Publications by Peter Borwein

             Available at

Only recent papers are available online in pdf form. For other papers or other formats please contact me. The online publications below are NOT necessarily authoritative nor necessarily final versions. They are listed in reverse chronological order of writing.


2007 The Riemann Hypothesis: A resource for the afficionado and virtuoso alike
2004 Pi: a source book (MR2065455)
2002 Computational excursions in analysis and number theory (MR1912495)
2000 Pi: a source book (MR1746004)
1998 Pi and the AGM. A study in analytic number theory and computational complexity (MR1641658)
1995 Polynomials and polynomial inequalities (MR1367960)
1990 A dictionary of real numbers (MR1030542)
1987 Pi and the AGM. A study in analytic number theory and computational complexity. (MR0877728)


    Other Manuscripts

  1. Goldbach Polynomials
  2. Transcendance of the Gaussian Liouville number and relatives
  3. Strong normality of numbers
  4. A variant of Liouville's lambda function: some surprising formulae
  5. Reverse symbolic computations, the identity function
  6. Challenges in mathematical computing
  7. The amazing number Pi
  8. Othogonality and irrationality
  9. Some observations on computer aided analysis


  10. Champernowne and Normality


  11. Completely Multiplicative Functions


  12. The Multivariate Integer Chebyshev Problem
  13. Transcendence of some power series


  14. On the zeros of cosine polynomials: solution to a problem of Littlewood
  15. Polynomials with coefficients from a finite set
  16. Sign changes in the sums of the Liouville function (MR2398787)
  17. Expected norms of 0-1 polynomials
  18. The merit factor problem
  19. Barker sequences and flat polynomials


  20. Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood (MR2320719)
  21. Irreducible polynomials and Barker sequences (MR2404490)
  22. Generalizations of Gonçalves' inequality (MR2280202)
  23. Lehmer's problem for polynomials with odd coefficients (MR2373144)
  24. Zeros of partial sums of the Riemann zeta function (MR2312975)
  25. The average norm of polynomials of fixed height (MR2255202)


  26. Nikolskii-type inequalities for shift invariant function spaces (MR2231907)
  27. An old conjecture of Erdös-Turàn on additive bases (MR2176410)


  28. Norms of cyclotomic Littlewood polynomials (MR2132173)
  29. Polyphase sequences with low autocorrelation (MR2241514)
  30. Explicit construction of general multivariate Padé approximants to an Appell function (MR2127575)


  31. The Mahler measure of polynomials with odd coefficients (MR2038720)
  32. Binary sequences with merit factor greater than 6.34 (MR2103494)
  33. A complete description of Golay pairs for lengths up to 100 (MR2031419)


  34. Newman polynomials with prescribed vanishing and integer sets with distinct subset sums (MR1954968)
  35. Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes (MR2019608)
  36. Monic integer Chebyshev problem (MR1986811)
  37. Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers (MR1997583)
  38. General forms for minimal spectral values for a class of quadratic Pisot numbers (MR1934431)
  39. Computational investigations of the Prouhet-Tarry-Escott problem (MR1986822)
  40. On the irrationality of a certain multivariate q series (MR1963741)
  41. Erratum to: "On the irrationality of a certain multivariate q series" (MR2063136)


  42. Paul Erdös and polynomials (MR1954691)
  43. Some computations on the spectra of Pisot and Salem numbers (MR1885627)
  44. Explicit merit factor formulae for Fekete and Turyn polynomials (MR1859033)


  45. An extremal property of Fekete polynomials (MR1784013)
  46. Merit factors of polynomials formed by Jacobi symbols (MR1814964)
  47. Ramanujan and π (MR1862751)
  48. The expected Lp norm of random polynomials (MR1814174)
  49. Trigonometric polynomials with many real zeros and a Littlewood-type problem (MR1801998)
  50. Visible structures in number theory (MR1954492)


  51. An efficient algorithm for the Riemann zeta function (MR1777614)
  52. Polynomials with height 1 and prescribed vanishing at 1 (MR1795875)
  53. Pointwise Remez- and Nikolskii-type inequalities for exponential sums (MR1735078)
  54. Merit factors of character polynomials (MR1766099)
  55. Rudin-Shapiro-like polynomials in L4 (MR1709147)
  56. Markov-Bernstein type inequalities under Littlewood-type coefficient constraints (MR1813157)


