Rosser, J. Barkley; Schoenfeld, Lowell Approximate formulas for some functions of prime numbers. (English) Zbl 0122.05001 Ill. J. Math. 6, 64-94 (1962). Reviewer: S. Knapowski Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 18 ReviewsCited in 546 Documents MathOverflow Questions: Can one show combinatorially how \(\operatorname{lcm}(1, \dotsc, n)\) grows? MSC: 11N05 Distribution of primes Keywords:approximation formulas; inequalities; distribution of primes PDF BibTeX XML Cite \textit{J. B. Rosser} and \textit{L. Schoenfeld}, Ill. J. Math. 6, 64--94 (1962; Zbl 0122.05001) OpenURL Online Encyclopedia of Integer Sequences: Euler totient function phi(n): count numbers <= n and prime to n. pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159... The composite numbers: numbers n of the form x*y for x > 1 and y > 1. Smallest prime that begins with n. Primorial numbers (second definition): n# = product of primes <= n. Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)). n such that pi(n) >= phi(n). The (10^n)-th composite number. Decimal expansion of -zeta’(2) (the first derivative of the zeta function at 2). Decimal expansion of constant B3 (or B_3) related to the Mertens constant. n-th composite number appears n times. n appears n-th composite number times. n-th composite number appears n-th composite number times. n-th prime appears n-th composite number times. n-th composite number appears n-th prime times. Decimal expansion of sum log(p)/p^2 over the primes p=2,3,5,7,11,... Decimal expansion of constant C = maximum value that psi(n)/n reaches where psi(n)=log(lcm(1,2,...,n)) and lcm(1,2,...,n)=A003418(n). Decimal expansion of constant C = maximum value that PrimePi(n)*log(n)/n reaches where PrimePi(n) is the number of primes less than or equal to n, A000720. Decimal expansion of constant C = maximum value that sigma(n)*log(n^2)/n^2 reaches where sigma(n) = (sum of primes <= n), A034387. Decimal expansion of -zeta’(3) (the first derivative of the zeta function at 3). Decimal expansion of zeta’(-5) (the derivative of Riemann’s zeta function at -5) (negated). Decimal expansion of -zeta’(4). Decimal expansion of Rosser’s constant. Define P = e^gamma*loglog(n) and Q = 3/loglog(n), where gamma is Euler’s constant A001620. Then a(n) = phi(n) - ceiling(n/(P + Q)), where phi(n) is Euler’s function A000010. Decimal expansion of Sum_{p prime} log(p)/p^3. Decimal expansion of Sum_{p prime} log(p)/p^4. Decimal expansion of -Zeta’(2)/Zeta(2). Integers n such that lcm(1, ..., n) > exp(n). a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n). a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).