Related papers: Sieve Method and Prime Gaps via Probabilistic Meth…
In this paper, the method of gaps, a technique for deriving closed-form expressions in terms of information measures for the generalization error of supervised machine learning algorithms is introduced. The method relies on the notion of…
We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $\lambda^{s_n}$. In particular, we obtain a result which is…
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set…
Recent decades have seen an interest in prediction problems for which Bayesian methodology has been used ubiquitously. Sampling from or approximating the posterior predictive distribution in a Bayesian model allows one to make inferential…
This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang \cite{Z} had revived in us an intention to produce a second edition of our book "Opera de Cribro", one…
We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…
The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation. This paper focuses on refining the power posterior approach…
We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…
This article solves the Hume's problem of induction using a probabilistic approach. From the probabilistic perspective, the core task of induction is to estimate the probability of an event and judge the accuracy of the estimation.…
One field of particular interest in Number Theory concerns the gaps between consecutive primes. Within the last few years, very important results have been achieved on how small these gaps can be. The strongest of these results were…
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
The prime number problem falls within the realm of number theory, specifically elementary number theory. Current research approaches have unnecessarily complicated this matter. In contrast to more advanced mathematical tools, the methods of…
A Bayesian inference method for problems with small samples and sparse data is presented in this paper. A general type of prior ($\propto 1/\sigma^{q}$) is proposed to formulate the Bayesian posterior for inference problems under small…
We analyze algorithms for computing the $n$th prime $p_n$ and establish asymptotic bounds for several approaches. Using existing results on the complexity of evaluating the prime-counting function $\pi(x)$, we show that the binary search…
This paper is an attempt to bring together two approaches to language analysis. The possible use of probabilistic information in principle-based grammars and parsers is considered, including discussion on some theoretical and computational…
Bayesian predictive probabilities are commonly used for interim monitoring of clinical trials through efficacy and futility stopping rules. Despite their usefulness, calculation of predictive probabilities, particularly in pre-experiment…
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function.…