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Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus Q is a polynomial in q [ T ] such that Q (0) ≠ 0. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms f and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

Let p be a prime such that p ≡ 3 (mod 4). Then, we show that there is no imaginary cyclic quartic extension K of ℚ whose class number is p. Suppose L / ℚ is a cyclic extension of number fields with an odd degree. Then, we show that 2 does not divide the class number of L if the class group of L is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime. [ABSTRACT FROM AUTHOR]

Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form y 2 = x 3 − m x , and quartic curves of the form X 2 − d Y 4 = k. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients d , k. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of k. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of k. [ABSTRACT FROM AUTHOR]

Let U = (U n) n ∈ ℕ be a fixed binary recurrence with real characteristic roots α , β satisfying | α | > | β | and let p 1 , ... , p s be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants C + and C − such that the Diophantine equation U n − p 1 x 1 ⋯ p s x s = c has at most s solutions (n , x 1 , ... , x s) ∈ ℕ s + 1 if c > C + and at most s + 1 solutions (n , x 1 , ... , x s) ∈ ℕ s + 1 if c < − C − . In order to demonstrate the strength of our method we show that for the binary recurrence sequence A = (A n) n ∈ ℕ , with A n + 2 = 3 A n + 1 + A n for n ≥ 2 , A 0 = 0 and A 1 = 1 , the Diophantine equation A n − 2 x 1 3 x 2 5 x 3 1 7 x 4 1 9 x 5 = c has at most six solutions (n , x 1 , ... , x 5) unless c = − 3 5 in which case it has exactly seven solutions. [ABSTRACT FROM AUTHOR]

For a function f : ℕ → ℕ , define N f × (x) = # { n ≤ x : n = k f (k)  for some  k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ≤ k ≤ n : (k , n) = 1 } be Euler's totient function. We prove that (1) N τ × (x) ≍ x (log x) 1 / 2 ; (2) N ω × (x) = (1 + o (1)) x log log x ; (3) N φ × (x) = (c 0 + o (1)) x 1 / 2 , where c 0 = 1. 3 6 5 .... [ABSTRACT FROM AUTHOR]

For any > 0 , there exists q 0 () such for any q ≥ q 0 () and any invertible residue class a modulo  q , there exists a natural number that is congruent to a modulo  q and that is the product of exactly three primes, all of which are below q 3 2 + . If we restrict our attention to odd moduli q that do not have prime factors congruent to 1 mod 4, we can find such primes below q 1 1 8 + . If we further restrict our set of moduli to prime q that are such that (q − 1 , 4 ⋅ 7 ⋅ 1 1 ⋅ 1 7 ⋅ 2 3 ⋅ 2 9) = 2 , we can find such primes below q 6 5 + . Finally, for any > 0 , there exists q 0 () such that when q ≥ q 0 () , there exists a natural number that is congruent to a modulo  q and that is the product of exactly four primes, all of which are below q (log q) 6 . [ABSTRACT FROM AUTHOR]

Given any cube-free integer λ > 0 , we study the 3 -adic valuation of the algebraic part of the central L -value of the elliptic curve X 3 + Y 3 = λ Z 3. We give a lower bound in terms of the number of distinct prime factors of λ , which, in the case 3 divides λ , also depends on the power of 3 in λ. This extends an earlier result of the author in which it was assumed that 3 is coprime to λ. We also study the 3 -part of the Tate–Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate–Shafarevich group. [ABSTRACT FROM AUTHOR]

Let  p be a prime number and r a positive even integer less than  p − 1. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of Gal ( ℚ ¯ p / ℚ p) with Hodge–Tate weights (0 , r) whose mod- p reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the irreducible components of the semi-stable deformation rings in Hodge–Tate weights (0 , r) of the non-split reducible residual representations of Gal ( ℚ ¯ p / ℚ p). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish upper bounds for some short sums of a class of multiplicative functions over integers with certain restrictions on the number of prime factors. More precisely, we consider sums expressed in the form ∑ x − y < n ≤ x ω (n) ≤ s f (n) ; or more generally ∑ x − y < n ≤ x ω (G (n)) = ℓ f (| G (n) |) where f is a positive multiplicative function, G is a polynomial with integer coefficients, and as usual, ω (m) denotes the number of distinct prime factors of the integer m. [ABSTRACT FROM AUTHOR]

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places. [ABSTRACT FROM AUTHOR]

