Math 259: Introduction to Analytic Number Theory Elementary
... Meanwhile we briefly consider upper bounds. Trivially π(x) < x for all x. Since all primes except 2 are odd (and 1 is not prime), we have in fact π(x) ≤ (x+1)/2, and π(2m) ≤ m for m = 1, 2, 3, . . .. Likewise since all but two primes are congruent to 1 or 5 mod 6 we have π(6m) ≤ 2m + 1 for natural n ...
... Meanwhile we briefly consider upper bounds. Trivially π(x) < x for all x. Since all primes except 2 are odd (and 1 is not prime), we have in fact π(x) ≤ (x+1)/2, and π(2m) ≤ m for m = 1, 2, 3, . . .. Likewise since all but two primes are congruent to 1 or 5 mod 6 we have π(6m) ≤ 2m + 1 for natural n ...
Digital properties of prime numbers
... Remark 2 Our approach can be summarized in a few steps: 1. A first step is reducing the problem to an exponential sum. 2. Then we remove some digits, namely the upper range and the lower range, using Van der Corput’s inequality, and this leads to focus on the digits in the middle range only. 3. Sep ...
... Remark 2 Our approach can be summarized in a few steps: 1. A first step is reducing the problem to an exponential sum. 2. Then we remove some digits, namely the upper range and the lower range, using Van der Corput’s inequality, and this leads to focus on the digits in the middle range only. 3. Sep ...
