We establish an asymptotic formula for the number of integer solutions to the Markoff-
Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11% of the time, as one would expect if all digits were equally likely, but rather 30%. This phenomenon is known as Benford's Law. Analyzing which datasets adhere to Benford's Law and how quickly Benford behavior sets in are the two most important problems in the field. Most previous work studied systems of independent random variables, and relied on the independence in their analyses.Inspired by natural processes such as particle decay, we study the dependent random variables that emerge from models of decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow, and give several conjectures for other fragmentation processes. The main difficulty is that the resulting random variables are dependent, which we handle by using tools from Fourier analysis and irrationality exponents to obtain quantified convergence rates. Our method can be applied to many other systems; as an example, we show that the n! entries in the determinant expansions of n × n matrices with entries independently drawn from nice random variables converges to Benford's Law.
In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x/2, x]. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that π(x) − π(x/2) ≥ n for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n ∼ p 2n (where p m is the mth prime). He and Laishram proved the bounds p 2n < R n < p 3n , respectively, for n > 1. In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c,n such that for all x ≥ R c,n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c,n exists for all n and all c, that R c,n ∼ p n 1−c as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes, and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.
In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x/2, x]. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that π(x) − π(x/2) ≥ n for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n ∼ p 2n (where p m is the mth prime). In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c,n such that for all x ≥ R c,n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c,n exists for all n and all c, that R c,n ∼ p n 1−c as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes, and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case c = 1/2. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.
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