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An Elementary Proof of the Prime Number Theorem AN ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM ABHIMANYU CHOUDHARY Abstract. This paper presents an "elementary" proof of the prime number theorem, elementary in the sense that no complex analytic techniques are used. First proven by Hadamard and Valle-Poussin, the prime number the- orem states that the number of primes less than or equal to an integer x x asymptotically approaches the value ln x . Until 1949, the theorem was con- sidered too "deep" to be proven using elementary means, however Erdos and Selberg successfully proved the theorem without the use of complex analysis. My paper closely follows a modified version of their proof given by Norman Levinson in 1969. Contents 1. Arithmetic Functions 1 2. Elementary Results 2 3. Chebyshev's Functions and Asymptotic Formulae 4 4. Shapiro's Theorem 10 5. Selberg's Asymptotic Formula 12 6. Deriving the Prime Number Theory using Selberg's Identity 15 Acknowledgments 25 References 25 1. Arithmetic Functions Definition 1.1. The prime counting function denotes the number of primes not greater than x and is given by π(x), which can also be written as: X π(x) = 1 p≤x where the symbol p runs over the set of primes in increasing order. Using this notation, we state the prime number theorem, first conjectured by Legendre, as: Theorem 1.2. π(x) log x lim = 1 x!1 x Note that unless specified otherwise, log denotes the natural logarithm. 1 2 ABHIMANYU CHOUDHARY 2. Elementary Results Before proving the main result, we first introduce a number of foundational definitions and results. Definition 2.1. An arithmetical function or a sequence, is a function whose domain is the natural numbers, and codomain is either the real numbers or the complex numbers. Definition 2.2. We define the divisor sum of an arithmetic function a to be: X a(d) djn where the symbol d ranges over the set of positive divisors of n. Definition 2.3. We define the Dirichlet product or Dirichlet Convolution of two arithmetic functions f; g as: X n (f ∗ g)(n) = f(d)g d djn Note that the Dirichlet Product is commutative and associative. Moreover, the set of Arithmetical functions has an identity I over this product, and every arith- metical function with the property that f(1) 6= 0 has an inverse f −1 such that f ∗ f −1 = I. It is easy to verify that the identity function I is given by: ( 1 1 if n = 1 I(n) = = n 0 otherwise Definition 2.4. We define the Mobius function, µ as: 8 1 if n = 1 <> k µ(n) = (−1) if n = p1; :::; pk for primes p1; :::; pk :>0 otherwise Thus, the Mobius function allows us to determine a "parity" of sorts for any squarefree integer. Theorem 2.5. The divisor sum of the mobius function is given by: ( X 1 1 if n = 1 µ(d) = = = I(n) n 0 otherwise djn This can be verified using the fundamental theorem of arithmetic and the bino- mial theorem. Note that this divisor sum yields the identity function, an important property we will use momentarily. Definition 2.6. We define the unit function by: u(n) = 1 for all natural n. We see that the divisor sum in 2.5 can be rewritten as: X X X n 1 µ(d) = µ(d)1 = µ(d)u = (µ ∗ u)(n) = d n djn djn djn AN ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM 3 Thus, the mobius and unit functions are inverses of each other. We can use this property to derive a powerful formula, known as the Mobius inversion formula. Theorem 2.7 (Mobius inversion formula). If f; g are arithmetical functions and: X g(d) = f(n) djn then: X n f(d)µ = g(n) d djn Proof. We have by 2.3 that: g ∗ u = f Taking the convolution of both sides with µ we have: µ ∗ (g ∗ u) = µ ∗ f Using associativity and commutativity, we can write the above expression as: g = µ ∗ f = f ∗ µ as required. Corollary 2.8 (Generalized Mobius Inversion). If f; g are arithmetical functions and: X x g(x) = f