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Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

Let r 1 , ... , r t be positive integers and let R 2 (r 1 , ... , r t) be the set of positive odd integers that can be represented as p + 2 k 1 r 1 + ⋯ + 2 k t r t , where p is a prime and k 1 , ... , k t are positive integers. It is easy to see that if r 1 − 1 + ⋯ + r t − 1 < 1 , then the set R 2 (r 1 , ... , r t) has asymptotic density zero. In this paper, we prove that if r 1 − 1 + ⋯ + r t − 1 ≥ 1 , then the set R 2 (r 1 , ... , r t) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p -adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m ∈ N such that p ℓ α is an infinite loop mod p m , for all ℓ ∈ N ∪ { 0 }. [ABSTRACT FROM AUTHOR]

Let F be a totally real number field. Dasgupta conjectured an explicit p -adic analytic formula for the Gross–Stark units of F. In a later paper, Dasgupta–Spieß conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of F. Dasgupta–Spieß conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when F is a cubic field. For a video summary of this paper, please visit https://youtu.be/hlBRUIOke04. [ABSTRACT FROM AUTHOR]

For a finite group G , we denote by d (G) and by E (G) , respectively, the small Davenport constant and the Gao constant of G. Let C n be the cyclic group of order n and let G m , n , s = C n ⋊ s C m be a metacyclic group. In [2, Conjecture 17] , Bass conjectured that d (G m , n , s) = m + n − 2 and E (G m , n , s) = m n + m + n − 2 provided ord n (s) = m. In this paper, we show that the assumption ord n (s) = m is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for G m , n , s and the mn -product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G 2 m , 2 n , r , where r 2 ≡ s (mod n). [ABSTRACT FROM AUTHOR]

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑ m ≥ 1 , n ≥ 1 λ π (1 , m) λ ⋆ (n) λ f (m + p n) U (m / X) V (n / H) in the level aspect are derived, where π is any G L 3 -Maaß cuspidal form, f ∈ B k ⁎ (p) , any Hecke newform of prime level p and weight k ∈ N + , and λ ⋆ (n) , n ≥ 1 , are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3- significance level. We study the strength of the result by specifying ⋆ being the cusp forms on G L 3 and G L 2 , respectively, and further obtain the more significant cancellations in these sums. [ABSTRACT FROM AUTHOR]

In a paper of Kedlaya and Medvedovsky [KM19] , the number of distinct dihedral mod 2 modular representations of prime level N was calculated, and a conjecture on the dimension of the space of level N weight 2 modular forms giving rise to each representation was stated. In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

Let 1 ⩽ a < b be two relatively prime integers. Sylvester found that a b − a − b is the largest integer which can not be represented by a x + b y (x , y ∈ Z ⩾ 0) about 160 years ago and this number shall be denoted by g a , b. Let N (a , b) = { n : n ⩽ g a , b , n = a x + b y , x , y ∈ Z ⩾ 0 } and π a , b be the number of primes in N (a , b). Recently, Ramírez Alfonsín and Skałba proved that π a , b ≫ ε g a , b (log ⁡ g a , b) 2 + ε for any fixed ε > 0. They further conjectured that the order of the magnitude of π a , b is 1 2 π (g a , b) , where π (x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a , b with 1 ⩽ a < b and (a , b) = 1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

Let E / Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of E / Q divides the product of the order of the Shafarevich–Tate group of E / Q , the (global) Tamagawa number of E / Q , and the Tamagawa number of E / Q at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, including the case where the curve E / Q is semi-stable. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number and k an imaginary quadratic field in which p splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for p and k , which states that the non-trivial Iwasawa module of the maximal multiple Z p -extension field over k has a non-trivial pseudo-null submodule. We prove this conjecture for p and k under the assumption that the Iwasawa λ -invariants vanish for the Z p -extensions over k in which one of the prime ideals of k lying above p do not ramify and that the characteristic ideal of the Iwasawa module associated to the cyclotomic Z p -extension over k has a square-free generator. [ABSTRACT FROM AUTHOR]

A famous conjecture of Parkin-Shanks predicts that p (n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any A ≡ ± 1 (mod 6) the multipartition function p A (n) has positive odd density, then so does p (n). Similarly, the positive odd density of any p A (n) with A ≡ 3 (mod 6) would imply that of p 3 (n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ 0 (N) with level N and nebentypus χ N. Let Q (x) be a positive definite integral quadratic form, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λ f (n) and r (n , Q). We generalize and improve previous results without the Ramanujan conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we formally introduce the notion of Apéry-like sums and we show that every multiple zeta values can be expressed as a Z -linear combination of them. We even describe a natural way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier's formulas for ζ (2 , ... , 2 , 3 , 2 , ... , 2) as well as of similar ones in the context of Apéry-like sums. The relations between Apéry-like sums themselves still remain rather mysterious, but we get significant results and state some conjectures about their pattern. [ABSTRACT FROM AUTHOR]

