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This paper develops a new description of the asymptotics for the empirical distributions of significands and significant digits associated with (pn), where pn denotes the nth prime number. The work utilizes the space of probability measures on the significand, endowed with a suitable Kantorovich metric, as well as finite-dimensional projections thereof. For sequences sufficiently close to (pn), it is shown that the limit points of the associated empirical distributions form a circle that is made up of all rescalings of a single absolutely continuous distribution, and is centered at a distribution known as Benford’s law (BL). The precise rate of convergence to that circle is determined. Moreover, even in the infinite-dimensional setting of significands the convergence is seen to occur along a distinguished low-dimensional object, in fact, along a smooth curve intimately related to BL. By connecting (pn) and BL in a new way, the results rigorously confirm well-documented experimental observations and complement known facts in the literature. [ABSTRACT FROM AUTHOR]

Let Ψ=(ψ1,⋯,ψt):'Zd→Rt$\Psi =(\psi _1,\hdots, \psi _t):'\mathbb {Z}^d\rightarrow \mathbb {R}^t$ be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system Ψ$\Psi$: ∑x∈[−N,N]d∏i=1t1P(ψi(x))∼(2N)d(logN)t∏pβp,$$\begin{equation*} \sum _{x\in [-N,N]^d} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) \sim \frac{(2N)^d}{(\log N)^t} \prod _{p} \beta _p, \end{equation*}$$where βp$\beta _p$ are the corresponding local densities. In this paper, we demonstrate limits to equidistribution of these primes on small scales; we show the analog to Maier's result on primes in short intervals. In particular, we show that for all λ>t/d$\lambda > t/d$, there exist δλ±>0$\delta _\lambda ^\pm > 0$ such that for N$N$ sufficiently large, there exist boxes B±⊂[−N,N]d$B^\pm \subset [-N, N]^d$ of sidelengths at least (logN)λ$(\log N)^\lambda$ such that ∑x∈B+∏i=1t1P(ψi(x))>(1+δλ+)vol(B+)(logN)t∏pβp,$$\begin{equation*} \sum _{x\in B^+} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) > (1+\delta _{\lambda }^+) \frac{\textrm {vol}(B^+)}{(\log N)^t} \prod _{p}\beta _p, \end{equation*}$$∑x∈B−∏i=1t1P(ψi(x))<(1−δλ−)vol(B−)(logN)t∏pβp.$$\begin{equation*} \sum _{x\in B^-} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) < (1-\delta _{\lambda }^-) \frac{\textrm {vol}(B^-)}{(\log N)^t} \prod _{p}\beta _p. \end{equation*}$$ [ABSTRACT FROM AUTHOR]

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev's estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%. [ABSTRACT FROM AUTHOR]

This paper gives an explicit version of Selberg's mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32\,277\log ^2 y]$ for at least half the $y\in [x,2x]$ for all $x\geq 2$ , and at least one Goldbach number in $(x,x+9696 \log ^2 x]$ for all $x\geq 2$. [ABSTRACT FROM AUTHOR]

A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer q 0 , which is the product of several prime numbers: q 0 = p 1 p 2 ... p n . It is shown that it is appropriate to use the expansion of the number q 0 , to q = p 0 p 1 p 2 ... p n , where p 0 is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo q 0 , is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed. [ABSTRACT FROM AUTHOR]

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet's theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might "know" anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias. [ABSTRACT FROM AUTHOR]

For x ≥ y > 1 and u : = log ⁡ x / log ⁡ y , let Φ (x , y) denote the number of positive integers up to x free of prime divisors less than or equal to y. In 1950 de Bruijn [4] studied the approximation of Φ (x , y) by the quantity μ y (u) e γ x log ⁡ y ∏ p ≤ y (1 − 1 p) , where γ = 0.5772156... is Euler's constant and μ y (u) : = ∫ 1 u y t − u ω (t) d t. He showed that the asymptotic formula Φ (x , y) = μ y (u) e γ x log ⁡ y ∏ p ≤ y (1 − 1 p) + O (x R (y) log ⁡ y) holds uniformly for all x ≥ y ≥ 2 , where R (y) is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result. [ABSTRACT FROM AUTHOR]

In our previous article (Camargo and Martin in Bull Braz Math Soc New Ser 53:501–522, 2022), we presented some families of sets Θ x ⊂ { 1 , 2 , ⋯ , ⌊ x ⌋ } such that the sum of the Möbius function over Θ x is constant and equals to - 1 and we showed that the existence of such sets is intimately connected with the existence of the alternating series used by Tschebyschef and Sylvester to bound the prime counter function Π (x) . In this note, we answer two open questions stated in the last section of (Camargo and Martin 2022) about the general structure of these constant functions. In particular, we show that every such constant function x ⟼ ∑ j ∈ Θ x μ (j) can be characterized by Tschebyschef–Sylvester alternating series. We also show that the asymptotic sizes of the sets Θ x connects to the Sylvester’s Stigmata of the Tschebyschef–Sylvester series. [ABSTRACT FROM AUTHOR]

