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The systematic generation of prime numbers has been almost ignored since the 1990s, when most of the IT research resources related to prime numbers migrated to studies on the use of very large primes for cryptography, and little effort was made to further the knowledge regarding techniques like sieving. At present, sieving techniques are mostly used for didactic purposes, and no real advances seem to be made in this domain. This systematic review analyzes the theoretical advances in sieving that have occurred up to the present. The research followed the PRISMA 2020 guidelines and was conducted using three established databases: Web of Science, IEEE Xplore and Scopus. Our methodical review aims to provide an extensive overview of the progress in prime sieving—unfortunately, no significant advancements in this field were identified in the last 20 years.

We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

In this paper, we mainly study the monomial-prime numbers, which are of the form p n k for primes p and integers k ≥ 2 . First, we give an asymptotic estimate on the number of numbers of a general form pf ( n ) for arithmetic functions f satisfying certain growth conditions, which generalizes Bhat’s recent result on the Square-Prime Numbers. Then, we prove three Mertens-type theorems related to numbers of a more general form, partially extending the recent work of Qi-Hu, Popa and Tenenbaum on the Mertens sum evaluations. At the end, we evaluate the average and variance of the number of distinct monomial-prime factors of positive integers by applying our Mertens-type theorems.

We study the asymptotic behavior of the sequence$ \\{\\Omega (n) \\}_{ n \\in \\mathbb {N} } $from a dynamical point of view, where$ \\Omega (n) $denotes the number of prime factors of$ n $counted with multiplicity. First, we show that for any non-atomic ergodic system$(X, \\mathcal {B}, \\mu , T)$, the operators$T^{\\Omega (n)}: \\mathcal {B} \\to L^1(\\mu )$have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along$\\Omega (n)$. Second, we show that the behaviors of$\\Omega (n)$captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.

The distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative. We prove that periodic numbers are composite in all bases except for a single trivial case and establish a set of twelve theorems governing the behavior of primes and composites in terms of binary periodicity. Building upon these results, we introduce a novel scale-invariant entropic measure of primality, denoted p(s′), which provides an exact and unconditional entropic probability of primality derived solely from the periodic structure of a binary number and its binary derivatives. We show that p(s′) is quadratic, statistically well-defined, and strongly correlated with our earlier BiEntropy measure of binary disorder. Empirical analyses across several numerical ranges demonstrate that the variance in prime density relative to quadratic expectation is small, binormal, and constrained by the central limit theorem. These findings reveal a deep connection between entropy and the randomness of the primes, offering new insights into the entropic structure of number theory, with implications for the Riemann Hypothesis, special classes of primes, and computational applications in cryptography.

We show that there exists some $\\delta > 0$ such that, for any set of integers B with $|B\\cap[1,Y]|\\gg Y^{1-\\delta}$ for all $Y \\gg 1$ , there are infinitely many primes of the form $a^2+b^2$ with $b\\in B$ . We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \\leq X$ with $b \\in B$ for any $|B|=X^{1/2-\\delta}$ with $\\delta < 1/10$ and $B \\subseteq [\\eta X^{1/2},(1-\\eta)X^{1/2}] \\cap {\\mathbb{Z}}$ , in terms of zeros of Hecke L-functions on ${\\mathbb{Q}}(i)$ . We obtain the expected asymptotic formula for the number of such primes provided that the set B does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if B is a sparse subset of primes. For an arbitrary B we obtain a lower bound for the number of primes with a weaker range for $\\delta$ , by bounding the contribution from potential exceptional characters.

Consider both the Logarithmic integral, Li(x)=limϵ→0∫01−ϵdulnu+∫1+ϵxdulnu, and the prime counting function π(x)=∑p≤x1. From several recently developed known effective bounds on the prime counting function of the general form |π(x)−Li(x)|Li−1n1−a(ln[nlnn])b+1exp−cln[nlnn] for n≥n∗. Herein, the range of validity is explicitly bounded by some calculable constant n∗ satisfying n∗≤maxπ(x0),π(17),π((1+e−1) exp2(b+1)c2). These bounds provide very clean and up-to-date and explicit information on the location of the nth prime number. Many other fully explicit bounds along these lines can easily be developed. Overall this article presents a general algorithmic approach to converting bounds on |π(x)−Li(x)| into somewhat clearer information regarding the primes.

Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.

We show that the sum function of the Möbius function of a Beurling number system must satisfy the asymptotic bound M ( x ) = o ( x ) if it satisfies the prime number theorem and its prime distribution function arises from a monotone perturbation of either the classical prime numbers or the logarithmic integral.

Let λ1,λ2,λ3 be nonzero real numbers, not all of the same sign; let λ1/λ2 be irrational; and let η be any real number. We investigate the solvability of the inequality |λ1p1+λ2p2+λ3p32+η|<(maxpj)−1/12+θ, θ>0 in the prime variables p1, p2, and p3. We require that p1+2 and p2+2 have no more than 20 prime factors, while p3+2 has no more than 42 prime factors.