Topics in Analytic Number Theory

I. Analytic tools.- 1. Bernoulli polynomials and Bernoulli numbers.- 1. The binomial coefficients.- 2. The Bernoulli polynomials.- 3. Zeros of the Bernoulli polynomials.- 4. The Bernoulli numbers.- 5. The von Staudt-Clausen theorem.- 6. A multiplication formula for the Bernoulli polynomials.- 2. The Euler-MacLaurin sum formula.- 7. Use of the Bernoulli polynomials.- 8. Fourier expansions of the Bernoulli polynomials.- 9. Sums of reciprocal powers.- 10. The generating function of the Bernoulli numbers.- 11. Tangent and cotangent coefficients.- 12. A theorem by Frobenius about the numerators of the Bernoulli numbers.- 13. The generating function of the Bernoulli polynomials.- 14. The secant coefficients or Euler numbers.- 15. Stirling's formula.- 16. A further application.- 17. A historical remark.- 3. The ?-function and Mellin's theorem.- 18. Definition of the ?-function.- 19. Functional equations of ?(s).- 20. Application of the Euler-MacLaurin sum formula.- 21. Asymptotic behavior of ?(s).- 22. A lemma.- 23. The Mellin formula.- 24. Hankel's formula.- 25. An application to Bessel functions.- 26. The Fourier integral.- 27. Mellin's formulae.- 28. Some further examples of Mellin's formulae.- 4. The Phragmen-Lindelof theorem.- 29. The main theorem.- 30. A theorem of the Phragmen-Lindelof type for subharmonic functions.- 31. The Poisson integral formula for a strip.- 32. A lemma.- 33. A generalization of the Phragmen-Lindelof theorem.- 34. Applications to the ?-function.- 5. The Poisson sum formula and applications.- 35. The theorem.- 36. Application: A transformation formula for a ?-function.- 37. Lipschitz's formula.- II. Special functions.- 6. The Riemann ?-function.- 38. Definition of the ?-function and its analytic continuation.- 39. Two special integrals.- 40. Riemann's functional equation for ? (s).- 41. Another proof for the functional equation of ? (s).- 42. Connection between the ?-function and a ?-function.- 43. Estimation of ? (s) in a vertical strip.- 7. About the prime-number theorem and the zeros of the ?-function.- 44. The Euler product.- 45. The borders of the critical strip are free of zeros of ? (s).- 46. Preparation for the proof of the prime-number theorem.- 47. A lemma.- 48. Expression of a function ?(x) connected with ? (x) by means of an integral.- 49. Some estimates for ?(s), ?'(s), 1/? (s).- 50. The prime-number theorem.- 51. The error term in the prime-number theorem.- 52. Caratheodory's lemma.- 53. Application of Caratheodory's lemma.- 54. The error term r (x).- 55. Existence of infinitely many non-trivial zeros.- 56. Additional remarks.- 57. Dirichlet series and the best order of the error term in the prime number theorem.- 8. The Eisenstein series.- 58. Definition of the Eisenstein series and of ? (u).- 59. Expansion of ? (u) in a Laurent series.- 60. Lambert series.- 61. Some arithmetical consequences.- 62. Modular forms.- 63. Definition of G2 (?1, ?2).- 64. The modular invariance of G2 (?1, ?2).- 65. Dedekind function ? (?) and the discriminant ?(?).- 9. The transformation of log ?(?) and the theory of the Dedekind sums.- 66. A formula of Iseki.- 67. Application of Iseki'I. Analytic tools.- 1. Bernoulli polynomials and Bernoulli numbers.- 1. The binomial coefficients.- 2. The Bernoulli polynomials.- 3. Zeros of the Bernoulli polynomials.- 4. The Bernoulli numbers.- 5. The von Staudt-Clausen theorem.- 6. A multiplication formula for the Bernoulli polynomials.- 2. The Euler-MacLaurin sum formula.- 7. Use of the Bernoulli polynomials.- 8. Fourier expansions of the Bernoulli polynomials.- 9. Sums of reciprocal powers.- 10. The generating function of the Bernoulli numbers.- 11. Tangent and cotangent coefficients.- 12. A theorem by Frobenius about the numerators of the Bernoulli numbers.- 13. The generating function of the Bernoulli polynomials.- 14. The secant coefficients or Euler numbers.- 15. Stirling's formula.- 16. A further application.- 17. A historical remark.- 3. The ?-function and Mellin's theorem.- 18. Definition of the ?-function.- 19. Functional equations of ?(s).- 20. Application of the Euler-MacLaurin sum formula.- 21. Asymptotic behavior of ?(s).- 22. A lemma.- 23. The Mellin formula.- 24. Hankel's formula.- 25. An application to Bessel functions.- 26. The Fourier integral.- 27. Mellin's formulae.- 28. Some further examples of Mellin's formulae.- 4. The Phragmen-Lindelof theorem.- 29. The main theorem.- 30. A theorem of the Phragmen-Lindelof type for subharmonic functions.- 31. The Poisson integral formula for a strip.- 32. A lemma.- 33. A generalization of the Phragmen-Lindelof theorem.- 34. Applications to the ?-function.- 5. The Poisson sum formula and applications.- 35. The theorem.- 36. Application: A transformation formula for a ?-function.