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The prime number theorem
Thesis (M.A.)--Boston Universit
The prime number theorem
Thesis (M.A.)--Boston University.In Chapter 1 of this thesis we give some elementary definitions and prove the following three theorems:
1.1 Every positive integer n greater than one can be expressed in the form n=p1p2...pk where each of the pi is a prime number.
1.2 Every integer n greater than one can be expressed in standard form in one and only one way. If we write n=(p1^a1)(p2^a2).....(pj^aj), where p1< p2 <...< pj and each ai is greater than 0, then n is expressed in standard form.
1.3 The number of prime numbers is infinite [TRUNCATED
The error term in the prime number theorem
We make explicit a theorem of Pintz concerning the error term in the prime
number theorem. This gives an improved version of the prime number theorem with
error term roughly square-root of that which was previously known. We apply
this to a long-standing problem concerning an inequality studied by Ramanujan.Comment: To appear in Math. Com
Harmonic sets and the harmonic prime number theorem
We restrict primes and prime powers to sets H(x)= Uān=1 (x/2n, x/(2n-1)). Let ĪøH(x)= ā pĪµH(x)log p. Then the error in ĪøH(x) has, unconditionally, the expected order of magnitude ĪøH (x)= xlog2 + O(āx). However, if ĻH(x)= āpmĪµ H(x) log p then ĻH(x)= xlog2+ O(log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the āharmonic prime number theoremā Ļ H(x)/ Ļ(x) ā log2
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