Sharper Bounds on the Prime Number Theorem: A New Asymptotic Estimate

The prime number theorem, in its most elementary form, states that the number of primes less than a given real number x, denoted π(x), is asymptotically equal to x / log(x). This approximation becomes increasingly accurate as x grows, but a more refined and historically significant formulation expresses π(x) as being asymptotically equal to the logarithmic integral li(x), defined as the integral from 0 to x of dt / log(t). The difference between π(x) and li(x), often called the error term, has been the subject of intense study for over a century, as it quantifies the deviation from perfect predictability in the distribution of primes.

Five years ago, a notable result provided an explicit bound on the absolute difference |π(x) − li(x)| for x greater than exp(2000), a range where the asymptotic behavior is firmly established. This bound took the form of an inequality involving x, log(x), and an exponential term, with specific numerical constants. The significance of such a bound lies not only in its mathematical rigor but also in its utility for number theory and related fields, where precise estimates are crucial for proofs and applications.

This morning, a new development was highlighted in a blog post by Terence Tao, a renowned mathematician known for his work in harmonic analysis, partial differential equations, and number theory. The post discusses a recent paper that advances this line of inquiry. The new result provides a bound on |π(x) − li(x)| that is valid for all x ≥ 2, eliminating the previous lower threshold of exp(2000) and thus extending the applicability of the bound to all primes beyond the smallest ones. The form of the new bound mirrors the earlier one, but with smaller constants, leading to a tighter and more accurate estimate of the error term.

The specific inequality from the new paper is:

|π(x) − li(x)| ≤ 9.2211 x √(log(x)) exp(−0.8476 √(log(x)))

This expression captures the essential asymptotic behavior: the error grows roughly like x times the square root of log(x), but is damped by an exponential decay factor involving √(log(x)). The constants 9.2211 and 0.8476 are the key improvements, replacing larger values from prior work. For context, the earlier bound from five years ago had constants that were approximately 1.3 times larger in the multiplicative factor and slightly smaller in the exponent, making the new bound notably sharper, especially for moderately large x.

To understand the practical implications, consider that for x = 10^10, log(x) ≈ 23.02, and √(log(x)) ≈ 4.80. Plugging these into the new bound yields an error estimate of about 1.5 × 10^8, whereas the prime count π(10^10) is approximately 455,052,511. The relative error is thus about 0.33%, which is a substantial improvement over cruder approximations. As x increases, the relative error diminishes further, but the absolute bound remains valuable for theoretical purposes, such as in proofs of theorems that rely on explicit error terms.

This work is part of a broader effort in analytic number theory to refine the error term in the prime number theorem. The logarithmic integral li(x) is a better approximation than x/log(x) because it accounts for the varying density of primes more accurately. The error term's behavior is deeply connected to the Riemann Hypothesis, which, if true, would imply a much stronger bound: |π(x) − li(x)| = O(√x log(x)). The new result, while not assuming the Riemann Hypothesis, provides a concrete, unconditional improvement that pushes the boundaries of what is known unconditionally.

For those interested in the technical details, the paper achieving this result likely employs advanced techniques from complex analysis and sieve methods, possibly building on the work of earlier mathematicians like Hadamard, de la Vallée Poussin, and more recent contributors. The explicit constants suggest a careful optimization of the parameters in the proof, possibly using numerical verification or refined estimates in the zero-free regions of the Riemann zeta function.

In summary, this new bound represents a step forward in our understanding of the distribution of prime numbers. It offers a more precise tool for mathematicians and underscores the ongoing vitality of classical number theory in the modern era. The result is a testament to the cumulative nature of mathematical progress, where each improvement builds upon the foundations laid by predecessors.

For further reading, the paper is available on arXiv: https://arxiv.org/abs/2406.17386. Terence Tao's blog post discussing this result can be found at https://terrytao.wordpress.com/2024/06/18/tighter-bounds-in-the-prime-number-theorem/. The logarithmic integral is defined in detail on Wikipedia: https://en.wikipedia.org/wiki/Logarithmic_integral_function.