The n-th prime exponentially

2025/04/20 by Visser, Matt
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.14458

Abstract

From known effective bounds on the prime counting function of the form |π(x)-Li(x)| lt; a x (ln x)^b exp(-c √(ln x)); (x ≥ x₀); it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem: For x ≥ x_* where x_*≤ max\x₀,17\ we have: Li \over 1+a (ln x)^b+1 exp(-c √(ln x)) lt; π(x) lt; Li \over 1-a (ln x)^b+1 exp(-c √(ln x)). Furthermore, it is possible to establish exponentially tight effective upper and lower bounds on the location of the n^th prime. Specifically: p_n lt; Li^-1 ( n [1+ a (ln[nln n])^b+1 exp(-c √(ln[nln n]))] ); (n≥ n_*). p_n gt; Li^-1 ( n [1- a (ln[nln n])^b+1 exp(-c √(ln[nln n]))] ); (n≥ n_*). Here the range of validity is explicitly bounded by some n_* satisfying n_* ≤ max\π(x₀),π(17), π( (1+e^-1) exp( [2(b+1)\over c]²)) \. Many other fully explicit bounds along these lines can easily be developed.

Citations

• Sharper bounds for the error term in the Prime Number Theorem
• Some explicit estimates for the error term in the prime number theorem
• Updating the error term in the prime number theorem
• The n-th prime asymptotically
• Approximate formulas for some functions of prime numbers
• Explicit Bounds for Some Functions of Prime Numbers
• The n -th Prime is Greater than n log n

Related

• Sharper bounds for the error term in the Prime Number Theorem
• Improving bounds on prime counting functions by partial verification of the Riemann hypothesis
• Algorithm, probability, and prime numbers
• A Note on Algorithms for Computing p_n
• Some explicit estimates for the error term in the prime number theorem