We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

Recently, Hirschhorn and the first author considered the parity of the function a(n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a(2m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n)

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of ℝan,exp-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in ℝan,exp that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function 2x, Wilkie showed that it must be eventually

For an algebraic number field K, let ζK(s) be the associated Dedekind zeta-function. It is conjectured that ζK(m) is transcendental for any positive integer m>1. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when K is a totally real number field, ζK(2n) is an algebraic multiple of π2n[K:ℚ] and hence, is transcendental. If K is not totally real,

The minimal excludant of a partition λ, mex(λ), is the smallest positive integer that is not a part of λ. For a positive integer n, σmex(n) denotes the sum of the minimal excludants of all partitions of n. Recently, Andrews and Newman obtained a new combinatorial interpretation for σmex(n). They showed, using generating functions, that σmex(n) equals the number of partitions of n into distinct parts

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 X0(14), using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of X0(14) whose Tate–Shafarevich group is of odd cardinality.

In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial and non-zeros located on the unit circle and a lower bound on the number of zeros of a polynomial in the disk {|x−1|<1}.

Let Fq denote the finite field with q elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over Fq in terms of the number of rational points on elliptic curves. In the case where q is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of

We study the period integrals of Maass forms restricted to Hirzebruch–Zagier cycles of Hilbert surfaces. In particular, we shall prove an upper bound for such integrals with respect to Laplace eigenvalues of Maass forms. In a special case, this leads to an upper bound for certain special L-values.

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but six candidates.

Let F be a finite extension of Qp and let q be the cardinality of its residue field. The Breuil-Schneider conjecture for G=GLn(F) [BS07] gives a necessary and sufficient condition for the existence of an invariant norm on ρ⊗π, where ρ is an irreducible algebraic representation of G and π is an irreducible smooth representation of G over F‾. The conjecture is still open, even when n=2, if π is a principal

Let rQ(n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2k variables. Let N be the level of Q. For a fixed prime p∤N, we assume that rQ(n) satisfies the relation rQ(1)rQ(p2n)=rQ(p2)rQ(n) for all positive integers n with p∤n. We actually show that this relation holds for any prime p∤N when (−1)kN is a fundamental discriminant.

For every positive integer N and every α∈[0,1), let B(N,α) denote the probabilistic model in which a random set A⊂{1,…,N} is constructed by choosing independently every element of {1,…,N} with probability α. We prove that, as N⟶+∞, for every A in B(N,α) we have |AA|∼|A|2/2 with probability 1−o(1), if and only iflog⁡(α2(log⁡N)log⁡4−1)log⁡log⁡N⟶−∞. This improves on a theorem of Cilleruelo, Ramana and

We prove Skolem's conjecture for the exponential Diophantine equation an+tbn=±cn under some assumptions on the integers a,b,c,t. In particular, our results together with Wiles' theorem imply that for fixed coprime integers a,b,c Fermat's equation an+bn=cn has no integer solution n≥3 modulo m for some modulus m depending only on a,b,c. We also provide a generalization where in the equation bn is replaced

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function ψ:N→(0,1/2) with ∑qψ(q)=∞ and any number γ, the following setW(ψ,γ)={x∈[0,1]:|qx−p−γ|<ψ(q) for infinitely many q,p∈N} has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In

K. Inkeri and his student S. Hyyrö have gained in the 50-es lower bounds for the possible solutions of Fermat's Last Theorem and of Catalan's equation, respectively. These bounds have not been replaced by stronger ones ever since, and doing so may appear as obsolete, now that it was proved, by Wiles and Taylor and by this author, respectively, that both equations have no solutions except the known

We prove a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on the upper half-plane H. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincaré series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincaré series.

In this paper we consider wildly ramified power series, i.e., power series defined over a field of positive characteristic, fixing the origin, where it is tangent to the identity. In this setting we introduce a new invariant under change of coordinates called the second residue fixed point index, and provide a closed formula for it. As the name suggests this invariant is closely related to the residue

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves.

In this paper, we proved the following theorem: Let β and 251252<γ<1 be two real numbers. Then for almost all irrational α>0 (in the sense of Lebesgue measure) there exist infinitely many Piatetski-Shapiro primes p∈{[n1/γ]:[n1/γ]is a prime for somen∈N} such that [αp+β] is also a prime.

We study the unitary principal series of the split group Spinm(F), where F is a p-adic field. Let χ˜ be a unitary character of a maximal F-split torus T˜ of GSpinm(F), and let χ be its restriction to T=T˜∩Spinm(F). The R-groups Rχ˜ and Rχ of the corresponding principal series representations fit in the exact sequence 0→Rχ˜→Rχ→Rχ/Rχ˜→0. We give a complete answer to the question of splitting of this

For 0≤α≤1, letLα(x)=∑n≤xλ(n)nα, where λ(n) is the Liouville function. Then famous criteria of Pólya and Turán claim that the eventual sign constancy of each of L0(x) and L1(x) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of Lα(x), Lα′(x), Lα″(x) for all 1/2<α<1 is equivalent

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian

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

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least 0.0254905.

