
    J. Hastad, B. Just, J. C. Lagarias and C. P. Schnorr,
    SIAM J. Computing 18 (1989), pp. 859-881.
    [Preliminary version in: STACS '86 , Lecture Notes in Computer Science No. 210, Springer-Verlag: New York 1986, pp. 105-118.]

  • Single suit two person card play III. The misere game ,
    J. Kahn, J. C. Lagarias and H. S. Witsenhausen,
    SIAM J. Discrete Math. 2 (1989), pp. 329-343.

  • On Lasker's card game ,
    J. Kahn, J. C. Lagarias and H. S. Witsenhausen,
    In: Proc. 3rd Intl. Conf. on Differential Games and Applications,
    (T. Basar and P. Bernard, Ed.), Lecture Notes in Control and Information Science No. 223,
    Springer-Verlag: New York 1989, pp. 1-8.

  • Algorithms for square packing: A probabilistic analysis ,
    E. G. Coffman, Jr. and J. C. Lagarias,
    SIAM J. Computing (1989), pp. 166-185.

  • Power series variants of Karmarkar-type algorithms ,
    N. Karmarkar, J. C. Lagarias, L. Slutsman and P. Wang,
    AT&T Technical Journal 68 , No. 3 (1989), pp. 20-36.

    [1988]

  • Unique Extrapolation of Polynomial Recurrences ,
    J. C Lagarias and J. A. Reeds,
    SIAM J. Computing 17 (1988), pp. 342-362.

  • Reconstructing truncated linear variables satisfying linear congruences ,
    A. Frieze, J. Hastad, R. Kannan, J. C. Lagarias, and A. Shamir,
    SIAM J. Computing 17 (1988), pp. 262-280.

  • Single-suit two person card play. II Domination ,
    J. Kahn, J. C. Lagarias, and H. S. Witsenhausen,
    Order 5 (1988), pp. 45-60.

    [1987]

  • Computing pi(x): An analytic method,
    J. C. Lagarias and A. M. Odlyzko,
    J. Algorithms, 8 (1987), pp. 173-191. [PostScript-Truncated] [Pdf]

  • On the distribution in short intervals of integers having no large prime factor ,
    J. B. Friedlander and J. C. Lagarias,
    J. Number Theory 25 (1987), pp. 249-273. [PostScript-Truncated] [Pdf]

  • Single-Suit Two-Person Card Play ,
    J. Kahn, J. C. Lagarias and H. S. Witsenhausen,
    Intl. J. Game Theory 16 (1987), pp. 291-320.

  • One-way functions and circuit complexity ,
    R. Boppana and J. C. Lagarias,
    Information and Computation 74 (1987), pp. 226-240.
    [Preliminary version in: Structure in Complexity Theory ,
    Lecture Notes in Computer Science No. 223, Springer-Verlag: New York 1986, pp. 51-65.] [PostScript-Truncated] [Pdf]

    [1986]

  • Simultaneous Diophantine approximation of rationals by rationals ,
    J. Hastad and J. C. Lagarias,
    J. Number Theory 24 (1986), pp. 200-228. [PostScript-Truncated] [Pdf]

  • Probabilities for intersecting systems and random subsets of finite sets,
    P. C. Fishburn, P. Frankl, D. Freed, J. C. Lagarias, and A. M. Odlyzko,
    SIAM J. Alg. Discr. Methods, 7 (1986), pp. 73-79.

    [1985]

  • The 3x+1 problem and its generalizations ,
    Jeffrey C. Lagarias,
    Amer. Math. Monthly 92 (1985) 3-23.
    [On-line version] (Postscript version also downloadable)

  • The computational complexity of simultaneous Diophantine approximation problems ,
    J. C. Lagarias,
    SIAM J. Computing 14 (1985), pp. 196-209.
    [Preliminary version in: Proc 23rd Annual Symposium on Foundations of Computer Science, IEEE Press 1982, pp. 32-39.]

  • The set of primes dividing the Lucas numbers has density 2/3 ,
    J. C. Lagarias,
    Pacific J. Math. 118 (1985), pp. 449-461.
    [PostScript-Truncated] [Pdf] [ Errata , Pacific J. Math. 162 (1994), pp. 393-396.]

