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Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions g2(x,q) and g3(x,q). In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan’s two-parameter mock theta functions are the special cases of these functions. [ABSTRACT FROM AUTHOR]

Many papers have been recently devoted to the study of the radial behavior as z → 1− of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like (1 + o(1))C(z)/(1 − z)ρ for some ρ ∈ ℂ and C(z) is a real analytic function of z ∈ (0, 1) such that C(z) = C(zr). They did not provide a formula for C(z) which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of C(z) as an exponential of a Fourier series in the variable loglog(1/z)/log(r) for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute C(z) associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in ℚ[[z]]), we provide an explicit expression of C(z) for every r-Mahler function of order 2 under mild assumptions on the coefficients in ℝ(z) of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1. [ABSTRACT FROM AUTHOR]

Let P (x) : = a d x d + ⋯ + a 0 ∈ ℚ [ x ] , a d > 0 , be a polynomial of degree d ≥ 2. Let (x n) be a sequence of integers satisfying x n + 1 = P (x n) for all  n = 0 , 1 , 2 , ... and x n → ∞ as  n → ∞. Set α : = lim n → ∞ x n d − n . Then, under the assumption a d 1 / (d − 1) ∈ ℚ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either α is transcendental or α can be an integer or a quadratic Pisot unit with α − 1 being its conjugate over ℚ. In this paper, we study the nature of such α without the assumption that a d 1 / (d − 1) is in ℚ , and we prove that either the number α is transcendental, or α h is a Pisot number with h being the order of the torsion subgroup of the Galois closure of the number field ℚ α , a d − 1 d − 1 . Other results presented in this paper investigate the solutions of the inequality | | q 1 α 1 n + ⋯ + q k α k n + β | | < n in (n , q 1 , ... , q k) ∈ ℕ × (K ×) k , considering whether β is rational or irrational. Here, K represents a number field, and ∈ (0 , 1). The notation | | x | | denotes the distance between x and its nearest integer in ℤ. [ABSTRACT FROM AUTHOR]

Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [ABSTRACT FROM AUTHOR]

In a recent paper by Curran and Goldmakher, the cardinality of h -fold sumsets h A was given when A ⊂ ℤ d has d + 2 elements. In this paper we provide a different method for doing this and obtain a more general result. We also obtain an upper bound for the value of | h A | when A ⊂ ℤ d is a set of d + 3 elements with simplicial hull. [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]

In this paper, we prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree. [ABSTRACT FROM AUTHOR]

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of q-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels 2, 4, and 6. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple proof of a famous result of Newton on unimodal sequences associated to polynomials with real roots. We then give a brief exposition of the recent work of Huh on such sequences arising from combinatorial geometries. [ABSTRACT FROM AUTHOR]

In this paper, we establish the expected order of magnitude of the k th-moment of central values of the family of Dirichlet L -functions to a fixed prime modulus over function fields for all real k ≥ 0. [ABSTRACT FROM AUTHOR]

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

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 show that the Ruzsa number R m is bounded by 1 9 2 for any positive integer m , which improves the prior bound R m ≤ 2 8 8 given by Chen in 2008. [ABSTRACT FROM AUTHOR]

Let Dλd denote the family of monomial deformations of diagonal hypersurface over a finite field 픽q given by Dλd : X 1d + X 2d + ⋯ + X nd = λdX 1h1X2h2⋯Xnhn, where d,n ≥ 2, hi ≥ 1, ∑i=1nh i = d, and gcd(d,h1,h2,…,hn) = 1. The Dwork hypersurface is the case when d = n, that is, h1 = h2 = ⋯ = hn = 1. Formulas for the number of 픽q-points on the Dwork hypersurfaces in terms of McCarthy’s p-adic hypergeometric functions are known. In this paper, we provide a formula for the number of 픽q-points on Dλd in terms of McCarthy’s p-adic hypergeometric function which holds for d ≥ n. [ABSTRACT FROM AUTHOR]

For a prime p, positive integers r,n, and a polynomial f with coefficients in 픽pr, let Wp,r,n(f) = fn 픽 pr ∖ fn+1 픽 pr. As n varies, the Wp,r,n(f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on 픽pr. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of 픽pr lying in a given Wp,r,n(f). Moreover, when we generalize our definition of Wp,r,n(f), we obtain both upper and lower bounds for the resulting averages. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences {un}n≥1−s of a fixed order s ≥ 2. These sequences satisfy the recurrence relation un = c1un−1 + c2un−2 + ⋯ + csun−s with initials u1−s = u2−s = ⋯ = u−1 = 0,u0 = 1. Here, the coefficients of the recurrence relation are positive integers satisfying c1 > 1 + c2 + ⋯ + cs or c1 ≥ c2 ≥⋯ ≥ cs. In both these conditions, c1 ≥ 4. [ABSTRACT FROM AUTHOR]

This is the first part of a series of papers where the behavior of the invariants under twist by Dirichlet characters is studied for L-functions of degree 2. Here we show, under suitable conditions, that degree and internal shift remain unchanged under twist. The ultimate goal of the series is to prove a general version of Weil converse theorem with minimal assumptions on the shape of the functional equation of the twists. [ABSTRACT FROM AUTHOR]

