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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]

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]

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]

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]

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]

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]

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

Let G ¯ n = ∏ k = 0 n n k , the product of the elements of the n th row of Pascal's triangle. This paper studies the partial factorizations of G ¯ n given by the product G (n , x) of all prime factors p of G ¯ n having p ≤ x , counted with multiplicity. It shows log G (n , α n) ∼ f G (α) n 2 as n → ∞ for a limit function f G (α) defined for 0 ≤ α ≤ 1. The main results are deduced from study of functions A (n , x) , B (n , x) , that encode statistics of the base p radix expansions of the integer n (and smaller integers), where the base p ranges over primes p ≤ x. Asymptotics of A (n , x) and B (n , x) are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

This paper touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

The "red" cluster indicates a research-intensive institute; the "blue" cluster indicates universities that provide technical training; the "green" cluster covers both: Red Cluster University of the Witwatersrand Stellenbosch University University of Cape Town University of Pretoria Rhodes University Green Cluster University of the Free State University of KwaZulu-Natal North-West University University of Fort Hare University of Limpopo University of Western Cape University of Johannesburg Nelson Mandela Metropolitan University University of Zululand Black Cluster Vaal University of Technology Central University of Technology Durban University of Technology Mangosuthu University of Technology Tshwane University of Technology Cape Peninsula University of Technology University of Venda Walter Sisulu University Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. In 1918, the University of Cape Town received its full status as a university [[50], p. 672], and on the same occasion, Victoria College became Stellenbosch University, thanks to a HT ht donation by a local benefactor, Jan Marais. Led by the South-African-born physicist Neil Turok (b. 1958), the institute opened in 2003 in Cape Town and enabled the universities in the Cape region (Stellenbosch, Cape Town, and Western Cape) to collaborate with three European universities - Oxford, Cambridge, and Paris-Sud - to train mathematicians. These universities and colleges thus became the home of such educated mathematicians and their students. The two colleges decided, however, to remain separate universities [[50], p. 624]. [Extracted from the article]

Investigation has been made regarding the properties of the ℿp≤n (1 ± 1/ps) products over the prime numbers, where we fix the s ∈ ℝ exponent, and let the n ≥ 2 natural bound grow toward positive infinity. The nature of these products for the s ≥ 1 case is known. We get approximations for the case when s ∈ [1/2, 1), furthermore different observations for the case when s

Dirichlet matrices are strongly connected to some important problems in Number Theory, such as the Prime Number Theorem, the Riemann Hypothesis and Kalmár's Factorisatio Numerorum problem. In this note, we obtain some recurrences for the determinants and permanents of Dirichlet matrices that allows us to obtain information about the asymptotic behavior of the permanents of certain classes of Dirichlet matrices. A generalization of Kalmár's Factorisatio Numerorum problem is obtained. [ABSTRACT FROM AUTHOR]

Infinitely many elliptic curves over Q have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let N j (X) denote the number of elliptic curves over Q with at least j pairs of Galois-stable cyclic subgroups of order 4, and height at most X. In this article we show that N 1 (X) = c 1 , 1 X 1 / 3 + c 1 , 2 X 1 / 6 + O (X 0.105) . We also show, as X → ∞ , that N 2 (X) = c 2 , 1 X 1 / 6 + o (X 1 / 12) , the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann zeta-function in the critical strip. Here, c 1 , 1 = 0.95740 ... , c 1 , 2 = - 0.87125 ... , and c 2 , 1 = 0.035515 ... are calculable constants. Lastly, we show no elliptic curve over Q has more than 2 pairs of Galois-stable cyclic subgroups of order 4. [ABSTRACT FROM AUTHOR]

The article presents the discussion on thesis showing new explicit resulting on the distribution of prime numbers. Topics include largely falling into the categories of error estimates for the prime number theorem (PNT) and interval estimates for primes; and Riemann hypothesis (RH) proving an explicit error estimate and an explicit mean-value estimate for the PNT in short intervals.

After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed. [ABSTRACT FROM AUTHOR]

In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained. [ABSTRACT FROM AUTHOR]

The aim of this expository article is to introduce the reader to some of the fundamental milestones in the study of prime numbers across several centuries. Among the important developments in the study of prime numbers, we review the history of the prime number theorem, the Riemann zeta function (in relation to prime number theory), and some recent investigations into spacings between consecutive primes. We also present an important application of prime numbers in safe data transmission, namely the 'RSA public key cryptosystem'. [ABSTRACT FROM AUTHOR]

Let be a positive integer, and define for , where denotes the number of distinct prime factors of , and represents the total number of prime factors of (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that which is similar to the known identity equivalent to the Prime Number Theorem. For , we prove that We also raise a hypothesis on the parities of which implies the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

The number of primes less or equal to a real number x, π(x), has been approximated in the past by the reciprocal of the logarithm of the number x. Such an approximation works well when x is large but it can be poor when x is small. This paper introduces a fractal formalism to provide more fl exible approximation to the density of primes less or equal to a number x using the λ(s)-fractal spectrum. Results revealed that the density of primes less than or equal to x can be modeled as a monofractal probability mass function with high fractal dimension for large x. High fractal dimensions can often be decomposed to form a multifractal representation. The fractal density representation of the density of primes is closely linked to the Riemann zeta function and, thus, to the famous unsolved Riemann hypothesis.

The article offers information on the Riemann zeta zeros to give a distribution of prime numbers. Topics discussed include the prime number theorem given by mathematician Bernard Riemann, a paper published by Riemann related to the zeros of the Riemann zeta function, and efforts of Riemann to derive an integration to give a count of prime numbers through the Riemann hypothesis.

In this article we provide new explicit Chebyshev bounds for the prime counting function ψ(x). The proof relies on two new arguments: smoothing the prime counting function which allows one to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function. [ABSTRACT FROM AUTHOR]