  57. On cyclotomic polynomials with ±1 coefficients (MR1737235)
  58. On the irrationality of a certain q$ series (MR1485462)
  59. Littlewood-type problems on [0,1] (MR1687555)


  60. On the irrationality of {i=0}q-i{j=0}i(1+q-jr+q-2js) (MR1742990)
  61. Some old problems on polynomials with integer coefficients (MR1742989)
  62. Growth of the product j=1n(1-xaj) (MR1654470)
  63. Power series with restricted coefficients and a root on a given ray (MR1468939)
  64. Multiple roots of [-1,1] power series (MR1624809)
  65. Muntz's theorem on compact subsets of positive measure (MR1625223)
  66. Remez-type inequality for non-dense Müntz spaces with explicit bound (MR1624834)


  67. What is organic mathematics? (MR1483911)
  68. Littlewood-type problems on subarcs of the unit circle (MR1631600)
  69. On the zeros of polynomials with restricted coefficients (MR1468873)
  70. The quest for pi (MR1439159)
  71. Polynomials with {0,+1,-1} coefficients and a root close to a given point (MR1604114)
  72. Markov- and Bernstein-type inequalities for polynomials with restricted coefficients (MR1606930)
  73. On the rapid computation of various polylogarithmic constants (MR1415794)
  74. Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces (MR1415318)
  75. Sharp extensions of Bernstein's inequality to rational spaces (MR1433285)
  76. Ramanujan, modular equations, and approximations to pi or How to compute one billion digits of pi (MR1483913)


  77. Questions about polynomials with {0,-1,+1} coefficients: Research Problems 96-3 (MR1405009)
  78. Making sense of experimental mathematics (MR1413248)
  79. A sharp Bernstein-type inequality for exponential sums (MR1401698)
  80. Newman's inequality for Müntz polynomials on positive intervals (MR1385812)
  81. The Lp version of Newman's inequality for lacunary polynomials (MR1285974)
  82. Giuga's conjecture on primality (MR1369150)
  83. The full Müntz theorem in C[0,1] and L1[0,1] (MR1395070)
  84. Rational approximation with varying weights. I (MR1393288)
  85. The integer Chebyshev problem (MR1333305)


  86. Matrix transformations of series of orthogonal polynomials (MR1376739)
  87. Müntz spaces and Remez inequalities (MR1273395)
  88. The arc length of the lemniscate { |p(z)| =1} (MR1223265)
  89. Incomplete rational approximation in the complex plane (MR1323965)
  90. Zeros of iterated integrals of polynomials (MR1319690)
  91. Dense Markov spaces and unbounded Bernstein inequalities (MR1323752)
  92. Upper bounds for the derivative of exponential sums (MR1232137)
  93. Corrigendum: "The density of rational functions in Markov systems: a counterexample to a conjecture of D. J. Newman" (MR1323967)
  94. Markov and Bernstein type inequalities in Lp for classes of polynomials with constraints (MR1332893)


  95. The Prouhet-Tarry-Escott problem revisited (MR1279058)
  96. Moment-matching and best entropy estimation (MR1288228)
  97. Markov and Bernstein type inequalities on subsets of [-1,1] and [-π,π] (MR1278768)
  98. Chebyshev polynomials and Markov-Bernstein type inequalities for rational spaces (MR1299454)
  99. Müntz systems and orthogonal Müntz-Legendre polynomials (MR1227091)
  100. Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros (MR1291056)
  101. Some cubic modular identities of Ramanujan (MR1243610)
  102. Exact inequalities for the norms of factors of polynomials (MR1289054)


  103. The density of rational functions in Markov systems: a counterexample to a conjecture of D. J. Newman (MR1198525)
  104. On the generating function of the integer part: [nα +γ] (MR1212686)
  105. An interesting infinite product (MR1244958)
  106. Some restricted partition functions (MR1242717)
  107. Lacunary Müntz systems (MR1242750)
  108. Inequalities for compound mean iterations with logarithmic asymptotes (MR1231502)
  109. Hypergeometric analogues of the arithmetic-geometric mean iteration (MR1237931)
  110. Class number three Ramanujan type series for 1/π. Computational complex analysis (MR1222488)
  111. The usual behavior of rational approximation. II (MR1209968)


  112. On the denseness of weighted incomplete approximations (MR1240793)
  113. On the irrationality of certain series (MR1162938)
  114. Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros (MR1164074)
  115. Strange series and high precision fraud (MR1176589)
  116. Rational approximation to Lipschitz and Zygmund classes (MR1194026)
  117. On approximation by trigonometric Lagrange interpolating polynomials. II (MR1155479)


