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In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

For K / F a finite Galois extension of number fields, the relative Pólya group Po (K / F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K / F. The notion of Ostrowski quotient Ost (K / F) , as the cokernel of the capitulation map into Po (K / F) , has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define Po (K / F) and Ost (K / F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F , we define the Ostrowski quotient Ost (E , K / F) and the coarse Ostrowski quotient Ost c (E , K / F) of E relative to K / F , for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group Ost c (E , K / F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K / F. [ABSTRACT FROM AUTHOR]

A Carmichael number is a composite number such that divides for all integers coprime to . Korselt discovered that is a Carmichael number if and only if is square-free and for each prime divisor of . Let , a -number is defined to be a composite number , such that and for each prime . The set of all such that is a -number is called the Korselt set of and we denote this set by . In this paper, we investigate some properties of when is a power of a prime. [ABSTRACT FROM AUTHOR]

In a recent paper, Sun posed six conjectures on the relations between T (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; n) , where T (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 (x 1 + 1) 2 + a 2 x 2 (x 2 + 1) 2 + ⋯ + a k x k (x k + 1) 2 , where a 1 , a 2 , ... , a k are positive integers, n , x 1 , x 2 , ... , x k are arbitrary nonnegative integers, and N (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , where this time x 1 , x 2 , ... , x k are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]

The goal of this paper is to prove several identities for sums of the type where is the number of divisors of and is the number of representations of as a sum of two squares. Some of our identities involve weighted divisor functions, and most of our identities are new. [ABSTRACT FROM AUTHOR]

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc.27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Epipelagic L -packets and rectifying characters, Invent. Math.202(1) (2015) 1–89, MR 3402796]. The interesting feature of our computation is that we find simplifications within L -packets of the two novel invariants introduced in the above-mentioned paper of Kaletha, the toral invariant and the admissible L-embedding. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

Let gcd (j , k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd -sum function f (gcd (j , k)) and the function ∑ d | k , d s | j (f ∗ μ) (d) for any positive integers j and k , namely ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) and ∑ k ≤ x 1 k s (r + 1) ∑ j = 1 k s j r ∑ d | k d s | j (f ∗ μ) (d) with any fixed integer s > 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd -sum functions f (gcd (j , k)). [ABSTRACT FROM AUTHOR]

For positive integers m , n , k with m ≥ 2 and ⌈ n / 2 ⌉ ≤ k ≤ n , let E { m } (m , n , k) be the sum of multiple zeta values of depth k and weight m n with arguments m or 2 m , i.e. E { m } (m , n , k) = ∑ | α | = n α j = 1 or 2 ζ (m α 1 , m α 2 , ... , m α k). In this paper, we are going to evaluate E { m } (m , n , k). As an application, we produce the stuffle relations from k identical Riemann zeta values ζ (2 m) as well as k identical Riemann zeta values ζ (2 m) and ζ (m). [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]

The Minkowski question mark function ? : [ 0 , 1 ] → [ 0 , 1 ] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus ? (x) is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]

We give a q -analogue of the supercongruence (I.2) of Van Hamme. As a conclusion, we confirm a recent conjecture of Swisher. We also give a q -analogue of the corresponding π series (I.1) along with some similar results. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

We give an alternative and relatively simple proof of the functional equation (Funct. Approx. Comment. Math. 55(2) (2016) 227–241) of Mordell–Tornheim multiple zeta function. [ABSTRACT FROM AUTHOR]

Let denote the classical Dedekind sum, where is a positive integer and , . For a given positive integer , we describe a set of at most numbers for which may be , provided that is sufficiently large. For the numbers not in this set, . [ABSTRACT FROM AUTHOR]

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between the modular surface ℳ = P S L (2 , ℤ) \ ℍ , cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits (1 , + 1) and (2 , − 1) in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that d x d y (1 + x y) 2 is invariant under the natural extension map. [ABSTRACT FROM AUTHOR]

