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In this paper, we establish a new mean value theorem of Bombieri–Vinogradov's type over Piatetski–Shapiro sequence. Namely, it is proved that for any given constant A > 0 and any sufficiently small ε > 0 , there holds ∑ d ⩽ x ξ (d , l) = 1 | ∑ A 1 (x) ⩽ a < A 2 (x) (a , d) = 1 g (a) (∑ a p ⩽ x a p ≡ l (mod d) a p = [ k 1 / γ ] 1 - 1 φ (d) ∑ a p ⩽ x a p = [ k 1 / γ ] 1) | ≪ x γ (log x) A , provided that 1 ⩽ A 1 (x) < A 2 (x) ⩽ x 1 - ε and g (a) ≪ τ r s (a) , where l ≠ 0 is a fixed integer and ξ : = ξ (γ) = 2 38 + 17 38 γ - 2 38 - 1 38 - ε with 1 - 18 2 38 + 17 < γ < 1. Moreover, for γ satisfying 1 - 0.03208 2 38 + 17 < γ < 1 , we prove that there exist infinitely many primes p such that p + 2 = P 2 with P 2 being Piatetski–Shapiro almost–primes of type γ , and there exist infinitely many Piatetski–Shapiro primes p of type γ such that p + 2 = P 2 . These results generalize the result of Pan and Ding [37] and constitute an improvement upon a series of previous results of [29, 31, 39, 47]. [ABSTRACT FROM AUTHOR]

Let ⌊ t ⌋ denote the integer part of t ∈ R and ‖ x ‖ the distance from x to the nearest integer. Suppose that 1 / 2 < γ 2 < γ 1 < 1 are two fixed constants. In this paper, it is proved that, whenever α is an irrational number and β is any real number, there exist infinitely many prime numbers p in the intersection of two Piatetski–Shapiro sets, i.e., p = ⌊ n 1 1 / γ 1 ⌋ = ⌊ n 2 1 / γ 2 ⌋ , such that ‖ α p + β ‖ < p - 12 (γ 1 + γ 2) - 23 38 + ε , provided that 23 / 12 < γ 1 + γ 2 < 2 . This result constitutes an generalization upon the previous result of Dimitrov (Indian J Pure Appl Math 54(3):858–867, 2023). [ABSTRACT FROM AUTHOR]

In this article, we prove that an asymptotic formula for the prime number race with respect to Fermat curves of prime degree is equivalent to part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of partial Euler products of L-functions on the critical line. We also show that such an equivalence holds for some quotients of Fermat curves. As an application, we compute the order of zero at s = 1 for the second moment L-functions of those curves under DRH. [ABSTRACT FROM AUTHOR]

We show that under certain conditions, nested Beatty sequences assume infinitely many prime values. [ABSTRACT FROM AUTHOR]

We define a Bessel type equation in multiplicative calculus by some algebraic structures. Eigenfunctions of constructed problem are obtained by power series solution technique. While making these solutions, multiplicative Bessel functions are used strongly. We get a generator for that problem and construct integration representations for multiplicative Bessel functions. Finally, some basic spectral properties of multiplicative Bessel problem are examined in detail. [ABSTRACT FROM AUTHOR]

In this article, we investigate a non-Archimedean analogue of a question of Atkin and Serre. More precisely, we derive lower bounds for the largest prime factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms of even weight k ≥ 2 , level N with integer Fourier coefficients. In particular, we show that for such a form f and for any real number ϵ > 0 , the largest prime factor of the p-th Fourier coefficient a f (p) of f, denoted by P (a f (p)) , satisfies P (a f (p)) > (log p) 1 / 8 (log log p) 3 / 8 - ϵ for almost all primes p. This improves on earlier bounds. We also investigate a number field analogue of a recent result of Bennett, Gherga, Patel and Siksek about the largest prime factor of a f (p m) for m ≥ 2. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Denote by D1/k the least integer d such that for every set A of nonnegative integers with the lower density l/k, the set (k + l)A contains an infinite arithmetic progression with difference at most d, where (k +l)A is the set of all sums of k + l elements (not necessarily distinct) of A. Chen and Li (2019) conjectured that D1/k = k2 + o(k2). The purpose of this paper is to confirm the above conjecture. We also prove that D1/k is a prime for all sufficiently large integers k. [ABSTRACT FROM AUTHOR]

In this work and its sister paper (Friedlander and Iwaniec (2023)), we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dispense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and relied on in earlier proofs. [ABSTRACT FROM AUTHOR]

