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Student Number 92221006
Author We-Min Ke(柯偉民)
Author's Email Address b8536028@yahoo.com.tw
Statistics This thesis had been viewed 1341 times. Download 640 times.
Department Mathematics
Year 2004
Semester 2
Degree Master
Type of Document Master's Thesis
Language English
Title On problems of certain arithmetic functions
Date of Defense 2005-07-04
Page Count 22
Keyword
  • arithmetic functions
  • Abstract 論文摘要:
      在1952年, Erdös 和Mirsky 提問是否存在無限多個正整數n, 使得d(n) = d(n +1), 其中d(n)的函數值為n 的正因數個數。在1984年, Heath-Brown證得對於n≦x, d(n) = d(n +1) 中n的解的量至少為x/(log x)7, 因此完整地回答了Erdös 和Mirsky 的提問。關於上述的問題, 解的上界和下界的量化估計, 經過多次的改進, 在1987年, 數學家們把估計推進至 ─── 在prime k-tuples conjecture的假設下, 解的量化估計正是 。除了d(n) = d(n +1) 這個問題之外, 關於其他arithmetic functions 的類似問題, 亦引起一些數學家的興趣, 並陸續得到一些結果。
    此篇論文主要分成兩個部分, 第一個部分是從Heath-Brown [8,Lemma1], 我們觀察其他的arithmetic functions, 並得到一些類似的部分結果。第二個部分是我們把[5]中Theorem 2.1證明的兩個缺口(不詳盡之處)補上。
    Table of Content Contents
    1Introduction        1
    1.1 Historical notes…………………………………1
    1.2 An outline of this thesis……………………2
    2 Prerequisit                 4
    2.1 Definitions and basic properties of
      certain arithmetic functions and series…4
    2.2 A theorem from sieve theory…………………6
    3 Main results                8

    3.1 Problems motivated by a result of
      Heath- Brown……………………………………8
    3.2 Filling gaps in the proof of a result of
      Erdös,Pomerance and Sárközy………………14
    References                 22      
    Reference References
    1.T. M. Apostol, Introduction to analytic number theory , Springer - Verlag, New York, 1986.
    2. R. Bellman, Analytic number theory an introduction, The Benjamin Cummigs Publishing Company, Inc, London, 1980.
    3. P. Erdös and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2(1952), 257-271.
    4. P. Erdös, C. Pomerance and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Acad. Sci. Hungar. 49 (1987), 251-259.
    5. P. Erdös, C. Pomerance and A. Sárközy, On locally repeated values of certain arithmetic functions, III, Proc. Amer. Math. Soc. 101(1987), 1-7.
    6. H. Halberstam and H. E. Richert, Sieve methods, Academic Press, London, 1974.
    7. D. R. Heath-Brown, A parity problem from sieve theory, Mathematika 29(1982), 1-6.
    8. D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31(1984), 141-149.
    9. A. Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129(1987), 307-319.
    10. D. Redmond, Number theory: an introduction, Dekker, New York, 1996.
    11. C. Spiro, Thesis, Urbana, 1981.
    12. W. R. Wade, An introduction to analysis, Prentice-Hall, Englewood Cliffs, 1995.
    Advisor
  • Ming-Guang Leu(呂明光)
  • Files
  • 92221006.pdf
  • approve immediately
    Date of Submission 2005-07-06

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