For arbitrary real t > 1 we examine the set { ⌊ x / n t ⌋ : n ≤ x }. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article informs about new effective results in the theory of the Riemann zeta-function, focusing on four groups providing estimates for the zeta-function and associated functions under the assumption of the Riemann hypothesis. Topic include explicit and RH estimates for various functions related to the zeta-function, including their applications to the distribution of prime numbers and other arithmetic properties, emphasizing the importance of these findings in mathematical research.

For a natural number n , let Z 1 (n) : = ∑ ρ n ρ ρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q ≥ 3 , we define Z 1 (s , χ) : = ∑ n = 1 ∞ χ (n) Z 1 (n) n s for ℜ (s) > 2 and obtain the meromorphic continuation of the function Z 1 (s , χ) to the region ℜ (s) > 1 2 . Our main result indicates that the poles of Z 1 (s , χ) in the region 1 2 < ℜ (s) < 1 , if they exist, are related to the zeros of many Dirichlet L -functions in the same region. [ABSTRACT FROM AUTHOR]

In this paper, we can show that S Λ (x) = ∑ 1 ≤ n ≤ x Λ ([ x n ]) = ∑ n = 1 ∞ Λ (n) n (n + 1) x + O (x 7 / 15 + 1 / 195 + ε) , where Λ (n) is the von Mangdolt function. Moreover, we can also give similar results related to the divisor function, which improve previous results. [ABSTRACT FROM AUTHOR]

In this paper, we prove a prime number theorem in short intervals for the Rankin-Selberg $ L $-function $ L(s, \phi\times\phi) $, where $ \phi $ is a fixed dihedral Maass newform. As an application, we give a lower bound for the proportion of primes in a short interval at which the Hecke eigenvalues of the dihedral form are greater than a given constant.

The document titled "Problems and Solutions" is an article from the American Mathematical Monthly that provides a list of proposed mathematical problems along with their solutions. The problems cover various topics and are contributed by different mathematicians from around the world. The article also includes instructions for submitting proposed problems and solutions. Additionally, it features a solution to a specific problem related to logarithmic trigonometric integrals. The given text presents a mathematical equation and its evaluation. The equation involves integrals and substitutions, and it is used to calculate a value called "I." The text provides step-by-step calculations and uses mathematical identities to simplify the equation. The final result is that I equals a specific value. The text also mentions the use of Fourier sine series in the calculations. The given text contains mathematical formulas and solutions to mathematical problems. The first part of the text presents a formula for the integral of a sine function, depending on whether the value of n is even or odd. The second part discusses the solution to a problem involving operator norms. The solution involves applying the Gram-Schmidt process to a basis in a certain space and obtaining an orthonormal basis. The text also mentions the names of individuals and groups who have solved the problems. The given text discusses the properties of a function f that belongs to a set S. It states that for any function f in S, the sum of the squares of the values of f and its derivative is equal to a constant. It also mentions that the supremum of [Extracted from the article]

A ring R is called radically z-covered (resp. radically z ◦-covered) if every √ zideal (resp. √ z◦-ideal) in R is a higher order z-ideal (resp. z◦-ideal). In this article we show with a counter-example that a ring may not be radically z-covered (resp. radically z◦-covered). Also a ring R is called z◦-terminating if there is a positive integer n such that for every m ≥ n, each z◦m-ideal is a z◦n-ideal. We show with a counter-example that a ring may not be z◦-terminating. It is well known that whenever a ring homomorphism φ: R → S is strong (meaning that it is surjective and for every minimal prime ideal P of R, there is a minimal prime ideal Q of S such that φ-1[Q] = P), and if R is a z◦-terminating ring or radically z◦-covered ring then so is S. We prove that a surjective ring homomorphism φ: R → S is strong if and only if ker(φ) ⊆ rad(R). [ABSTRACT FROM AUTHOR]

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet L -functions is true, we then establish explicit formulae for |$\psi(x,\chi)$|⁠ , |$\theta(x,\chi)$| and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli |$q\geq 3$|⁠. Finally, we restrict our attention to |$q\leq 10\,000$| and use an exact computation to refine these results. [ABSTRACT FROM AUTHOR]

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.