A finite set of prime numbers S is called unavoidable with respect to b > 1 if for each ξ ∈ ℝ the sequence of integer parts ⌊ ξ b n ⌋ , n = 0 , 1 , 2 , ... , contains infinitely many elements divisible by at least one prime number p from the set S. It is known that an unavoidable set exists with respect to b = 2 , 3 , 4 , 6 and that it does not exist if b > 1 is an integer such that b − 1 is not square free. In this paper, we show that no finite unavoidable sets exist with respect to b if b > 3 is a prime number or b belongs to some explicitly given arithmetic progressions, for instance, 2 4 k + 1 1 and 2 4 k + 1 9 , k = 0 , 1 , 2 , .... [ABSTRACT FROM AUTHOR]

In this paper, we consider the Diophantine equation in the title, where p , q are distinct odd prime numbers and a , b are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of p , q , a and b. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials. [ABSTRACT FROM AUTHOR]

Let a , b be fixed positive integers such that a is not a perfect square and b is squarefree, and let ω (b) denote the number of distinct prime divisors of b. Let (u 1 ,   v 1) denote the least solution of Pell equation u 2 − a v 2 = 1. Further, for any positive integer n , let u n = α n + α ̄ n 2 and v n = α n − α ̄ n 2 a , where α = u 1 + v 1 a and α ̄ = u 1 − v 1 a . In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations (∗) x 2 − a y 2 = 1 and y 2 − b z 2 = v 1 2 to have positive integer solutions (x , y , z) is obtained. By this result, we prove that if (∗) has a positive integer solution (x , y , z) for ω (b) ≤ 2 or 3 according to 2 ∤ b or not, then 4 u 1 2 − δ = b g 2 and (x , y , z) = (u r , v r , g v [ r + 1 2 ]) , where g is a positive integer, δ = 1 or 2 and r = 2 or 3 according to 2 ∤ b or not, [ r + 1 2 ] is the integer part of r + 1 2 , except for (a , b , x , y , z) = (2 4 , 2 1 3 4 , 4 7 5 2 5 , 9 7 0 1 , 2 1 0) [ABSTRACT FROM AUTHOR]

We examine oscillations in a number of sums of arithmetic functions involving Ω (n) , the total number of prime factors of n , and ω (n) , the number of distinct prime factors of n. In particular, we examine oscillations in S α (x) = ∑ n ≤ x (− 1) n − Ω (n) / n α and in H α (x) = ∑ n ≤ x (− 1) ω (n) / n α for α ∈ [ 0 , 1 ] , and in W (x) = ∑ n ≤ x (− 2) Ω (n) . We show for example that each of the inequalities S 0 (x) < 0 , S 0 (x) > 3. 3 x , S 1 (x) > 0 , and S 1 (x) x < − 3. 3 is true infinitely often, disproving some hypotheses of Sun. [ABSTRACT FROM AUTHOR]

Let k be a number field, O k its ring of integers, Cl (k) its classgroup and h the class number of k. Let Γ be a finite group. Let ℳ be a maximal O k -order in the semi-simple algebra k [ Γ ] containing O k [ Γ ] , and Cl (ℳ) its locally free classgroup. Let A = ℤ ∪ { 1 m , m ∈ ℤ ∖ { 0 , 1 , − 1 } } and n ∈ A. We define the set ℛ ( n , ℳ) of Galois module classes realizable by the n th power of the different to be the set of classes c ∈ Cl (ℳ) such that there exists a Galois extension N / k with Galois group isomorphic to Γ (Γ -extension), which is tamely ramified, and for which the class of ℳ ⊗ O k [ Γ ] N / k n is equal to c , where we clarify that if n = 1 m , where m ∈ ℤ ∖ { 0 , 1 , − 1 } , N / k n is the | m | th root of the inverse different N / k − 1 (respectively, the different N / k ) if m < 0 (respectively, m > 0) when it exists. Let l be a prime number and ξ be a primitive l th root of unity. In this article, we suppose that Γ is cyclic of order l and k / ℚ and ℚ (ξ) / ℚ are linearly disjoint. We prove, sometimes under an assumption on h , that ℛ ( n , ℳ) is a subgroup of Cl (ℳ) , by an explicit description using a Stickelberger ideal. In addition, for each n ∈ A , we determine the set of the Steinitz classes of N / k n , N / k runs through the tame Γ -extensions of k , and prove that it is a subgroup of Cl (k) , also sometimes under an hypothesis on h. [ABSTRACT FROM AUTHOR]

For an algebraic number field K and a prime number p , let K ̃ / K be the maximal multiple ℤ p -extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro- p extension of K ̃ is pseudo-null over the completed group ring ℤ p [ [ Gal (K ̃ / K) ] ]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers. [ABSTRACT FROM AUTHOR]