- 37. Lipschitz's formula.- II. Special functions.- 6. The Riemann ?-function.- 38. Definition of the ?-function and its analytic continuation.- 39. Two special integrals.- 40. Riemann's functional equation for ? (s).- 41. Another proof for the functional equation of ? (s).- 42. Connection between the ?-function and a ?-function.- 43. Estimation of ? (s) in a vertical strip.- 7. About the prime-number theorem and the zeros of the ?-function.- 44. The Euler product.- 45. The borders of the critical strip are free of zeros of ? (s).- 46. Preparation for the proof of the prime-number theorem.- 47. A lemma.- 48. Expression of a function ?(x) connected with ? (x) by means of an integral.- 49. Some estimates for ?(s), ?'(s), 1/? (s).- 50. The prime-number theorem.- 51. The error term in the prime-number theorem.- 52. Caratheodory's lemma.- 53. Application of Caratheodory's lemma.- 54. The error term r (x).- 55. Existence of infinitely many non-trivial zeros.- 56. Additional remarks.- 57. Dirichlet series and the best order of the error term in the prime number theorem.- 8. The Eisenstein series.- 58. Definition of the Eisenstein series and of ? (u).- 59. Expansion of ? (u) in a Laurent series.- 60. Lambert series.- 61. Some arithmetical consequences.- 62. Modular forms.- 63. Definition of G2 (?1, ?2).- 64. The modular invariance of G2 (?1, ?2).- 65. Dedekind function ? (?) and the discriminant ?(?).- 9. The transformation of log ?(?) and the theory of the Dedekind sums.- 66. A formula of Iseki.- 67. Application of Iseki's formula to the transformation of log ?(?).- 68. The Dedekind sums.- 69. The formula of reciprocity of the Dedekind sums.- 70. A direct proof of the reciprocity formula for Dedekind sums.- 71. Composition of modular transformations of ? (?).- 72. A group-theoretical remark.- 73. The Dedekind sums and the Jacobi residue symbol.- 74. Again the transformation of ?(?).- 10. The ?- functions.- 75. Introduction of the ?-functions.- 76. Definition of the ?-functions.- 77. Zeros of the ?-functions.- 78. Product expansions of the ?-functions.- 79. Transformation of the ?-functions.- 80. Transformation of ?1(?/?), continued.- 81. Transformation of ?2(?|?), ?3(?|?),?4(?|?).- 11. Elliptic functions and their applications to number theory.- 82. Construction of elliptic functions from the ?-functions.- 83. Sums of four Squares.- 84. Sums of two Squares.- 85. Lambert series for fa (v).- 86. Lambert series for f2?(?).- 87. Some addition formulae for -functions.- 88. Formulae of differentiation.- 89. Even powers of ?3 expressed by derivatives of f?(v) and f2?(v).- 90. Lambert series for the even powers of ?3.- 91. Sums of an even number of Squares.- 92. Discussion of the foregoing results.- 93. Further discussion of ? (n).- III. Formal power series.- 12. Formal power series and the theory of partitions.- 94. Introduction and definitions.- 95. Some elementary identities.- 96. Partitions with restricted size or number of parts.- 97. Some similar theorems.- 98. Unrestricted partitions.- 99. Formal differentiation and its application.- 100. Jacobi's triple product.- 101. Another proof of the pentagonal numbers theorem.- 102. A Jacobi formula.- 103. An identity of Euler.- 13. Ramanujan's congruences and identities.- 104. Some divisibility properties of p (n).- 105. Two Ramanujan identities.- 106. Relations between the Gs, Hs and ?.- 107. The Rogers-Ramanujan identities. Introductory remarks.- 108. Arithmetical statement of the identities.- 109. Reformulation of the problem.- 110. The Gaussian polynomials.- 111. Schur's functions.- 112. Linear combinations of Schur's functions.- 113. Determination of D1(x) and D2(x).- 114. A digression, concerning a further proof of the pentagonal number theorem.- 115. A further remark.- IV. The circle method.- 14. Analytic theory of partitions.- 116. A Cauchy integral and a special path of Integration.- 117. An expression for p (n).- 118. Application of the transformation formula for ?(?).- 119. Estimates and evaluations.- 120. Continuation of estimates and evaluations. The final formula for p (n).- 121. A partial sum with error term.- 122. Discussion of the sums Ak (n), A new expression for ?hk.- 123. A lemma by Whiteman and the Seiberg sum.- 124. Different cases of Bk (v) according to k.- 125. Multiplicativity of Bk(v).- 126. Evaluation of Bk (v) for a prime power.- 127. Estimations of Ak (n).- 128. The generating function f(x) for p (n).- 129. Discussion of ?k (z).- 130. Decomposition of f(x) into partial fractions.- 15. Application of the circle method to modular forms of positive dimension.- 131. Generalized modular forms.- 132. Computation of the coefficients of the modular form.- 133. Estimations.- 134. The final formula for the coefficients.- 135. The series for the modular form F (?).- Editor's notes.