Researchers Identify a Previously Unnoticed Prime Number Pattern

May 21, 2009

Bartolo Luque and Lucas Lacasa, of the Universidad Politécnica de Madrid, have identified a new pattern in the distribution of prime numbers.

The researchers say that a generalization of Benford's law (BL), which addresses the rate of appearance of a given leading digit d in datasets, describes the statistical distribution of leading digits in the prime number sequence. Moreover, the researchers claim, a version of this pattern is found in the sequence of nontrivial Riemann zeta zeros.

By examining multiple datasets as d increases, the researchers found that the first-digit distribution of primes becomes more uniform. This finding, they said, follows a trend described by a Generalized BL (GBL), which can be explained by the prime number theorem. In other words, the shape of the mean local density of the sequences is responsible for the pattern.

Luque and Lacasa also investigated the sequence of nontrivial Riemann zeta zeros, which are related to the distribution of primes, and whose distribution is an important mathematical problem. Although the distribution of zeros does not follow BL, the researchers discovered that it does follow a size-dependent GBL—just as in the case of the primes.

"Mathematicians have studied prime numbers for centuries," Lacasa observed. "New insights and concepts coming from nonlinear science, such as multiplicative processes, help us to look at prime numbers from a different perspective. According to this focus, it becomes significant that even today it is still possible to discover unnoticed hints of statistical regularity in such sequences, without being an expert in number theory."

However, the most significant issue in their work, Lacasa said, is "to understand the reason and implications of such unexpected structure, not just for number theoretical issues but, interestingly, for other disciplines as well."

Their paper, "The first digit frequencies of primes and Riemann zeta zeros," appeared in the Proceedings of the Royal Society A.

Source: PhysOrg.com, May 8, 2009.