Every odd prime number p has exactly (p + 1) / 2 different expressions as a sum ab + cd of two ordered products ab and cd such that min (a , b) > max (c , d) . An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4 N as sums of two squares. [ABSTRACT FROM AUTHOR]

Let $[t]$ be the integral part of the real number t and let $\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by $\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by $S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence $\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman ['Primes in floor function sets', Integers 22 (2022), Article no. A59], we show $$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$ for $x\to \infty $ , where $C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$ , $c>0$ is a positive constant and $\varepsilon $ is an arbitrarily small positive number. [ABSTRACT FROM AUTHOR]

This article has a pedagogical aim. Indeed, we want to tackle the twin primes conjecture with an elementary and intuitive approach suitable for high schools. Perhaps it is needless to point out that this article does not have such a high goal of proving the conjecture but to give an intuitive idea of its validity. Research on prime numbers and related conjectures is so vast and advanced that it is impossible to fully address it in a high school. Hence, we want to analyze the twin primes from a didactic point of view and with the knowledge of high schools. Even a simple approach seems to indicate that there are infinitely many twin primes. [ABSTRACT FROM AUTHOR]

In 1845, Bertrand conjectured what became known as Bertrand's postulate or the Bertrand-Chebyshev theorem: twice and prime strictly exceeds the next prime. Surprisingly, a stronger statement seems not to be well-known: the sum of any two consecutive primes strictly exceeds the next prime, except for the only equality 2 + 3 = 5 . Our main theorem is a much more general result, perhaps not previously noticed, that compares sums of any number of primes. We prove this result using only the prime number theorem. We also give some numerical results and unanswered questions. [ABSTRACT FROM AUTHOR]

We obtain bounds for the error term in the prime number theorem of the form π (x) - Li (x) ≤ 9.2211 x log (x) exp - 0.8476 log (x) for all x ≥ 2 , as well as other classical forms, improving upon the various constants and ranges compared to those in the literature. The strength and originality of our methods come from leveraging numerical results for small x in order to improve both the asymptotic and numerical bounds one obtains. We develop algorithms and formulas optimizing the conversion of both asymptotic and explicit numerical bounds from the prime counting function ψ (x) to both θ (x) and π (x) . [ABSTRACT FROM AUTHOR]

Continuing previous studies of the Beurling zeta function, here, we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero |$\rho _0$| of the Beurling zeta function |$\zeta _{{\mathcal {P}}}$| for a given number system generated by the primes |${\mathcal {P}}$|⁠ , the corresponding error term |$\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$|⁠ , where |$\psi _{{\mathcal {P}}}(x)$| is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as |$\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$|⁠. The somewhat mysterious appearance of the constant |$\pi /2$| is explained in the study. Finally, we prove as the next main result of the paper the following: given |$\varepsilon>0$|⁠ , there exists a Beurling number system with primes |${\mathcal {P}}$|⁠ , such that |$|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$|⁠. In this 2nd part, a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery, and Vorhauer. [ABSTRACT FROM AUTHOR]

We provide an explicit O (x log x / T) error term for the Riemann–von Mangoldt formula by making results of Wolke [On the explicit formula of Riemann–von Mangoldt, II, J. London Math. Soc.2(3) (1983) 406–416] and Ramaré [Modified truncated Perron formulae, Ann. Math. Blaise Pascal23(1) (2016) 109–128] explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan. [ABSTRACT FROM AUTHOR]

This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between n 155 and (n + 1) 155 for all n ≥ 1. This result is in part obtained with a new explicit version of Goldston's 1983 estimate for the error in the truncated Riemann–von Mangoldt explicit formula. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e and R be a multiplicative hyperring. We introduce and study the notions of graded 2-absorbing and graded 2-absorbing primary hyperideals of a graded multiplicative hyperring R which are generalizations of prime hyperideals. We present basic properties and characterizations of these graded hyperideals and homogeneous components. Among various results, we prove that the intersection of two graded prime hyperideals is a graded 2-absorbing hyperideal. [ABSTRACT FROM AUTHOR]

We show that, for the Möbius function μ(n), we have ... μ(n) = o(xϑ) for any ϑ > 0.55. This improves on a result of Motohashi and Ramachandra from 1976, which is valid for ϑ > 7/12. Motohashi and Ramachandra's result corresponded to Huxley's 7/12 exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré's identity to extract a small prime factor from the n-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the Möbius function as well as some results on multiplicative functions and almost primes in short intervals. [ABSTRACT FROM AUTHOR]

Copyright of Journal of South China Normal University (Natural Science Edition) / Huanan Shifan Daxue Xuebao (Ziran Kexue Ban) is the property of Journal of South China Normal University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$. In 1978, Borwein ['A sum of reciprocals of least common multiples', Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$. Hong ['A sharp upper bound for the sum of reciprocals of least common multiples', Acta Math. Hungar. 160 (2020), 360–375] improved Borwein's upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$. In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained. [ABSTRACT FROM AUTHOR]

