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  "body": "\n## Statement\n\nLet $p$ run over the [[prime numbers|01826KMG2H]], let $\\log$ be the [[natural logarithm|04DXAFN2ES]], and let $x \\ge 2$ be real.\n\n1. $\\displaystyle \\sum_{p \\le x} \\frac{\\log p}{p} = \\log x + O(1)$.\n2. There is a constant $M$ such that $\\displaystyle \\sum_{p \\le x} \\frac{1}{p} = \\log \\log x + M + O\\!\\left(\\frac{1}{\\log x}\\right)$.\n3. $\\displaystyle \\prod_{p \\le x} \\left(1 - \\frac{1}{p}\\right) \\sim \\frac{e^{-\\gamma}}{\\log x}$ as $x \\to \\infty$, where $\\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].\n\nHere $O$ is [[big O notation|0TGV80E6A4]] as $x \\to \\infty$, and $\\sim$ is [[asymptotic equivalence|0QR92XBV6H]].\n\n## Remarks\n\nFor the first theorem, compare two expressions of $\\log(n!)$.\nLegendre's formula gives $\\log(n!) = \\sum_{p^k \\le n} \\floor{n/p^k} \\log p = n \\sum_{p \\le n} \\frac{\\log p}{p} + O(n)$, using the bound $\\psi(n) = O(n)$ of Chebyshev for the [[Chebyshev function|B8AKSVBE5Z]] $\\psi$.\n[[Stirling's formula|0KC0F51T6A]] gives $\\log(n!) = n \\log n + O(n)$, and dividing by $n$ gives the result.\nThe second theorem follows from the first by partial summation.\nThe third follows from the second by taking logarithms; identifying the constant $e^{-\\gamma}$ needs the behaviour of the [[Riemann zeta function|1XWBY4N0VT]] near $s = 1$.\nThe constant $M \\approx 0.2615$ is the Meissel–Mertens constant.\nThe theorems do not need the [[prime number theorem|0EHBY5Y84N]]; they are elementary.\n\n## Explanations\n\n### L1 human\n\nThe sum of the reciprocals of the primes diverges, but very slowly, like $\\log \\log x$.\nMertens' theorems give the precise size of this sum and of two related quantities.\nThe product $\\prod_{p \\le x} (1 - 1/p)$ is the proportion of integers that have no prime factor up to $x$, in a heuristic sense, and it decays like $e^{-\\gamma}/\\log x$.\n\n### L2 human\n\nThe theorems are typical of elementary prime number theory: they use only Chebyshev bounds and the factorization of $n!$.\nThe third theorem is used in sieve methods, where products over primes such as $\\prod_{p \\le z} (1 - 1/p)$ give the main term.\nA naive sieve heuristic with $z = \\sqrt{x}$ would predict about $2e^{-\\gamma} x/\\log x \\approx 1.12\\, x/\\log x$ primes up to $x$, while the prime number theorem gives $x/\\log x$.\nThe difference shows that divisibility by different large primes is not independent.\n",
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