We obtain global explicit numerical bounds, with best possible constants, for the differences 1 n ∑ k ⩽ n ω (k) - log log n and 1 n ∑ k ⩽ n Ω (k) - log log n , where ω (k) and Ω (k) refer to the number of distinct prime divisors, and the total number of prime divisors of k, respectively. [ABSTRACT FROM AUTHOR]

We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function an with the periodic Bernoulli polynomial weight and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order 1 of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order 1 subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer's Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [ 1 , x ] . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set. [ABSTRACT FROM AUTHOR]

Let P be the set of all primes and π (x) be the number of primes up to x. For any n ≥ 2 , let P + (n) be the largest prime factor of n. For 0 < c < 1 , let T c (x) = # p ≤ x : p ∈ P , P + (p - 1) ≥ p c. In this note, we prove that there exists some c < 1 such that lim sup x → ∞ T c (x) π (x) < 1 2 , which disproves a conjecture of Chen and Chen. [ABSTRACT FROM AUTHOR]

Let ([ n c ]) n = 1 ∞ be the Piatetski-Shapiro sequences. In this paper, it is proved that there exist infinitely many Piatetski-Shapiro primes in arithmetic progressions for 1 < c < 12 11 . Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from Piatetski-Shapiro sequences for 1 < c < 344 337 . These two theorems constitute improvements upon the previous results of Baker et al. (Acta Arith 157(1):37–68, 2013). [ABSTRACT FROM AUTHOR]

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 regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions. [ABSTRACT FROM AUTHOR]

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs α 1 , α 2 which are product of two primes in the imaginary quadratic field K such that | σ (α 1 - α 2) | ≤ 2 for all embeddings σ of K if the class number of K is one and | σ (α 1 - α 2) | ≤ 8 for all embeddings σ of K if the class number of K is two. [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]

Gross and Smith have put forward generalizations of Hardy–Littlewood twin prime conjectures for algebraic number fields. We estimate the behaviour of sums of a singular series that arises in these conjectures, up to lower-order terms. More exactly, where S (η) is the singular series, we find asymptotic formulas for smoothed sums of S (η) - 1 . Based upon Gross and Smith's conjectures, we use our result to suggest that for large enough 'short intervals' in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramér model prediction by a universal factor, independent of K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers. Numerical data are provided supporting the conjecture. [ABSTRACT FROM AUTHOR]

For an elliptic curve E / Q , let a p denote the trace of its Frobenius endomorphism over F p , where p is a prime of good reduction for E. Hasse's theorem asserts that | a p | ≤ 2 p . In this paper we establish average asymptotics for primes p for which a p ∈ 2 p - f (p) , 2 p or a p ∈ c p - f (p) , c p , where f (x) = o (x) is a function satisfying mild growth conditions and 0 < c < 2 is a constant. [ABSTRACT FROM AUTHOR]

Let g ⩾ 2 be an integer and S g (n) denote the sum of the digits in base g of the positive integer n. We obtain asymptotic formulas for the number of odd integers up to x that can be written as a product pq, with p, q both primes, satisfy S g (p) ≡ S g (q) ≡ a mod b (b ⩾ 2 , a ∈ Z) . So, we observe a large bias amongst such integers. [ABSTRACT FROM AUTHOR]

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for x ≥ 2 . Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at s = 1 . [ABSTRACT FROM AUTHOR]

Let V(T) denote the number of sign changes in ψ (x) - x for x ∈ [ 1 , T ] . We show that lim inf T → ∞ V (T) / log T ≥ γ 1 / π + 1.867 · 10 - 30 , where γ 1 = 14.13 ... denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski. [ABSTRACT FROM AUTHOR]

Let K be a cyclic number field of odd degree over Q with odd narrow class number, such that 2 is inert in K / Q . We define a family of number fields { K (p) } p , depending on K and indexed by the rational primes p that split completely in K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in K (p) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals. [ABSTRACT FROM AUTHOR]

Given β ∈ (0 , 1) , we show the existence of a Beurling generalized number system whose integer counting satisfies N (x) = a x + O (x exp (- c log β x)) for some a > 0 and c > 0 , and whose prime counting function satisfies π (x) = Li (x) + Ω (x exp (- c ′ (log x) β β + 1 )) for some c ′ > 0 . This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the prime number theorem with Malliavin-type remainder. [ABSTRACT FROM AUTHOR]

We discuss recent developments in the following important areas of Alfréd Rényi's research interest: axiomatization of quantitative dependence measures, qualitative independence in combinatorics, conditional qualitative independence in statistics/data science and in measure theory/probability theory, and finally, prime gaps that are responsible for Rényi's early career reputation. Most authors of this paper are main contributors to the new developments. [ABSTRACT FROM AUTHOR]