In this paper we consider a tower of number fields ⋯ ⊇ K (1) ⊇ K (0) ⊇ K arising naturally from a continuous p -adic representation of Gal (Q ¯ / K) , referred to as a p -adic Lie tower over K. A recent conjecture of Daqing Wan hypothesizes, for certain p -adic Lie towers of curves over F p , a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the p -adic valuation of the discriminant of the extension K (i) / K is given by a polynomial in i , p i for i sufficiently large. This generalizes a previously known result on discriminant-growth in Z p -towers of local fields of characteristic zero. [ABSTRACT FROM AUTHOR]

Let f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity S f (x) : = ∑ n ⩽ x f ([ x n ]) for x → ∞ , where [ t ] is the integral part of the real number t. This generalises some recent results of Bordellès, Dai, Heyman, Pan & Shparlinski and of Zhai (f = φ = the Euler function), and of Zhao & Wu (f = σ = the sum-of-divisors function). [ABSTRACT FROM AUTHOR]

A Diophantine m -tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1) (y 2 − 1) (z 2 − 1) = k 2 , be an affine variety over K. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x , y , z , k) ∈ X (K) is equal to k. We denote by X ‾ the projective closure of X and for a fixed k by X k a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties X k , X and X ‾ tell us about the arithmetic of Diophantine triples. First, we prove that the variety X ‾ is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface X k for a given k ∈ F p × in the prime field F p of odd characteristic and an abelian surface which is a product of two elliptic curves E k × E k where E k : y 2 = x (k 2 (1 + k 2) 3 + 2 (1 + k 2) 2 x + x 2). We derive an explicit formula for N (p , k) , the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X ‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ‾ over an arbitrary finite field F q. Using it we reprove the formula for the number of Diophantine triples over F q from [DK21]. Curiously, from the interplay of the two (K3 and rational) fibrations of X ‾ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)). Finally, in the Appendix, Luka Lasić defines circular Diophantine m -tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples. [ABSTRACT FROM AUTHOR]

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of m -regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all m ≤ 28 , we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area. [ABSTRACT FROM AUTHOR]

In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 < | A | < p 2 , B ∩ − B = ∅ and | B | < κ | A | ln ⁡ (| A |) , we have that max b ∈ B ⁡ | (A + b) ∖ A | ≥ | B |. In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed. [ABSTRACT FROM AUTHOR]

The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled by the works of Beuzart-Plessis, Gan and Ichino, says that for a pair of generic L -parameters of (U (n + 1) , U (n)) , there is a unique pair of representations in their associated Vogan L -packets which produces the Bessel model. In this paper, we examined the conjecture for a pair of L -parameters of (U (n + 1) , U (n)) as fixing a special non-generic parameter of U (n + 1) and varying tempered L -parameters of U (n) and observed that there still exist Gan-Gross-Prasad type formulae depending on the choice of L -parameter of U (n). [ABSTRACT FROM AUTHOR]

We propose a function-field analog of Pisot's d -th root conjecture on linear recurrences, and prove it under some "non-triviality" assumption. Besides a recent result of Pasten-Wang on Büchi's d -th power problem, our main tool, which is also developed in this paper, is a function-field analog of a GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such a GCD estimate, we also obtain an asymptotic result. [ABSTRACT FROM AUTHOR]

Let f be any arithmetic function and define S f (x) : = ∑ n ⩽ x f ([ x / n ]). When f equals the Euler totient function φ , several authors studied the upper and lower bounds of S φ (x). In this paper we shall prove that S φ (x) has an asymptotic formula by the method of exponential sums. This result proves a conjecture proposed by Bordellés, Dai, Heyman, Pan and Shparlinski. Some other asymptotic formulas for arbitrary f are also given in this paper. [ABSTRACT FROM AUTHOR]

We study the maximum gap g (maximum of the differences between any two consecutive exponents) of cyclotomic polynomials. In 2012, Hong, Lee, Lee and Park showed that g (Φ p 1 p 2 ) = p 1 − 1 for primes p 2 > p 1. In 2017, based on numerous calculations, the following generalization was conjectured: g (Φ m p) = φ (m) for square free odd m and prime p > m. The main contribution of this paper is a proof of this conjecture. The proof is based on the discovery of an elegant structure among certain sub-polynomials of Φ m p , which are divisible by the m -th inverse cyclotomic polynomial Ψ m = x m − 1 Φ m . [ABSTRACT FROM AUTHOR]