Let 𝔸 [ n ] be the group of n -torsion points of a commutative algebraic group 𝔸 defined over a number field F. For a prime 𝔭 of F , we let N 𝔭 (𝔸 [ n ]) be the number of 𝔽 𝔭 -solutions of the system of polynomial equations defining 𝔸 [ n ] when reduced modulo 𝔭. Here, 𝔽 𝔭 is the residue field at 𝔭. Let π F (x) denote the number of primes 𝔭 of F such that N (𝔭) ≤ x. We then, for algebraic groups of dimension one, compute the k th moment limit M k (𝔸 / F , n) = lim x → ∞ 1 π F (x) ∑ N (𝔭) ≤ x N 𝔭 k (𝔸 [ n ]) by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of F on k copies of 𝔸 [ n ] by an application of Burnside's Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on k copies of a set. [ABSTRACT FROM AUTHOR]

In this paper, we study the Iwasawa λ -invariant of the cyclotomic ℤ 2 -extension of ℚ (p q) , where p , q are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the 2 -part of the class number of certain quartic number fields by calculating the Sinnott index explicitly. [ABSTRACT FROM AUTHOR]

We give a new proof of the norm relations for the Asai–Flach Euler system built by Lei–Loeffler–Zerbes. More precisely, we redefine Asai–Flach classes in the language used by Loeffler–Skinner–Zerbes for Lemma–Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular form over a quadratic real field F have been already proved by Lei–Loeffler–Zerbes for primes which are inert in F and for split primes satisfying some assumption; with this technique we are able to remove it and prove tame norm relations for all inert and split primes. [ABSTRACT FROM AUTHOR]

In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let p be an odd prime and let A be any integer. We determine completely the product f p (A) : = ∏ 1 ≤ i , j ≤ (p − 1) / 2 p ∤ i 2 − A i j − j 2 (i 2 − A i j − j 2) modulo p ; for example, if p ≡ 1 (mod  4) then f p (A) ≡ − (A 2 + 4) (p − 1) / 4 (mod  p) if A 2 + 4 p = 1 , (− A 2 − 4) (p − 1) / 4 (mod  p) if A 2 + 4 p = − 1 , where (⋅ p) denotes the Legendre symbol. We also determine ∏ i , j = 1 p ∤ 2 i 2 + 5 i j + 2 j 2 (p − 1) / 2 (2 i 2 + 5 i j + 2 j 2 ) and ∏ i , j = 1 p ∤ 2 i 2 − 5 i j + 2 j 2 (p − 1) / 2 (2 i 2 − 5 i j + 2 j 2 ) modulo p. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over ℚ of conductor N , c the Manin constant of E , and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | III (E / K) | 1 2 is divisible by | E (ℚ) tor |. In this paper, we show that this conjecture is true if E (ℚ) tor ≅ ℤ / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

We prove that for the Steinhaus Random Variable z (n) , 𝔼 ∑ n ∈ E N , m z (n) 6 ≍ | E N , m | 3 for  m ≪ (log log N) 1 3 , where E N , m : = { 1 ≤ n ≤ N : Ω (n) = m } and Ω (n) denotes the number of prime factors of n. [ABSTRACT FROM AUTHOR]

For k ∈ ℕ and p a prime number, define Γ ∗ (k , p) to be the smallest positive integer n ∈ ℕ such that any diagonal form f (x 1 , ... , x s) = a 1 x 1 k + ⋯ + a s x s k , with integer coefficients, has nontrivial zero over ℚ p whenever s ≥ n. A special case of a conjecture attributed to Artin states that Γ ∗ (k , p) ≤ k 2 + 1. It is well known that the equality occurs when p = k + 1. In this paper, we obtain the exact values of Γ ∗ (1 0 , p) for all primes p and, except for p = 1 1 , these values are much lower than those established in the conjecture, as might be expected. [ABSTRACT FROM AUTHOR]

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at s = 1. [ABSTRACT FROM AUTHOR]

𝜃 -congruent numbers are a generalization of congruent numbers. As in the classical case, there is a particular interest in finding infinite families of non- 𝜃 -congruent numbers with special properties, such as having an arbitrary number of prime factors. This paper presents methods to find families of non- 𝜃 -congruent numbers in the spirit of the initial result of Iskra, and extending the results of Girard, Lalín and Nair for 𝜃 = π 3 and 𝜃 = 2 π 3 by adapting Monsky matrices following the ideas of Reinholz, Spearman and Yang. This paper also presents induction theorems using adapted Monsky matrices in order to find more general non- 𝜃 -congruent families than those presently known. [ABSTRACT FROM AUTHOR]