  • On the capacity of disjointly shared networks,
    J. C. Lagarias, A. M. Odlyzko, and D. B. Zagier,
    Computer Networks and ISDN Systems, 10 (1985), pp. 275-285. [PostScript-Truncated] [Pdf]

  • Computing pi(x): The Meissel-Lehmer method,
    J. C. Lagarias, V. S. Miller, and A. M. Odlyzko,
    Math. Comp., 44 (1985), pp. 537-560. [PostScript]

  • Solving low-density subset sum problems,
    J. C. Lagarias and A. M. Odlyzko,
    J. ACM, 32 (1985), pp. 229-246.
    [Preliminary version in Proc. 24th Annual Symposium on Foundations Computer Science ,IEEE Press 1983, pp. 1-10.]

    [1984]

  • Is there a density for the set of primes p such that the class number of Q(sqrt(-p)) is divisible by 16? ,
    H. Cohn and J. C. Lagarias,
    in: Topics in Classical Number Theory ,
    (G. Halasz, Ed.), Colloquium Soc. Janos Bolyai No. 34, 1984, pp. 257-279.

  • Knapsack Public Key Cryptosystems and Diophantine Approximation ,
    J. C. Lagarias,
    in: Advances in Cryptology: Proceedings of Crypto 83 , (D. Chaum, ed.), Plenum Press 1984, pp. 3-24.
    [PostScript-Truncated] [Pdf]

  • Evaluation of the Adleman attack on multiply iterated knapsack cryptosystems,
    E. F. Brickell, J. C. Lagarias, and A. M. Odlyzko,
    in: Advances in Cryptology: Proceedings of Crypto 83, (D. Chaum, ed.) Plenum Press, 1984, pp. 39-42.

  • Performance analysis of Shamir's attack on the basic Merkle-Hellman knapsack cryptosystem ,
    J. C. Lagarias,
    in: ICALP 1984, Automata Languages and Programming, Eleventh Colloquium, Antwerp, Belgium, July 1984,
    Lecture Notes in Computer Science No. 172, Springer-Verlag: New York 1984, pp. 312-323.
    [PostScript-Truncated] [Pdf]

  • Linear congruential generators do not produce random sequences ,
    A. Frieze, R. Kannan, and J. C. Lagarias,
    Proc. 25th Annual Symposium on Foundations of Computer Science , IEEE Press 1984, pp. 480-484.

  • New algorithms for computing pi(x),
    J. C. Lagarias and A. M. Odlyzko,
    pp. 176-193 in Number Theory: New York 1982,
    (D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn and M. B. Nathanson,eds.),
    Springer-Verlag, Lecture Notes in Mathematics #1052, 1984.

    [1983]

  • On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt(dp)) as p varies ,
    H. Cohn and J. C. Lagarias,
    Math. Comp. 37 (1983), pp. 711-730.

  • Sets of primes determined by systems of polynomial congruences ,
    J. C. Lagarias,
    Illinois J. Math. 27 (1983), pp. 224-239.

  • Best Diophantine approximation to a set of linear forms ,
    J. C. Lagarias,
    J. Australian Math. Soc., Series A 34 (1983), pp. 114-122.

  • Multi-stage planning and the extended linear-quadratic-Gaussian control problem ,
    J. C. Lagarias and F. Aminzadeh,
    Math. Operations Research 8 (1983), pp. 42-63.

  • On the tightest packing of sums of vectors ,
    J. R. Griggs, J. C. Lagarias, A. M. Odlyzko, and J. B. Shearer,
    European J. Combinatorics , 4 (1983), pp. 231-236.

  • Approximation Algorithms for Maximizing the Number of Squares Packed in a Rectangle ,
    B. Baker, A. R. Calderbank, E. G. Coffman and J. C. Lagarias,
    SIAM J. Alg. Disc. Methods 4 (1983), pp. 383-397.

  • On the Density of Sequences of Integers the Sum of No Two of Which is a Square. II. General Sequences,
    J. C. Lagarias, A. M. Odlyzko, and J. B. Shearer,
    J. Comb. Theory A, 34 (1983), pp. 123-139. [Pdf]

    [1982]

  • Best simultaneous Diophantine approximations I. Growth of best approximation denominators ,
    J. C. Lagarias,
    Trans. Amer. Math. Soc. 272 (1982), pp. 545-554.

  • Best simultaneous Diophantine approximations II. Behavior of consecutive best approximations ,
    J. C. Lagarias,
    Pacific J. Math. 102 (1982), pp. 61-88.