Let S be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in S having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in S. In the appendix, we use the same strategy to give a proof of Huber's geometric prime number theorem. [ABSTRACT FROM AUTHOR]

The present work is the result of the exploration of teaching methodologies that seek a greater interaction of the student with the approached concept. During the text we present methods to obtain the famous Pythagorean triples from concepts of number theory and algebra. Our proposal is to provide the teacher (reader) with theoretical mathematical tools with which he can better develop his practice in various mathematical contents. The search for new Pythagorean cracks has always been the target of fascination and study, both by professional mathematicians and curious students, so the theme here is developed and results already known to the community are presented and demonstrated in an unprecedented way, that is, with approaches mathematics that were not found in previous works. [ABSTRACT FROM AUTHOR]

In classical analytic number theory, the Prime Number Theorem and the asymptotic estimate for the sum of the Moebius function can easily be deduced from one another. We show that the analogues of these relations do not necessarily imply one another in Beurling generalized number systems, and we give an explanation for this different behavior. [ABSTRACT FROM AUTHOR]

We establish an operator theoretic version of the Wiener–Ikehara Tauberian theorem and use it to obtain a short proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory and Fourier analysis. [ABSTRACT FROM AUTHOR]

We prove a restricted inverse prime number theorem for an arithmetical semigroup with polynomial growth of the abstract prime counting function. The adjective "restricted" refers to the fact that we consider the counting function of abstract integers of degree whose prime factorization may only contain the first abstract primes (arranged in nondescending order of their degree). The theorem provides the asymptotics of this counting function as . The study of the discussed asymptotics is motivated by two possible applications in mathematical physics: the calculation of the entropy of generalizations of the Bose gas and the study of the statistics of propagation of narrow wave packets on metric graphs. [ABSTRACT FROM AUTHOR]

The Goldbach Conjecture, one of the oldest problems in mathematics, has fascinated and inspired many mathematicians for ages. In 1742 German mathematician Christian Goldbach, in a letter addressed to Leonhard Euler, proposed a conjecture. The modern-day version of the Binary/Strong Goldbach conjecture asserts that: Every even integer greater than 2 can be written as the sum of two primes. The conjecture had been verified empirically up to 4 × 1018, its proof however remains elusive, which seems to confirm that: Some problems in mathematics remain buried deep in the catacombs of slow progress ... mind stretching mysteries await to be discovered beyond the boundaries of former thought. Avery Carr (2013) The research was aimed at exposition, of the intricate structure of the fabric of the Goldbach Conjecture problem. The research methodology explores several topics, before the definite proof of the Goldbach Conjecture can be presented. The Ternary Goldbach Conjecture Corollary follows the proof of the Binary Goldbach Conjecture as well as the representation of even numbers by the difference of two primes Corollary. The research demonstrates that the Goldbach Conjecture is a genuine arithmetical question.

What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10 − 4 . This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes. [ABSTRACT FROM AUTHOR]

In this paper, we give a new upper bound for the number NR which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function π(x) holds for every x ≥ NR. [ABSTRACT FROM AUTHOR]

Consider a ring R, which is semiprime and also having k-torsion freeness. If F, d : R → R are two additive maps fulfilling the algebraic identity for each x in R. Then F will be a generalized derivation having d as an associated derivation on R. On the other hand, in this article, it is also derived that f is a generalized left derivation having a linked left derivation δ on R if they satisfy the algebraic identity for each x in R and k {2, m, n, (n + m - 1)!} and at last an application on Banach algebra is presented. [ABSTRACT FROM AUTHOR]

We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms. [ABSTRACT FROM AUTHOR]

• The notion of an asymptotic continued fraction expansion of a complex-valued function is introduced. • Some general results about asymptotic continued fraction expansions are established. • Two continued fraction expansions of the exponential integral E n (z) are shown to yield two asymptotic continued fraction expansions of the prime counting function π (x). • All of the best rational function approximations of the function π (e x) / e x are determined. • Our results on π (x) are generalized to any arithmetic semigroup satisfying Axiom A, hence to any number field. We provide several asymptotic expansions of the prime counting function π (x) and related functions. We define an asymptotic continued fraction expansion of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer n , two well-known continued fraction expansions of the exponential integral function E n (z) correspondingly yield two asymptotic continued fraction expansions of π (x) / x. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the "best" rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function π (e x) / e x. Finally, we generalize our results on π (x) to any arithmetic semigroup satisfying Axiom A, and thus to any number field. [ABSTRACT FROM AUTHOR]

This paper gives an explicit bound for the prime number theorem in short intervals under the assumption of the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

We prove explicit versions of the Kronecker–Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis. [ABSTRACT FROM AUTHOR]