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial f , for every δ > 0 , and almost every prime q in [ Q , 2 Q ] , the number of primes from the interval [ x , x + x 1 2 + δ ] that are values of f modulo  q is the expected one, provided Q is not more than x 2 3 − 𝜖 . We obtain this via a variant of the classical truncated Perron's formula for the partial sums of the coefficients of a Dirichlet series. [ABSTRACT FROM AUTHOR]

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over ℚ of conductor N , c the Manin constant of E , and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | III (E / K) | 1 2 is divisible by | E (ℚ) tor |. In this paper, we show that this conjecture is true if E (ℚ) tor ≅ ℤ / 3 ℤ. [ABSTRACT FROM AUTHOR]

Let [ 𝜃 ] denote the integral part of the real number 𝜃. In this paper, it is proved that for 2 < c < 4 0 8 1 9 7 , the Diophantine equation [ p 1 c ] + [ p 2 c ] + [ p 3 c ] + [ p 4 c ] + [ p 5 c ] = N is solvable in prime variables p 1 , ... , p 5 for sufficiently large integer N. [ABSTRACT FROM AUTHOR]

A composite positive integer n has the Lehmer property if φ (n) divides n − 1 , where φ is an Euler totient function. In this paper, we shall prove that if n has the Lehmer property, then n ≤ 2 2 K − 2 2 K − 1 , where K is the number of prime divisors of n. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set g n − 1 g − 1 | n , g ∈ ℕ , ν 2 (g) + ν 2 (g + 1) ≤ L   , where ν 2 (g) denotes the highest power of 2 that divides g , and L ≥ 1 is a fixed real number. [ABSTRACT FROM AUTHOR]

Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p 1 a 1 p 2 a 2 ⋯ p k a k , where p 1 < p 2 ⋯ < p k , then one must have p k < 3 1 / 3 N 1 / 3 . Using methods similar to theirs, we show that p k − 1 < (2 N) 1 / 5 and that p k − 1 p k < 6 1 / 4 N 1 / 2. We also show that if p k and p k − 1 are close to each other, then these bounds can be further strengthened. [ABSTRACT FROM AUTHOR]

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers. [ABSTRACT FROM AUTHOR]

We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to x. [ABSTRACT FROM AUTHOR]

Denote by ℙ the set of all primes and by P (n) the largest prime factor of integer n ≥ 1 with the convention P (1) = 1. Let η 0 ≈ 2. 1 4 2 6 be the unique positive solution of the equation η − 1 − 4 η log (η − 1) = 0 in (1 , ∞). Very recently Wu proved that for η ∈ (3 2 1 7 , η 0) there is a constant c (η) > 1 such that for each fixed non-zero integer a ∈ ℤ ∗ the set 𝒫 a , c , η : = { p ∈ ℙ : p = P (q − a) for some prime  q  with  p η < q ≤ c (η) p η } has relative density 1 in ℙ. In this paper, we shall further extend the domain of η at the cost of obtaining a lower bound in place of an asymptotic formula, by showing that for each η ∈ [ η 0 , 1 + e 4) the set 𝒫 a , c , η has relative positive density in ℙ. [ABSTRACT FROM AUTHOR]

The distribution of the primes of the forms ⌊ n α ⌋ and ⌊ n α + β ⌋ are studied extensively, where ⌊ x ⌋ denotes the largest integer not exceeding x. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number 𝜃 with 0 < 𝜃 ≤ 1 , let π 𝜃 ′ (n) be the number of integers k with 1 ≤ k ≤ n 𝜃 such that ⌈ n / k ⌉ is prime and let π 𝜃 ′ ′ (n) be the number of primes p for which there exists an integer k with 1 ≤ k ≤ n 𝜃 such that p = ⌈ n / k ⌉ , where ⌈ x ⌉ denotes the least integer not less than x. These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial B n (x) − B n . In this paper, we study asymptotic properties of π 𝜃 ′ (n) and π 𝜃 ′ ′ (n). The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function. [ABSTRACT FROM AUTHOR]

An integer n is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers n ≤ x with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the n th (inverse) cyclotomic polynomials with ternary n. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction ≥ 0. 9 2 5 of ternary integers. [ABSTRACT FROM AUTHOR]