In this paper, we study some important means of Jordan's totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan's totient function and Carleman's inequality for this function. [ABSTRACT FROM AUTHOR]

We consider the number π(x, y; q, a) of primes p ⩽ x such that p ≡ a (mod q) and (p − a)/q is free of prime factors greater than y. Assuming a suitable form of Elliott-Halberstam conjecture, it is proved that π(x, y; q, a) is asymptotic to ρ(log(x/q)/log y)π(x)/φ(q) on average, subject to certain ranges of y and q, where ρ is the Dickman function. Moreover, unconditional upper bounds are also obtained via sieve methods. As a typical application, we may control more effectively the number of shifted primes with large prime factors. [ABSTRACT FROM AUTHOR]

In this note, we will show that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourths of primes and 24 powers of 2. This is an improvement of Z. Liu's result. [ABSTRACT FROM AUTHOR]

The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2. Unfortunately this long-standing conjecture remains open, but recently there has been several dramatic developments making partial progress. We survey the key ideas behind proofs of bounded gaps between primes (due to Zhang, Tao and the author) and developments on Chowla's conjecture (due to Matomäki, Radziwiłł and Tao). [ABSTRACT FROM AUTHOR]

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta function are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term. [ABSTRACT FROM AUTHOR]

In this short note, we give partial answers to two questions on shifted primes with large prime factors, posed by Luca et al. (Bull Belg Math Soc Simon Stevin 22:39-47, 2015) and by Chen and Chen (Acta Math Sin (Engl Ser) 33(3):377-382, 2017), respectively. [ABSTRACT FROM AUTHOR]

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term between the prime counting function and the logarithm integral having a bound lower than xlnx multiplied by a constant. An explicit value of this constant was found by Schoenfeld in 1976. We refine the work of Schoenfeld to find a bound for the kthprime number and find slightly more sharper bounds for sums and products over primes. [ABSTRACT FROM AUTHOR]

Erdős, Pólya and Turán conjectured 70 years ago that a linear combination of consecutive differences of primes takes infinitely often both positive and negative values if and only if the (fixed) coefficients of the linear combination do not have all the same sign. In this work we prove this conjecture in a somewhat more general form. Our proof is based on a method of Banks, Freiberg and Maynard which is again based on the method of Maynard, Tao and the Polymath 8 project which showed the existence of infinitely many prime gaps not exceeding 246. [ABSTRACT FROM AUTHOR]

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve. This allows us to derive a simplified proof of the Zhang and Maynard-Tao result on bounded gaps between primes. We also apply the sieve to other subsequences of the primes and obtain bounded gaps in various settings. [ABSTRACT FROM AUTHOR]

As usual, denote by P( n) the largest prime factor of the integer n ⩾ 1 with the convention P(1) = 1. For 0 < θ < 1, define In this paper, we obtain a new lower bound for T( x) as x → ∞, which improves some recent results of Luca et al. (2015) and of Chen and Chen (2017). As a corollary, we partially prove a conjecture of Chen and Chen (2017) about the size of T( x). [ABSTRACT FROM AUTHOR]

New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann hypothesis implies results about the distribution of prime numbers. We get better effective estimates of common number theoretical functions which are closely linked to $$\zeta $$ zeros like $$\psi (x),\vartheta (x),\pi (x)$$ , or the $$k\mathrm{{th}}$$ prime number $$p_k$$ . [ABSTRACT FROM AUTHOR]

Let $$k>-1$$ . The sum of the kth powers of the primes less than x is asymptotic to $$\pi (x^{k+1})$$ . We show that the sum is less than $$\pi (x^{k+1})$$ for arbitrarily large x, and the reverse inequality also holds for arbitrarily large x. When $$k>0$$ , there is a bias toward the first inequality, and we explain why this should be true and why the reverse bias holds when $$-1

Using $$ \mathfrak{G} $$ ( d, k) to denote the singular series for primes in arithmetic progressions and using the word 'tail' to denote the difference of $$ \mathfrak{G} $$ ( d, k) and its partial sums, we establish some asymptotic formulas for weighted sums of the square of the tail and so give more information on the singular series $$ \mathfrak{G} $$ ( d, k). [ABSTRACT FROM AUTHOR]

For any integer n ≥ 2, let P( n) be the largest prime factor of n. In this paper, we prove that the number of primes p ≤ x with P( p−1) ≥ p is more than (1− c+ o(1)) π( x) for 0 < c < 1/2. This extends a recent result of Luca, Menares and Madariaga for 1/4 ≤ c ≤ 1/2. We also pose two conjectures for further research. [ABSTRACT FROM AUTHOR]

