Ⅰ．INTRODUCTION 1
1．1 Abelian transformations 1
1．2 Tauberian theorems 2
1．3 Mercerian theorems 4
1．4 miscellaneous tauberian theorems and the prime number theorem 4
Ⅱ．ELEMENTARY TAUBERIAN THEOREMS 6
2．1 Tauberian conditions 6
2．2 Elementary general tauberian theorems 17
Ⅲ．CLASSICAL TAUBERIAN THEOREMS 29
3．1 Transformation of the kernel 29
3．2 Cesaro and Riesz summation 30
3．3 Abel summation 34
3．4 Borel summation 37
Ⅳ．WIENER＇S THEORY 43
4．1 Fourier integrals 43
4．2 Wiener＇s tauberian theorems 50
4．3 Some extensions of Wiener＇s theorems 57
4．4 Application to special tauberian theorems 78
Ⅴ．MERCERIAN THEOREMS 93
5．1 Fourier-Stieltjes transforms 93
5．2 General mercerian theorems 114
5．3 Some special mercerian theorems 121
5．4 Further extensions and applications to integrodifferential equations 126
Ⅵ．TAUBERIAN THEOREMS AND THE PRIME NUMBER THEOREM 130
6．1 The Landau-Ikehara theorem 130
6．2 Classical proofs of the prime number theorem 138
6．3 The elementary proof of the prime number theorem and related tauberian theorems 152
BIBLIOGRAPHY 167