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Automaticity IV : sequences, sets, and diversity

Jeffrey Shallit (1996)

Journal de théorie des nombres de Bordeaux

This paper studies the descriptional complexity of (i) sequences over a finite alphabet ; and (ii) subsets of N (the natural numbers). If ( s ( i ) ) i ≥ 0 is a sequence over a finite alphabet Δ , then we define the k -automaticity of s , A s k ( n ) , to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 ≤ i ≤ n , takes i expressed in base k as input and computes s ( i ) . We give examples of sequences that have high automaticity in all bases k ; for example, we show that the characteristic...

Bounds for the counting function of the Jordan-Pólya numbers

Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala, William Verreault (2020)

Archivum Mathematicum

A positive integer n is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number x .

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple ( x ) = g x + h j j = 1 k of linear forms in ℤ[x], the set ( n ) = g n + h j j = 1 k contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that ( n ) = g n + h j j = 1 k contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing (resp....

Discretization of prime counting functions, convexity and the Riemann hypothesis

Emre Alkan (2023)

Czechoslovak Mathematical Journal

We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free...

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