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Sharper bounds for Chebyshev's θ(x) function

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Sharper bounds for Chebyshev's θ(x) function
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CC Attribution - NonCommercial - NoDerivatives 4.0 International:
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In 1792, Gauss conjectured that the primes occur with a density of 1logx around x. Therefore, when developing explicit results relating to the Prime Number Theorem, it is useful to study Chebyshev‚Äôs θ(x) function, given by ∑p≤xlogp. Over summer 2017, I worked on a joint project supported by NSERC USRA to develop an effective version of the Prime Number Theorem. In this talk, I present our results which are the current best results for the prime counting function θ(x) for various ranges of x. We developed these results by first surveying existing explicit results from the past 60 years on prime counting functions. Our results are based on a recent zero density result for the zeroes of the Riemann Zeta function (due to H. Kadiri, A. Lumley, and N. Ng).