The use of this function of n in the statement of the prime number theorem of Hadamard and de la Vallée Poussin (“duh lah vah-LAY poo-SAHN”) improves on Chebyshev’s result that the prime counting function of n is theta of this function of n. For 10 points each:
[10m] Give this function of natural number n that asymptotically equals the prime counting function, denoted pi, of n in the common statement of the prime number theorem.
ANSWER: n over log n [accept equivalents such as n divided by log n; accept “l·n” or “natural logarithm” instead of “log”]
[10h] This mathematician and Korobov independently proved the most recent significant improvement to the error term of the PNT. This mathematician proved a weaker version of the Goldbach conjecture for extremely large numbers.
ANSWER: Ivan Vinogradov [or Ivan Matveevich Vinogradov]
[10e] A proof of this conjecture would mean that the error term in the PNT is big O of x to the power of one-half plus epsilon, the best bound possible. Per this conjecture, all nontrivial zeros of a certain function have real part one-half.
ANSWER: Riemann hypothesis [prompt on Riemann]
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