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  "body": "\n## Summary\n\nThe Chebyshev functions $\\vartheta(x)$ and $\\psi(x)$ count the [[prime numbers|01826KMG2H]] up to $x$ with the weight $\\log p$.\nThe first sums $\\log p$ over primes $p \\le x$, and the second also counts the prime powers $p^k \\le x$.\nThey are easier to handle analytically than the [[prime-counting function|0RQSNVTFVZ]] $\\pi(x)$, and the [[prime number theorem|0EHBY5Y84N]] is equivalent to $\\psi(x) \\sim x$.\n\n## Explanations\n\n### L1 human\n\nInstead of counting each prime as $1$, give the prime $p$ the weight $\\log p$.\nThen large primes count more, in the right amount to make the total grow like $x$ itself.\nFor $x = 10$, the primes $2, 3, 5, 7$ give $\\vartheta(10) = \\log 210 \\approx 5.35$.\nThe function $\\psi$ also counts $4, 8, 9$, and $\\psi(10) = \\log 2520 \\approx 7.83$, the logarithm of the least common multiple of $1, \\dots, 10$.\n",
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