Arithmetic functions and integer products
著者
書誌事項
Arithmetic functions and integer products
(Die Grundlehren der mathematischen Wissenschaften, 272)
Springer-Verlag, c1985
- : us
- : gw
- : pbk
大学図書館所蔵 全77件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Bibliography: p. [449]-457
Includes index
内容説明・目次
- 巻冊次
-
: us ISBN 9780387960944
内容説明
Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory.
目次
Duality and the Differences of Additive Functions.- First Motive.- 1 Variants of Well-Known Arithmetic Inequalities.- 2 A Diophantine Equation.- 3 A First Upper Bound.- 4 Intermezzo: The Group Q*/?.- 5 Some Duality.- Second Motive.- 6 Lemmas Involving Prime Numbers.- 7 Additive Functions on Arithmetic Progressions with Large Moduli.- 8 The Loop.- Third Motive.- 9 The Approximate Functional Equation.- 10 Additive Arithmetic Functions on Differences.- 11 Some Historical Remarks.- 12 From L2 to L?.- 13 A Problem of Kátai.- 14 Inequalities in L?.- 15 Integers as Products.- 16 The Second Intermezzo.- 17 Product Representations by Values of Rational Functions.- 18 Simultaneous Product Representations by Values of Rational Functions.- 19 Simultaneous Product Representations with aix + bi.- 20 Information and Arithmetic.- 21 Central Limit Theorem for Differences.- 22 Density Theorems.- 23 Problems.- Supplement Progress in Probabilistic Number Theory.- References.
- 巻冊次
-
: pbk ISBN 9781461385509
内容説明
Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = +/- I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x". Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory.
目次
- Duality and the Differences of Additive Functions.- First Motive.- 1 Variants of Well-Known Arithmetic Inequalities.- Multiplicative Functions.- Generalized Turan-Kubilius Inequalities.- Selberg's Sieve Method.- Kloosterman Sums.- 2 A Diophantine Equation.- 3 A First Upper Bound.- The First Inductive Proof.- The Second Inductive Proof.- Concluding Remarks.- 4 Intermezzo: The Group Q*/?.- 5 Some Duality.- Duality in Finite Spaces.- Self-adjoint Maps.- Duality in Hilbert Space.- Duality in General.- Second Motive.- 6 Lemmas Involving Prime Numbers.- The Large Sieve and Prime Number Sums.- The Method of Vinogradov in Vaughan's Form.- Dirichlet L-Series.- 7 Additive Functions on Arithmetic Progressions with Large Moduli.- Additive Functions on Arithmetic Progressions.- Algebraicanalytic Inequalities.- 8 The Loop.- Third Motive.- 9 The Approximate Functional Equation.- 10 Additive Arithmetic Functions on Differences.- The Basic Inequality.- The Decomposition of the Mean.- Concluding Remarks.- 11 Some Historical Remarks.- 12 From L2 to L?.- 13 A Problem of Katai.- 14 Inequalities in L?.- 15 Integers as Products.- More Duality
- Additive Functions as Characters.- Divisible Groups and Modules.- Sets of Uniqueness.- Algorithms.- 16 The Second Intermezzo.- 17 Product Representations by Values of Rational Functions.- A Ring of Operators.- Practical Measures.- 18 Simultaneous Product Representations by Values of Rational Functions.- Linear Recurrences in Modules.- Elliptic Power Sums.- Concluding Remarks.- 19 Simultaneous Product Representations with aix + bi.- 20 Information and Arithmetic.- Transition to Arithmetic.- Information as an Algebraic Object.- 21 Central Limit Theorem for Differences.- 22 Density Theorems.- Groups of Bounded Order.- Measures on Dual Groups.- Arithmic Groups.- Concluding Remarks.- 23 Problems.- Exercises.- Unsolved Problems.- Supplement Progress in Probabilistic Number Theory.- Analogues of the Turan-Kubilius Inequality.- References.
「Nielsen BookData」 より