For the ℤ-lift Xℤ of a monoid scheme X of finite type, Deitmar et al. calculated its absolute zeta function by interpolating #Xℤ(픽q) for all prime powers q using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence (#Xℤ(픽q))q, which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme X of finite type over ℚ by means of a pair of Puiseux polynomials which estimate “#X(픽pm)” for sufficiently large p. We call them the ceiling and floor Puiseux polynomials of X. In particular, if X is an elliptic curve, then our absolute zeta functions of X do not depend on its isogeny class. [ABSTRACT FROM AUTHOR]

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties.In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations — one relating to supercharacter theory, the other relating to class field theory — along with discussions and visual examples of each. [ABSTRACT FROM AUTHOR]

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In [a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641], Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin–Schreier covers. [ABSTRACT FROM AUTHOR]

Let K be a totally real number field of degree n over ℚ, with discriminant and regulator ΔK,RK, respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by |ΔK|−1/2γ Knexp(2nρ ∞(ΛK)), where ΔK is the discriminant of the field K, γK is the additive Hermite–Humbert constant over positive-definite unary forms for K and ρ∞(ΛK) is the covering radius of the log-unit lattice. In particular, when K is Galois over ℚ and n is a prime number, the number of homothety classes of unary forms is upper bounded by (2 3n)nexp(O(n3 2log (n)RK 1 n−1))|ΔK|1/2, where RK is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by (2 3n)nexp(O(n3 2log (n)))|ΔK|1/2. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group SL(2, ℤ), respectively. In this paper, we investigate the sign changes of the sequence {λf×f×f(n)}n∈ℕ,{λf×f×f(n)}n∈ℕ in short intervals, where λf×f×f(n),λf×f×g(n) denote the normalized coefficients of triple product L-functions L(f × f × f,s),L(f × f × g,s) attached to f and f,g, respectively. We provide quantitative results for the number of sign changes of the sequences mentioned above in short intervals. [ABSTRACT FROM AUTHOR]

This paper extends Bombieri and Pila’s estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the ℓ2 decoupling inequality for non-degenerate curves in ℝn. Additionally, we review the curve-lifting method introduced in Bombieri and Pila’s work and establish the estimate of lattice points in the neighborhood of a planar curve. [ABSTRACT FROM AUTHOR]

In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials. [ABSTRACT FROM AUTHOR]

Let m and N be positive integers and k be a negative real number. Let χ be a multiplier system of weight k on Γ0(N). In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight k, index m, and multiplier system χ on Γ0(N). As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we provide an algorithm to detect linear congruences of p l k (n) , the number of MacMahon's k -rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the k -rowed plane partition functions. We also show p (n , m) that the number of partitions at most m parts always admits linear congruences. [ABSTRACT FROM AUTHOR]

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus 2. We also extend separable integer partition classes with modulus 1 to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the set S (n) of positive integer solutions of the title Diophantine equation. In particular, for a given n we prove boundedness of the number of solutions, give precise upper bound on the common value of σ 2 ( X ¯ n) and σ n ( X ¯ n) together with the biggest value of the variable x n appearing in the solution. Moreover, we enumerate all solutions for n ≤ 1 6 and discuss the set of values of x n / x n − 1 over elements of S (n). [ABSTRACT FROM AUTHOR]

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa's result on the p -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties. [ABSTRACT FROM AUTHOR]

In this paper, we study Lehmer-type bounds for the Néron–Tate height of K̄-points on abelian varieties A over number fields K. Then, we estimate the number of K-rational points on A with Néron–Tate height ≤log B for B ≫ 0. This estimate involves a constant C, which is not explicit. However, for elliptic curves and the product of elliptic curves over K, we make the constant explicitly computable. [ABSTRACT FROM AUTHOR]

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris 360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given. [ABSTRACT FROM AUTHOR]

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums W d 1 , d 2 , d 3 r 1 , r 2 , r 3 (n) = ∑ l 1 , l 2 , l 3 ∈ ℕ d 1 l 1 + d 2 l 2 + d 3 l 3 = n σ r 1 (l 1) σ r 2 (l 2) σ r 3 (l 3) , for odd integers r 1 , r 2 , r 3 ≥ 1 , and d 1 , d 2 , d 3 , n ∈ ℕ , where σ r (n) is the sum of the r th powers of the positive divisors of n. We consider four cases, namely (i) r 1 = r 2 = r 3 = 1 , (ii) r 1 = r 2 = 1 ; r 3 ≥ 3 (iii) r 1 = 1 ; r 2 , r 3 ≥ 3 and (iv) r 1 , r 2 , r 3 ≥ 3 , and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer n by certain positive definite quadratic forms. The existing formulas for W 1 , 1 , 1 (n) (in [20]), W 1 , 1 , 2 (n) , W 1 , 2 , 2 (n) , W 1 , 2 , 4 (n) (in [7]), W 1 , 1 , 1 1 , 3 , 3 (n) , W 1 , 1 , 3 1 , 3 , 3 (n) , W 1 , 3 , 3 1 , 3 , 3 (n) , W 3 , 1 , 1 1 , 3 , 3 (n) , W 3 , 3 , 1 1 , 3 , 3 (n) (in [35]), W d 1 , d 2 , d 3 (n) , lcm (d 1 , d 2 , d 3) ≤ 6 (in [30]) and lcm (d 1 , d 2 , d 3) = 7 , 8 , 9 (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method. [ABSTRACT FROM AUTHOR]