For any prime p , let R p be the set of all quadratic residues modulo p. In 2012, Sárközy proved that if p is a sufficiently large prime, then R p has no 3-decomposition A + B + C = R p with | A | , | B | , | C | ≥ 2. In this paper, we prove that for any prime p , R p has no additive 3-decomposition A + B + C = R p with | A | , | B | , | C | ≥ 2. Furthermore, for any prime p , if A + B = R p is a 2-decomposition, then 0.17 p + 1 < | A | , | B | < 2.8 p − 6.63. We also pose three related conjectures. [ABSTRACT FROM AUTHOR]

We prove a number of conjectures recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence. Furthermore, we study the regularity of sequences involved in the paper, and prove that for all q ≥ 2 , the sequence consisting of the positive integers whose odd part is of the form 4 k + 1 is not q -regular. Finally we establish the 2-regularity of two sequences of Hankel determinants. [ABSTRACT FROM AUTHOR]

In this paper, following Perrin-Riou's work, for any eigenform f of weight at least 2 and p -adic slope less than 1, we construct algebraic p -adic L -function L f (X) , and show that for eigenforms f 2 of weight 2 and f k of weight k , if f 2 ≡ f k (mod p N ′ ) and a p (f 2) / p ≡ a p (f k) / p (mod p N ⋅ p) for certain N , N ′ , then for every Dirichlet character χ of sufficiently large p -power conductor, χ (L f 2 (X)) and χ (L f k (X)) have the same p -adic valuation. Combined with our previous work with Choi on analytic congruences ([2]), this implies the following: Suppose E is an abelian variety over Q associated to f 2 , a p (E) = 0 (always true if E is an elliptic curve over Q and has good supersingular reduction at p > 3), and the above conditions for f 2 and f k hold true with sufficiently large N , N ′ (where the meaning of "sufficiently large" depends only on E). Then, the Main Conjecture of Iwasawa Theory for f k implies the Main Conjecture for E. [ABSTRACT FROM AUTHOR]

A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve X 0 (14) , using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of X 0 (14) whose Tate–Shafarevich group is of odd cardinality. [ABSTRACT FROM AUTHOR]

Denote by P + (n) the largest prime factor of an integer n. In this paper, we show that the Elliott-Halberstam conjecture for friable integers (or smooth integers) implies three conjectures concerning the largest prime factors of consecutive integers, formulated by Erdős-Turán in the 1930s, by Erdős-Pomerance in 1978, and by Erdős in 1979 respectively. More precisely, assuming the Elliott-Halberstam conjecture for friable integers, we deduce that the three sets E 1 = { n ⩽ x : P + (n) ⩽ x s , P + (n + 1) ⩽ x t } , E 2 = { n ⩽ x : P + (n) < P + (n + 1) x α } , E 3 = { n ⩽ x : P + (n) < P + (n + 1) } have an asymptotic density ρ (1 / s) ρ (1 / t) , ∫ T α u (y) u (z) d y d z , 1/2 respectively for s , t ∈ (0 , 1) , where ρ (⋅) is the Dickman function, and T α , u (⋅) are defined in Theorem 2. [ABSTRACT FROM AUTHOR]

We report here on the computational verification of Zhi-Wei Sun's "1-3-5 conjecture" for all natural numbers up to 105 103 560 126. This, together with a result of two of the authors, completes the proof of that conjecture. Furthermore, the computations made in the verification process of the 1-3-5 conjecture revealed a refinement, which we state as a separate conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhi-Wei Sun's "1-3-5 conjecture" for all integers, and reduce the general case to its verification up to 1.052 × 10 11. The computational verification was performed by the authors and a colleague, concluding the proof of Sun's 1-3-5 conjecture. We also establish some variations of this conjecture. [ABSTRACT FROM AUTHOR]

In this paper we apply the L -function Ratios Conjecture to compute the one-level density for a symplectic family of L -functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet L -functions [11,12]. We then directly calculate main and lower order terms for test functions f such that supp(f ˆ) ⊂ [ − α , α ] for some α < 1 , and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally, as a corollary, we deduce that, under GRH, at least 75% of these L -functions do not vanish at the central point. [ABSTRACT FROM AUTHOR]

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbers is bounded above by the sum of a constant and the logarithm of the product of the ramified primes. We prove the validity of Harbater's conjecture in a number of cases, including the situation where we restrict our attention to finite groups containing a nilpotent subgroup of index 1 , 2 , or 3. We also derive some consequences that are implied by the truth of this conjecture. [ABSTRACT FROM AUTHOR]

In this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part for the neutral Tannakian category mixed Tate motives over Z. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown's matrix conjecture and the nondegeneracy conjecture about depth-graded motivic Lie subalgebra generated by the depth one part. [ABSTRACT FROM AUTHOR]

In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst [3] , and give some general results. [ABSTRACT FROM AUTHOR]

