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  "body": "\n## Summary\n\nThe Riemann zeta function is given by the series $\\zeta(s) = \\sum_{n \\ge 1} n^{-s}$ for complex $s$ with $\\operatorname{Re} s \u003e 1$.\nIt extends to a [[meromorphic function|0ZZ7ZYDR8W]] on $\\C$ with a single pole at $s = 1$.\nThrough the [[Euler product formula|0C3PN4075R]] it encodes the [[prime numbers|01826KMG2H]], and the location of its zeros controls the error term in the [[prime number theorem|0EHBY5Y84N]].\nThe [[Riemann hypothesis|VCGF9YR36X]] about these zeros is one of the main open problems of mathematics.\n\n## Explanations\n\n### L1 human\n\nStart with the sum $1 + 1/2^s + 1/3^s + \\cdots$ of the reciprocals of the powers $n^s$.\nFor $s = 2$ the sum is $\\pi^2/6$, and for $s = 1$ it is the harmonic series, which diverges.\nLetting $s$ be a complex number turns this sum into a function with a rich structure.\nBecause each integer factors into primes in one way, the function carries information about all primes at once.\n",
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