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The Prime Number Theorem
Order of Magnitude of Arithmetic Functions
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0(x log log 2x log arithmetic functions arithmetic progression assume characters modulo Chinese Remainder Theorem clearly common multiple consider the following contradiction Corollary d-th power non-residue d-th power residues Definition 1.1 divide divisible divisor equal equation establish the theorem estimate evaluate finite number fixed modulus following theorem follows immediately integers less inversion formula Jacobi symbol Legendre symbol Lemma log pn log2 log2x mn<x mutually disjoint n<x n notation number of integers number of primes number of solutions odd prime polynomial congruence pq<x prime number theorem primitive root modulo progression of length Proof properties prove the theorem quadratic non-residue Quadratic Reciprocity Law reduced residue system relatively prime residue system modulo roots of unity solutions modulo square free Substituting Sum-Formula Suppose Theorem 2.1 true unique factorization theorem unique solution vectors write zero