For an irrational number θ , let 0 = a 0 < a 1 < a 2 < ... < a m < a m + 1 = 1 be the sequence of points { ℓ θ } , 1 ≤ ℓ ≤ m (where { x } denotes x − x , the fractional part of x) and define d θ (m) = max ⁡ { (a i − a i − 1) , 1 ≤ i ≤ m + 1 } Sarnak conjectured that sup ⁡ m d θ ⁎ m → ∞ (m) ≤ sup ⁡ m d θ m → ∞ (m) where θ ⁎ = 1 + 5 2 is the golden mean and θ is an arbitrary irrational number. In this paper we establish the conjecture, and we determine exactly sup ⁡ m d θ ⁎ (m). Four special properties of the golden mean are crucial to the proof. [ABSTRACT FROM AUTHOR]

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A = F q t be the ring of polynomials over the finite field F q and 0 ≠ a ∈ A. Let C be the A -Carlitz module. Let a be a fixed element in A. For n ∈ A , C (A / n A) is a finite A -module. The set of all annihilators of C (A / n A) is an ideal and generated by a monic polynomial, denoted by λ (n). Similarly, The set of all annihilators of the submodule of C (A / n A) generated by a is an ideal and let l a (n) be its monic generator. We say that a is a primitive root of n , if λ (n) = l a (n). Define N a (x) : = | { n ∈ A | deg ⁡ n = x , n is monic , a is a primitive root of n } | We prove that for a given non-constant a ∈ A , a ∉ E , an exceptional set, there exists an unbounded set V of integers such that lim inf x ∈ V N a (x) / q x = 0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields. [ABSTRACT FROM AUTHOR]

It has been known since Erdős that the sum of 1 / (n log ⁡ n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k , and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k = 6. [ABSTRACT FROM AUTHOR]

In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved. To cover levels that are important for elliptic curves, namely those that are not square-free, we establish results about L -functions with additive twists that are of independent interest. [ABSTRACT FROM AUTHOR]

We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture which empirically demonstrates remarkable accuracy even for modest values of primes. For a video summary of this paper, please visit https://youtu.be/mINg-0n7nOY. [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. For each η > 1 , let c = c (η) > 1 be some constant depending on η and P a , c , η : = { p ∈ P : p = P + (q − a) for some prime q with p η < q ⩽ c (η) p η }. In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞ , π a , c , η (x) : = | (1 , x ] ∩ P a , c , η | ∼ π (x) or π a , c , η (x) ≫ a , η π (x) according to values of η. These are complement for some results of Banks-Shparlinski [1] , of Wu [12] and of Chen-Wu [2]. [ABSTRACT FROM AUTHOR]

In this paper, we study Iwasawa theory for p = 2. First of all, we show that the classical Iwasawa main conjecture holds true even for p = 2 over a totally real field k assuming μ = 0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k , we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k = Q and K / Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal. [ABSTRACT FROM AUTHOR]

The paper generalizes the method of Zhao for an infinite family of Q -curves and establishes some analytic results on the 2-part of the Birch and Swinnerton-Dyer conjecture. We give lower bounds on the 2-adic valuations of the algebraic part of the L -values at s = 1 for those Q -curves. Moreover, we also discuss 2-descent on Q -curves and compute their Mordell-Weil group and Tate-Shafarevich group. [ABSTRACT FROM AUTHOR]

To extend Iwasawa's classical theorem from Z p -towers to Z p d -towers, Greenberg conjectured that the exponent of p in the n -th class number in a Z p d -tower of a global field K ramified at finitely many primes is given by a polynomial in p n and n of total degree at most d for sufficiently large n. This conjecture remains open for d ≥ 2. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on p -adic stability of zeta functions in a p -adic Lie tower of function fields. [ABSTRACT FROM AUTHOR]

In this paper, we provide an effective and practical method to find all three term arithmetic progressions in a given Lucas sequence of the first or second kind. Our interest is the case when the sequence has a negative discriminant, since the case of positive discriminant has recently been resolved by Hajdu et al. [11]. We present a conjecture on the maximal number and length of such arithmetic progressions based on computational evidence. [ABSTRACT FROM AUTHOR]

Abstract Let A an abelian variety of dimension r , defined over Q. For p a rational prime, we denote by F p the finite field of cardinality p. If A has good reduction at p , let A ¯ p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A (Q) , and let Γ p be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p , with p ≤ x , for which the quotient A ¯ p (F p) Γ p has at most 2 r − 1 cyclic components. [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove a conjecture that a D (4) -quintuple does not exist using both classical and new methods. Also, we give a new version of the Rickert's theorem that can be applied on some D (4) -quadruples. [ABSTRACT FROM AUTHOR]