Let K be a number field of degree n over ℚ and A its ring of integers. For a prime number p ≥ n , we determine the types of splittings of p in K for which the set J ¯ = { 𝜃 ¯ ∈ A / p A , p | I (𝜃) } is of cardinality a power of p. We prove that this necessary condition is also sufficient for J ¯ to be a subgroup of the additive group A / p A. Consequently, we show that, in this case, the subset of A , J = { 𝜃 ∈ A , p | I (𝜃) } is an order of the number field. [ABSTRACT FROM AUTHOR]

Denote by P + (n) (respectively, P − (n)) the largest (respectively, the smallest) prime factor of the integer n ≥ 1. In this paper, we prove a lower bound of almost-primes n ≤ x with P − (n) > x 1 / v , v > 4 such that P + (n − a) ≤ x 1 / u for 1 ≤ u ≪ 1 , a ∈ ℤ , a ≠ 0. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor. [ABSTRACT FROM AUTHOR]

Spiro proved that the identity function is the only multiplicative function with f (p) ≠ 0 for some prime p and f (p + q) = f (p) + f (q) for all prime p and q. We determine the sets S of primes for which restricting our condition to f (p + q) = f (p) + f (q) for all p , q ∈ S still implies that f is the identity function. We prove that S satisfies these conditions if and only if S contains every prime that is not the larger element of a twin prime pair and S contains 5 or 7. [ABSTRACT FROM AUTHOR]

We study generalized prime systems 𝒫 and generalized integer systems 𝒩 obtained from them. The asymptotic distribution of generalized integers is deduced assuming that the generalized prime counting function π 𝒫 (x) behaves as π 𝒫 (x) = b li (x) + O (x α) for some b > 0 and α ∈ (1 / 2 , 1). [ABSTRACT FROM AUTHOR]

In this paper, we consider the mean-value estimate for nonlinear Weyl sums over primes in short intervals and establish the related Bombieri-type theorems. These results have applications in additive problems with prime variables in short intervals. [ABSTRACT FROM AUTHOR]

Let p be a fixed prime, and let v (a) stand for the exponent of p in the prime factorization of the integer a. Let f and g be two monic polynomials with integer coefficients and nonzero resultant r. Write S for the maximum of v (gcd (f (n) , g (n))) over all integers n. It is known that S ≤ v (r). We give various lower and upper bounds for the least possible value of v (r) − S provided that a given power p s divides both f (n) and g (n) for all n. In particular, the least possible value is p s 2 − s for s ≤ p and is asymptotically (p − 1) s 2 for large s. [ABSTRACT FROM AUTHOR]

Let 𝒫 r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer N , the equation N = x 2 + p 1 3 + p 2 3 + p 3 3 + p 4 3 + p 5 3 is solvable with x being an almost-prime 𝒫 6 and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with 𝒫 3 6 in place of 𝒫 6 . [ABSTRACT FROM AUTHOR]

We show that integral solutions of x 3 + y 3 + z 3 = q , where q is prime, satisfy certain congruence conditions. [ABSTRACT FROM AUTHOR]

In this paper, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms f and g with Fourier coefficients a n and b n respectively, we consider the following questions: existence of infinitely many primes p such that a p b p ≠ 0 ; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals. [ABSTRACT FROM AUTHOR]

For each integer m ≥ 3 , let P m (x) denote the generalized m -gonal number (m − 2) x 2 − (m − 4) x 2 with x ∈ ℤ. Given positive integers a , b , c , k and an odd prime number p with p ∤ c , we employ the theory of ternary quadratic forms to determine completely when the mixed sum a x 2 + b y 2 + c P p k + 2 (z) represents all but finitely many positive integers. [ABSTRACT FROM AUTHOR]

Given positive integers a 1 , … , a k , we prove that the set of primes p such that p ≢ 1 mod a i for i = 1 , … , k admits asymptotic density relative to the set of all primes which is at least ∏ i = 1 k (1 − 1 φ (a i)) , where φ is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer n such that n ≢ 0 mod a i for i = 1 , … , k admits asymptotic density which is at least ∏ i = 1 k (1 − 1 a i ). [ABSTRACT FROM AUTHOR]

Let 1 < c < 4 3 3 6 . In this paper, it is proved that for every sufficiently large real number N , the Diophantine inequality | p 1 c + p 2 c + p 3 c − N | < N − 9 1 0 c (4 3 3 6 − c) is solvable in primes p 1 , p 2 , p 3 . This result constitutes an improvement upon that of Baker and Weingartner. [ABSTRACT FROM AUTHOR]