n n≤x then X x f(x) = µ(n)g n n≤x where the symbol n ranges over all integers not greater than x. We now introduce Von Mangoldt's function given by the symbol Λ. Definition 2.9 (Von Mangoldt's Function). For every integer n ≥ 1 we define: ( log(p) if n = pk for some prime p and k ≥ 1 Λ(n) = 0 otherwise The above definition is fairly powerful as it turns a multiplication problem (prime factorization), into an addition problem through the use of logarithms. We are also prohibited from "double counting" any prime factors, as we will see in the next theorem. Theorem 2.10 (Divisor sum of the Von Mangoldt Function). X Λ(d) = log n djn The proof for this result can be derived follows naturally from the fundamental theorem of arithmetic. Roughly speaking, the function counts each prime factor of n exactly as many times as it appears in the prime factorization of n. Through summing and properties of the logarithm, the result follows. 4 ABHIMANYU CHOUDHARY 3. Chebyshev's Functions and Asymptotic Formulae Definition 3.1. We say that a function f is "big-oh g(x) " for all x ≥ a or write that: f(x) = O(g(x)) if there exists a constant M such that for all x ≥ a we have: f(x) ≤ Mg(x) Corollary 3.2. Let f; g be Riemann Integrable functions such that f(t) = O(g(t)) for t ≥ a. Then we have: Z x Z x f(t)dt = O g(t)dt a a Definition 3.3. We say that f is asymptotic to g or f ∼ g, if: f(x) lim = 1 x!1 g(x) Definition 3.4. We define the extension of an arithmetic function a as a map R+ ! R given by: + a(x) = a (bxc) for all x 2 R We now have a suitable way to extend the domain of arithmetic functions to the positive reals. We now have a new set of tools to our disposal, namely those of cal- culus. We now supply a powerful summation formula that allows us to approximate the partial sums of arithmetic functions. Theorem 3.5 (Abel's Summation Formula). Let f be a real valued function with a Riemann-Integrable derivative for t ≥ 1. Let a(n) be an arithmetical function and let A(x) be the partial sum of a up to x. Then: X Z x a(n)f(n) = f(x)A(x) − f 0(t)A(t)dt n≤x 1 Proof. Taking suitable a, f, we have that: X A(n) − A(n − 1) = [A(1) − A(0)] + ::: + [A(n) − A(n − 1)] 1≤n≤x This sum clearly telescopes to the value A(n) − A(0). Because A(0) is an empty sum, we have that: X X a(n) = [A(n) − A(n − 1)] n≤x n≤x Mutliplying both sides by f(n): X X a(n)f(n) = [A(n) − A(n − 1)]f(n) n≤x n≤x Expanding the right hand side: X X X [A(n) − A(n − 1)]f(n) = A(n)f(n) − A(n − 1)f(n) n≤x n≤x n≤x Reindexing, it follows that: X X X X A(n)f(n) − A(n − 1)f(n) = A(n)f(n) − A(n)f(n + 1) n≤x n≤x n≤x n≤x−1 AN ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM 5 Note that: X X A(n)f(n) = A(n)f(n) + A(x)f(x) n≤x n≤x−1 So we have: X X X X A(n)f(n)− A(n)f(n+1) = A(x)f(x)+ A(n)f(n)− A(n)f(n+1) n≤x n≤x−1 n≤x−1 n≤x−1 Combining we have: X X X A(x)f(x) + A(n)f(n) − A(n)f(n + 1) = A(n)(f(n + 1) − f(n)) n≤x−1 n≤x−1 n≤x−1 Because f has Riemann integrable derivative, we can apply the fundamental theo- rem of calculus to it and say: X X Z n+1 A(n)(f(n + 1) − f(n)) = A(n) f 0(t)dt n≤x−1 n≤x−1 n Because A(n) is constant on the interval [n; n + 1) and takes on the value A(n) everywhere, we can place it inside the integral. Thus: X Z n+1 X Z n+1 A(n) f 0(t)dt = A(t)f 0(t)dt n≤x−1 n n≤x−1 n . Summing the integrals, we have: X Z n+1 Z x A(t)f 0(t)dt = A(t)f 0(t)dt n≤x−1 n 1 Thus, in conclusion, we see that X Z x a(n)f(n) = A(x)f(x) + A(t)f 0(t)dt n≤x 1 as we need. Corollary 3.6 (Euler's Summation Formula). Let f be a function with Riemann- integrable derivative defined on the interval [1; x]. Then: X Z x Z x f(n) = f(t)dt + f 0(t)dt + f(x)(bxc − x) n≤x 1 1 Proof. This result follows from the Abel summation formula. We will use these results to derive results about the asymptotic behavior of cer- tain arithmetic functions.
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