We give two variations on a result of Wilkie's [A. J. Wilkie, Complex continuations of ℝ an , exp -definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in ℝ an , exp that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function 2 x , Wilkie showed that it must be eventually equal to a polynomial. Assuming a stronger growth condition, but only assuming that the function takes values sufficiently close to integers at positive integers, we show that the function must eventually be close to a polynomial. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers (for instance the primes), again under a stronger growth bound than that in Wilkie's result. [ABSTRACT FROM AUTHOR]

Let f be a fixed self-dual Hecke–Maass form for S L (3 , ℤ) , and let u be an even Hecke–Maass form for S L (2 , ℤ) with Laplace eigenvalue 1 / 4 + k 2 , k > 0. A subconvexity bound for L (1 / 2 , f × u) is improved to O (k 2 1 / 1 6 + 𝜀) , and a subconvexity bound for L (1 / 2 + i t , f) is improved to O ((1 + | t |) 2 1 / 3 2 + 𝜀). New techniques employed include an application of an asymptotic formula by Salazar and Ye [Spectral square moments of a resonance sum for Maass forms, Front. Math. China 12(5) (2017) 1183–1200] to make error terms negligible, an iterative algorithm to locate stationary point, and a non-trivial estimation of Kloosterman sums. [ABSTRACT FROM AUTHOR]

Let 𝒮 be a commutative semigroup endowed with a binary associative operation +. An element e of 𝒮 is said to be idempotent if e + e = e. The Erdős–Burgess constant of 𝒮 is defined as the smallest ℓ ∈ ℕ ∪ { ∞ } such that any sequence T of terms from S and of length ℓ contains a nonempty subsequence, the sum of whose terms is idempotent. Let q be a prime power, and let 𝔽 q [ x ] be the polynomial ring over the finite field 𝔽 q . Let R = 𝔽 q [ x ] / K be a quotient ring of 𝔽 q [ x ] modulo any ideal K. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring R , in particular, we determined the Erdős–Burgess constant in the case when K is the power of a prime ideal or a product of pairwise distinct prime ideals in 𝔽 q [ x ]. [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]

We describe an algorithm that allows to compute the Euclidean minimum (for the norm form) of any order of a totally definite quaternion field over a number field K of degree strictly greater than 1. Our approach is a generalization of the previous work dealing with number fields. The algorithm was practically implemented when K has degree 2. [ABSTRACT FROM AUTHOR]

Using a q -series identity of Carlitz and a q -analogue of Morley's congruence due to Pan, we give a proof of a q -congruence conjectured by Tauraso. [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 determine primitive solutions to the equation (x − r) 2 + x 2 + (x + r) 2 = y n for 1 ≤ r ≤ 1 0 4 , making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier. [ABSTRACT FROM AUTHOR]

The purpose of this note is to give a short and elementary proof of the fact that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p -adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than log (3) / p. [ABSTRACT FROM AUTHOR]

We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to Whipple-type 3 F 2 -series identities and utilizing L'Hôspital rule, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

We establish several surjectivity theorems regarding the Galois groups of small iterates of ϕc(x)=x2+c for c∈ℚ. To do this, we use explicit techniques from the theory of rational points on curves, including the method of Chabauty–Coleman and the Mordell–Weil sieve. For example, we succeed in finding all rational points on a hyperelliptic curve of genus 7, with rank 5 Jacobian, whose points parametrize quadratic polynomials with a “newly small” Galois group at the fifth stage of iteration. [ABSTRACT FROM AUTHOR]

Let L be a degree-2L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t1+𝜖). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. [ABSTRACT FROM AUTHOR]

Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos.61(203) (1993) 117–130] investigated the Diophantine problem of representing integers n in the form (x+y+z)(1/x+1/y+1/z) for rationals x,y,z. For fixed n, the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each n, and remarked that it seemed plausible that there were always solutions with positivew,x,y,z when n≥16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m2, 4m2+4, where m≢2(mod4). Computations within our range seem to indicate that solutions exist for all other values of n. [ABSTRACT FROM AUTHOR]

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor for infinitely many dimensions . Here, we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A. We also prove a similar result for families of symplectic lattices. [ABSTRACT FROM AUTHOR]