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Observations in Pure Mathematics Dale A Miller November

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Niveau: Supérieur, Doctorat, Bac+8
Observations in Pure Mathematics Dale A. Miller November 1973 Part I. Introduction The true justification of mathematics is aesthetic †, and it is my hope that the essay which follows is to some degree justified. In Part II, I discuss a function which is easily understood in one sense and then describe it in several other ways. The beauty of that sec- tion lies in the relationship it has to other branches of mathematics. For this reason I have added a concluding section. The strength of Part II lies in the elementary argument which finds the Riemann zeta function for even, positive integers. This approach is contrasted to those found in most other books which use the notion of double series expansion. The concluding theorem in Part III is very intriguing because it joins two very different kinds of primes. This theorem, however, appears to be very limited. Part II. Functions Concerning the Powers of Two The following discussion deals with functions which are related to a number's repre- sentation in base two. For example, f(n) is defined so that it is the number of ones in the base two form of n. Thus f(45) = 4 since 45 = 1011012. It is the goal of this part of my paper to give formulas for f(n) and related functions. The ultimate goal is to express these functions in terms of number-theoretic functions, such as µ(n) and d(n) to be defined later.

  • then µ

  • pi ∫

  • zeta function

  • fermat primes

  • thus using

  • mobius inversion

  • primes pi


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Observations in Pure Mathematics Dale A. Miller November 1973 Part I.Introduction The true justication of mathematics is aesthetic, and it is my hope that the essay which follows is to some degree justied.In Part II, I discuss a function which is easily understood in one sense and then describe it in several other ways.The beauty of that sec-tion lies in the relationship it has to other branches of mathematics.For this reason I have added a concluding section.The strength of Part II lies in the elementary argument which nds the Riemannzetafunction for even, positive integers.This approach is contrasted to those found in most other books which use the notion of double series expansion.The concluding theorem in Part III is very intriguing because it joins two very dierent kinds of primes.This theorem, however, appears to be very limited. Part II.Functions Concerning the Powers of Two The following discussion deals with functions which are related to a number’s repre-sentation in base two.For example,f(n) is dened so that it is the number of ones in the base two form ofn. Thusf(45) = 4 since 45 = 1011012is the goal of this part of. It my paper to give formulas forf(n) and related functions.The ultimate goal is to express these functions in terms of number-theoretic functions, such asµ(n) andd(n) to be dened later. We make the following denitions: k k (1)an= 1 ifn= 2andan= 0 ifn6= 2for somek. k (2)θ(n) =kifn= 2mwherem1(mod 2). k k (3)f(2 +b) = 1 +f(b), if 0b <2 andf(0) = 0. Of these functionsanAlso,is the simplest because it assumes only the values 0 and 1. by inspection we notice that X θ(n) =ad1. d|n In order to nd a formula forf(n), its generating series is useful.Using only the denition off(n) it can be shown that k ∞ ∞ 2 X X 1x k f(k)x=k.(1) 2 1x1 +x k=1k=0 rd This paper contains a slightly modied version of a paper I sent to the 33West-inghouse Science Talent Search in 1974.The principal changes have been the correction of some spelling and grammar errors, and the inclusion of a few extra steps in certain proofs to make them easier to read.The principal reference book for this paper is Hardy and Wright’sAn Introduction to the Theory of Numbers. Oh, the simplicity of youth!