We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (n) . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define X (K , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]

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 methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function ζ (s) of the form σ > 1 − c (log τ) 2 / 3 (log log τ) 1 / 3 (τ : = | t | + 1 0 0). For many decades, the general shape of the zero-free region has not changed (although explicit known values for c have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of ζ (s) lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for L -functions associated to nonquadratic Dirichlet characters χ modulo q ≥ 2. [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]

In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt's subspace type theorem for hypersurfaces which are located in subgeneral position. [ABSTRACT FROM AUTHOR]

For a given sequence A of nonnegative integers, let P (A) be the set of all finite subsequence sums of A. A is called complete if P (A) contains all sufficiently large integers. A real number α > 0 is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of α. Hegyvári conjectured that A α , β is complete if α or β is i.d.f. and α / β ≠ 2 l (l ∈ ℤ) , where A α , β = { [ α ] , [ β ] , ... , [ 2 n α ] , [ 2 n β ] , ... } is a sequence of integers. In this paper, we give a partial result of Hegyvári's conjecture. [ABSTRACT FROM AUTHOR]

Let f be a normalized Hecke eigenform of weight k for the full modular group SL2(ℤ). In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions R(s,f × f), supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant D < 0. We improve previous results in the case when the reduced form is given by 풬(x1,x2) = x12 + x 22. [ABSTRACT FROM AUTHOR]

For an odd prime p, let 픽p be the finite field of p elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any a,b,c ∈ 픽p∗, we also obtain new upper bounds of the following double character sum Ta,b,c(χ,ℋ1,ℋ2) =∑h1∈ℋ1∑h2∈ℋ2χ(a + bh1 + ch2) and a triple character sum Sχ(a,b,ℋ1,ℋ2,풩) =∑x∈풩∑h1∈ℋ1∑h2∈ℋ2χ(x + ah1 + bh2) with 풩 = {1,…,N} and multiplicative subgroups ℋ1,ℋ2 ⊆ 픽p∗ of order H1 and H2, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers n such that n,n + h and n + k are all sums of two squares where h and k are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain n in parametric terms such that all the four integers n,n + 1,n + 2,n + 4 are sums of two squares. We also find infinitely many integers n such that all the five integers n,n + 1,n + 2,n + 4,n + 5 are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference 4, of five integers all of which are sums of two squares. [ABSTRACT FROM AUTHOR]

Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple. [ABSTRACT FROM AUTHOR]

In this paper, we establish an asymptotic formula for the twisted second moment of L -functions associated to Hecke–Maass forms, which can be used to deduce a zero-density estimate for these L -functions in the spectral aspect. [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]

We often use the plus/minus Selmer groups to study Iwasawa theory for elliptic curves with good supersingular reduction at p. But, if the plus/minus local conditions are not co-free, it can be difficult to use them effectively. In this paper, we introduce some technical improvements so that we can use the plus/minus Selmer groups even when the plus/minus local conditions are not co-free. In particular, we show that the plus/minus Selmer groups have no proper submodule of finite index and also they satisfy algebraic functional equations even when the said local conditions are not co-free, thus improve our previous results [B. Kim, The algebraic functional equation of an elliptic curve at supersingular primes, Math. Res. Lett. 15(1) (2008) 83–94; The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95(2) (2013) 189–200]. [ABSTRACT FROM AUTHOR]

Let F be a newform for S L 3 (ℝ) underlying the symmetric square lift of a (either holomorphic or Hecke–Maaß) newform f of square-free level and trivial nebentypus. In this paper, we present the classical level aspect version of the Voronoĭ formula for the symmetric square lift is established in an alternative way by tracing back to its geometric nature, compared with Zhou's work [F. Zhou, The Voronoi formula on G L (3) with ramification, preprint (2018), arXiv:1806.10786]. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from q = e 2 π i τ to the reciprocal of the elliptic modular j -function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of p -adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them. [ABSTRACT FROM AUTHOR]

In this paper, we determine all intermediate modular curves X Δ (N) that admit infinitely many cubic points over the rational field ℚ. [ABSTRACT FROM AUTHOR]

For k ≥ 1, let ℛk(x) denote the reciprocal sum up to x of numbers with k prime factors, counted with multiplicity. In prior work, the authors obtained estimates for ℛk(x), extending Mertens’ second theorem, as well as a finer-scale estimate for ℛ2(x) up to (log x)−N error for any N > 0. In this paper, we establish the limiting behavior of the higher Mertens constants from the ℛ2(x) estimate. We also extend these results to ℛ3(x), and we comment on the general case k ≥ 4. [ABSTRACT FROM AUTHOR]