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  "body": "\n## Statement\n\nLet $\\psi$ be the second [[Chebyshev function|B8AKSVBE5Z]] and $\\zeta$ the [[Riemann zeta function|1XWBY4N0VT]].\nFor $x \u003e 1$, let $\\psi_0(x) = \\psi(x)$ if $x$ is not a prime power, and $\\psi_0(x) = \\psi(x) - \\frac{1}{2} \\log p$ if $x = p^k$ with $p$ prime and $k \\ge 1$.\nLet $\\rho$ run over the nontrivial zeros of $\\zeta$, those with $0 \u003c \\operatorname{Re} \\rho \u003c 1$, each counted with its multiplicity.\nThen\n$$\\psi_0(x) = x - \\lim_{T \\to \\infty} \\sum_{\\abs{\\operatorname{Im} \\rho} \u003c T} \\frac{x^{\\rho}}{\\rho} - \\log(2\\pi) - \\frac{1}{2} \\log\\!\\left(1 - x^{-2}\\right).$$\n\n## Remarks\n\nThis is the explicit formula of Riemann and von Mangoldt.\nThe proof starts from Perron's formula: for $c \u003e 1$,\n$$\\psi_0(x) = \\frac{1}{2\\pi i} \\int_{c - i\\infty}^{c + i\\infty} \\left(-\\frac{\\zeta'(s)}{\\zeta(s)}\\right) \\frac{x^s}{s} \\, ds,$$\nbecause $-\\zeta'(s)/\\zeta(s) = \\sum_n \\Lambda(n) n^{-s}$ with the [[von Mangoldt function|0NK8JTE1NN]] $\\Lambda$.\nMoving the line of integration to the left picks up residues.\nThe pole at $s = 1$ gives $x$, and each nontrivial zero gives $-x^\\rho/\\rho$.\nThe point $s = 0$ gives $-\\zeta'(0)/\\zeta(0) = -\\log(2\\pi)$.\nEach trivial zero $-2n$ gives $x^{-2n}/(2n)$, and these terms add up to $-\\frac{1}{2}\\log(1 - x^{-2})$.\nThe sum over zeros converges only conditionally, so the order of summation matters.\nHeuristically, the [[prime number theorem|0EHBY5Y84N]] $\\psi(x) \\sim x$ reflects that each term $x^\\rho/\\rho$ is $o(x)$, since $\\operatorname{Re} \\rho \u003c 1$.\nA rigorous proof along these lines also needs a zero-free region to control the whole sum.\n\n## Explanations\n\n### L1 human\n\nThe formula writes a prime-counting function exactly as a main term $x$ plus a sum of waves, one for each zero of the zeta function.\nThe real part of a zero gives the size of its wave, and the imaginary part gives its frequency.\nThis is why the zeros of zeta control the distribution of the primes.\n\n### L2 human\n\nUnder the [[Riemann hypothesis|VCGF9YR36X]] each term has size $\\abs{x^\\rho} = \\sqrt{x}$, and a truncated form of the formula gives $\\psi(x) = x + O(\\sqrt{x} \\log^2 x)$.\nWeil generalized the formula to a symmetric identity between a sum over the zeros of an L-function and a sum over prime powers, for a class of test functions.\nSimilar formulas hold for [[Dirichlet L-functions|9D2Z543EJQ]], and they give the distribution of primes in arithmetic progressions.\n",
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