  • On the Density of Sequences of Integers the Sum of No Two of Which is a Square. I. Arithmetic Progressions ,
    J. C. Lagarias, A. M. Odlyzko and J. B. Shearer,
    J. Combinatorial Theory, Series A 33 (1982), pp. 167-185.

  • Optimal pairs of score vectors for positional scoring rules ,
    W. Gehrlein, B. Gopinath, J. C. Lagarias and P. Fishburn ,
    Appl. Math. Optim. 88 (1982), pp. 309-324.

  • Minimal Storage Representations for Binary Relations ,
    J. C. Lagarias and D. E. Swartwout,
    Information Processing Letters 14 (1982) 63-66.
    [Preliminary version in: Proc. ACM-SIGMOD 1981 International Conf. on
    the Management of Data     , (Y. E. Lien, ed.), ACM 1991, pp. 1-9.]

    [1981]

  • A complement to Ridout's p-adic generalization of the Thue-Siegel-Roth theorem ,
    J. C. Lagarias,
    in: Analytic Number Theory, Proceedings Philadelphia 1980 ,
    (M. I. Knopp, Ed.), Lecture Notes in Math. No. 899, Springer-Verlag, New York 1981, pp. 264-275.

  • Fibonacci and Lucas Cubes ,
    J. C. Lagarias and D. P. Weisser,
    Fibonacci Quarterly 19 (1981), pp. 39-43.

    [1980]

  • On the computational complexity of determining the solvability or unsolvability of the equation X^2 - dY^2 = -1 ,
    J. C. Lagarias,
    Trans. Amer. Math. Soc. 260 (1980), pp. 485-508.

  • Worst-case complexity bounds in the theory of integral quadratic forms ,
    J. C. Lagarias,
    J. Algorithms 1 (1980), pp. 142-186.

  • On determining the 4-rank of the ideal class group of a quadratic field ,
    J. C. Lagarias,
    J. Number Theory 12 (1980), pp. 191-196.

  • Signatures of Units and Congruences (mod 4) in Certain Real Quadratic Fields II ,
    J. C. Lagarias,
    J. reine angew. Math. 320 (1980), pp. 115-126.

  • Signatures of Units and Congruences (mod 4) in Certain Totally Real Fields ,
    J. C. Lagarias,
    J. reine angew. Math. 320 (1980), pp. 1-5.

  • Divisibility properties of some cyclotomic sequences,
    J. C. Lagarias and A. M. Odlyzko,
    Amer. Math. Monthly, 87 (1980), pp. 561-564.

  • Some new results in simultaneous Diophantine approximation ,
    J. C. Lagarias,
    in: Proc. Queens's University Number Theory Conference 1979 ,
    (P. Ribenboim, Ed.), Queen's Papers in Pure and Applied Math. No. 54, Queen's University, 1980, pp. 453-474.

    [1979]

  • A bound for the least prime ideal in the Chebotarev density theorem,
    J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko,
    Inventiones math., 54 (1979), pp. 271-296.

  • On computing Artin L-functions in the critical strip,
    J. C. Lagarias and A. M. Odlyzko,
    Math. Comp., 33 (1979), pp. 1081-1095.

  • Succinct Certificates for the Solvability of Binary Quadratic Diophantine Equations
    J. C. Lagarias,
    Proc. 20th IEEE Symposium on Foundations of Computer Science , IEEE Press 1979, pp. 47-54.

    [1978]

  • Signatures of Units and Congruences (mod 4) in Certain Real Quadratic Fields ,
    J. C. Lagarias,
    J. reine Angew. Math. , 301 (1978), pp. 142-146.

    [1977]

  • Effective versions of the Chebotarev density theorem,
    J. C. Lagarias and A. M. Odlyzko,
    pp. 409-464 in Algebraic Number Fields, A. Frohlich (ed.), Academic Press, 1977.

  • Discrete Balancing Games ,
    J. C. Lagarias,
    Bull. Inst. of Math. Acad. Sinica , 5 (1977), pp. 363-373.

  • Minimum Range Sequences of all k-subsets of a Set ,
    F. K. Hwang and J. C. Lagarias,
    Discrete Mathematics 19 (1977), pp. 257-264.

    [1974]

  • The 4-part of the ideal class group of a quadratic field ,
    Jeffrey C. Lagarias,
    PhD Thesis , M. I. T. (H.M. Stark, advisor).

    [1972]

  • Evaluation of Certain Character Sums ,
    Jeffrey C. Lagarias,
    Master's Thesis , M. I. T. (H. M. Stark, advisor).


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