For $$x>0$$ , let $$\pi (x)$$ denote the number of primes not exceeding x. For integers a and $$m>0$$ , we determine when there is an integer $$n>1$$ with $$\pi (n)=(n+a)/m$$ . In particular, we show that, for any integers $$m>2$$ and $$a\leqslant \lceil e^{m-1}/(m-1)\rceil $$ , there is an integer $$n>1$$ with $$\pi (n)=(n+a)/m$$ . Consequently, for any integer $$m>4$$ , there is a positive integer n with $$\pi (mn)=m+n$$ . We also pose several conjectures for further research; for example, we conjecture that, for each $$m=1,2,3,\ldots $$ , there is a positive integer n such that $$m+n$$ divides $$p_m+p_n$$ , where $$p_k$$ denotes the k-th prime. [ABSTRACT FROM AUTHOR]

We prove that every integer greater than two may be written as the sum of a prime and a square-free number. [ABSTRACT FROM AUTHOR]

The aim of this work is to estimate sums involving P( n), the largest prime factor of an integer $${n \geqq 2}$$ under digital constraints $${{f(P(n)) \equiv a}{\rm mod} b}$$ , for every $${a \in \mathbb{Z}}$$ and an integer $${b \geqq 2}$$ where f is a strongly q-additive function with integer values (i.e. $${f(aq^j + b) = f(a) + f(b)}$$ , with $${(a, b, j) \in \mathbb{N}^3}$$ , $${{0 \leqq b} < q^j}$$ ). We also estimate the cardinality of the set $${\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}$$ , where $${c \in \mathbb{Z}}$$ , $${k \geqq 2}$$ . [ABSTRACT FROM AUTHOR]

An improved estimate is given for $$|\theta (x) -x|$$ , where $$\theta (x) = \sum _{p\le x} \log p$$ . Four applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, the third to a conjecture by Pomerance and the fourth to an inequality studied by Ramanujan. [ABSTRACT FROM AUTHOR]

In this paper, we establish several results concerning the generalized Ramanujan primes. For $$n\in \mathbb {N}$$ and $$k \in \mathbb {R}_{> 1}$$ , we give estimates for the $$n$$ th $$k$$ -Ramanujan prime, which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $$k$$ -Ramanujan primes. In addition, we find explicit formulae for certain $$n$$ th $$k$$ -Ramanujan primes. As an application, we prove that a conjecture of Mitra et al. (, ) concerning the number of primes in certain intervals holds for every sufficiently large positive integer. [ABSTRACT FROM AUTHOR]

A prime number p is called strong if p − 1 and p + 1 each have a large prime factor. Let x be a large positive number. We show a lower bound on the number of strong primes p between 1 and x such that p−1 and p+1 each have a prime factor larger than $${\sqrt{p}}$$ . [ABSTRACT FROM AUTHOR]

Assuming a uniform version of Gross and Smith's generalized k-tuple conjecture, we prove that the prime elements in a ring of algebraic integers have Poisson distribution in small regions, by considering the sum of the corresponding singular series. [ABSTRACT FROM AUTHOR]

Let $$\left\lfloor \cdot \right\rfloor $$ be the floor function. In this paper, we show that for any fixed $$c\in \left( 1,\frac{77}{76}\right) $$ there are infinitely many primes of the form $$p=\left\lfloor n^c\right\rfloor $$ , where $$n$$ is a natural number with at most eight prime factors (counted with multiplicity). [ABSTRACT FROM AUTHOR]

Let $${q\geqq2}$$ be an integer and denote S( n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum $${\Sigma_{n\leqq x}\tilde{\omega}(n)}$$ where $${\tilde{\omega}(n)}$$ is the number of distinct prime factors p of n such that $${S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}$$ . [ABSTRACT FROM AUTHOR]

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $${\mathrm{Im }}\log \zeta (1/2+it)$$ . This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that $${\mathrm{Im }}\log \zeta (1/2+it)$$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow. [ABSTRACT FROM AUTHOR]

We give a heuristic proof of a conjecture of Hardy and Littlewood concerning the density of prime pairs to which twin primes and Sophie Germain primes are special cases. The method uses the Ramanujan-Fourier series for a modified von Mangoldt function and the Wiener-Khintchine theorem for arithmetical functions. The failing of the heuristic proof is due to the lack of justification of interchange of certain limits. Experimental evidence using computer calculations is provided for the plausibility of the result. We have also shown that our argument can be extended to the m-tuple conjecture of Hardy and Littlewood. [ABSTRACT FROM AUTHOR]

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of ℕ is a set of quadratic residues for infinitely many primes of the form [ n] with 1 ≤ c ≤ 243/205. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by Wright in certain aspects. [ABSTRACT FROM AUTHOR]