  118. Notes on lacunary Müntz polynomials (MR1177339)
  119. Variations on Müntz's theme (MR1127751)
  120. On the irrationality of ∑(1/(qn+r)) (MR1096442)
  121. A cubic counterpart of Jacobi's identity and the AGM (MR1010408)
  122. Quadratically converging rational mean iterations. (MR1088637)


  123. A remarkable cubic mean iteration (MR1071759)
  124. A survey of Sylvester's problem and its generalizations (MR1069788)
  125. Zeros of Chebyshev polynomials in Markov systems (MR1074081)
  126. Some questions of Erdös and Graham on numbers of the form ∑gn/2gn (MR0990598)


  127. Pi, Euler numbers, and asymptotic expansions (MR1019148)
  128. Hypertranscendence of the functional equation g(x2)=[g(x)]2+cx (MR0979226)
  129. On the mean iteration (a,b)← ((a+3b)/4,(√ab +b)/2) (MR0968148)
  130. Approximating π with Ramanujan's modular equations (MR1016163)
  131. The density of alternation points in rational approximation (MR0948147)
  132. Ramanujan, modular equations, and approximations to pi, or How to compute one billion digits of pi (MR0991866)


  133. More Ramanujan-type series for 1/π (MR0938974)
  134. On the complexity of familiar functions and numbers (MR0967961)
  135. Padé approximants for the q-elementary functions (MR0956175)
  136. Ramanujan's rational and algebraic series for 1/π (MR0988313)
  137. Explicit Ramanujan-type approximations to pi of high order (MR0983604)


  138. Quadratic and higher order Padé approximants (MR0899533)
  139. Unsolved Problems: The Way of All Means (MR1541118)
  140. Reduced complexity evaluation of hypergeometric functions (MR0892217)


  141. Quadratic Hermite-Padé approximation to the exponential function (MR0892157)
  142. An explicit cubic iteration for π (MR0833837)
  143. More quadratically converging algorithms for π (MR0815846)


  144. On the complexity of calculating factorials (MR0800727)
  145. Rational interpolation to ex. II (MR0783989)
  146. Markov's inequality for polynomials with real zeros (MR0766524)


  147. Explicit algebraic nth order approximations to pi (MR0786845)
  148. The arithmetic-geometric mean and fast computation of elementary functions (MR0750454)
  149. Cubic and higher order algorithms for π (MR0763041)
  150. On monochromatic triangles (MR0757616)
  151. The relationship between the zeros of best approximations and differentiability (MR0760939)
  152. Sylvester's problem and Motzkin's theorem for countable and compact sets (MR0733410)
  153. On the behavior of the MLE of the scale parameter of the Student family (MR0764853)


  154. The relationship between differentiability of a function and the zeros of its best approximations (MR0754361)
  155. A very rapidly convergent product expansion for π (MR0721199)
  156. The usual behaviour of rational approximations (MR0703403)
  157. The Desmic conjecture (MR0704251)
  158. Rational approximations with real poles to e-x and xn (MR0705546)
  159. Uniform approximation by polynomials with variable exponents (MR0717138)
  160. On Padé and best rational approximation (MR0681952)
  161. On Sylvester's problem and Haar spaces (MR0721919)
  162. Unsolved Problems: A Conjecture Related to Sylvester's Problem (MR1540214)
  163. Rational approximations to Stieltjes transforms (MR0733943)


  164. Approximations with negative roots and poles (MR0662161)
  165. On monochrome lines and hyperplanes (MR0665657)
  166. Rational interpolation to ex (MR0662162)
  167. On a method of Newman and a theorem of Bernstein (MR0647709)
  168. The size of {x:r′n/rn ≥ 1} and lower bounds for e-x-rn (MR0673858)


  169. Approximation of xn by reciprocals of polynomials (MR0641960)
  170. Markov's and Bernstein's inequalities on disjoint intervals (MR0608865)


  171. Approximations by rational functions with positive coefficients (MR0568377)
  172. Rational functions with positive coefficients, polynomials and uniform approximations (MR0595813)
  173. Inequalities and inverse theorems in restricted rational approximation theory (MR0571930)


  174. Arbitrarily slow rational approximations on the positive real line (MR0509563)