We prove the Marchenko–Pastur law for the eigenvalues of p×p sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the p coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In

In this paper, we propose a stochastic version of the Hawking–Penrose black hole model. We describe the dynamics of the stochastic model as a continuous-time Markov jump process of quanta out and in the black hole. The average of the random process satisfies the deterministic picture accepted in the physical literature. Assuming that the number of quanta is finite the proposed Markov process consists

A set of gauge transformations of a relativistic field of quantum harmonic oscillators is studied in a mathematically rigorous manner. Each wave function in the domain of the number operator of a single oscillator generates a Fréchet-differentiable field of wave functions. Starting from a coherent wave function one obtains a two-dimensional differentiable manifold of coherent vector states. As an illustration

The dynamical rules in an auxiliary stochastic process that generate the biased ensemble of rare events are nonlocal. For the systems with one type of particles, it is shown that one can find special cases for which the transition rates of the auxiliary stochastic generator can be local. In this paper, we investigate this possibility for a system of classical hard-core particles with more than one

To capture a multidimensional consistency feature of integrable systems in terms of geometry, we give a condition called geodesic compatibility implying the existence of integrals in involution of the geodesic flow. The geodesic compatibility condition is constructed from a concrete example namely the integrable Calogero's goldfish system through the Poisson structure and the variational principle

In this paper we propose and discuss the fractional diffusion equation with time-dependent diffusion coefficient, considering the Hilfer-type and Weyl fractional derivatives in the time-variable and space-variable, respectively. We apply the similarity method and Mellin transform methodology to find an explicit solution in terms of Fox H-function. We illustrate graphically the diffusive behaviour described

Let ℌ be a complex Hilbert space with dim ℌ ≥ 3 and ℬ(ℋ) the algebra of all bounded linear operators on ℌ. The effect algebra E (ℌ) on ℌ is the set of all positive contractions in ℬ(ℋ). We consider the automorphism of E(ℌ) with respect to convex sequential product ox on E(ℌ) for some λ ∈ [0, 1] defined by A∘λB=λA12BA12+(1−λ)B12AB12,∀A,B∈E(ℋ). We show that an automorphism of E (ℌ) with respect to convex

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside ℝ × ℝ3 = ℝ4, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or ‘linked’ with each other, and they are closely

We formulate the dynamics of an infinitely extended open dissipative quantum system, Σ, in the Schroedinger picture. The generic model on which this is based comprises a C★-algebra, A, of observables, a folium, ℱ, of states on this algebra and a one-parameter semigroup, τ, of linear transformations of ℱ that represents its dynamics and is given by a natural infinite-volume limit of the corresponding

Nonlinear coherent states or f -coherent states are one of the important class of quantum states of light attached to the f -deformed oscillators. They have been introduced in a pioneering work by Manko et al. and have been realized physically as the stationary states of the centre of mass motion of a trapped ion by de Matos Filho et al. To gain insight into the effectiveness of these states in quantum

In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.

Motivated by certain q-series of Ramanujan, we examine two overpartition difference functions. We give both combinatorial and asymptotic formulas for the differences and show that they are always positive. We also briefly discuss similar differences for some other types of partitions. Our main tools are elementary q-series transformations and Ingham’s Tauberian theorem.

Given a field K and n>1, we say that a polynomial f∈K[x] has newly reducible nth iterate over K if fn−1 is irreducible over K, but fn is not (here fi denotes the ith iterate of f). We pose the problem of characterizing, for given d,n>1, fields K such that there exists f∈K[x] of degree d with newly reducible nth iterate, and the similar problem for fields admitting infinitely many such f. We give results

Let p be an odd prime and q a power of p. We examine the deformation theory of reducible and indecomposable Galois representations ρ̄:Gℚ→GSp2n(𝔽q) that are unramified outside a finite set of primes S and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors W(𝔽q). The main theorem of this paper is a higher-dimensional

We show that if {Un}n≥0 is a Lucas sequence, then the largest n suc that |Un|=Cm1Cm2⋯Cmk with 1≤m1≤m2≤⋯≤mk, where Cm is the mth Catalan number satisfies n<6500. In case the roots of the Lucas sequence are real, we have n∈{1,2,3,4,6,8,12}. As a consequence, we show that if {Xn}n≥1 is the sequence of the X coordinates of a Pell equation X2−dY2=±1 with a nonsquare integer d>1, then Xn=Cm implies n=1.

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

In this paper, we establish a very flexible and explicit Voronoï summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic L-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural p-adic analogue to the well-studied t-aspect.

We explore the possibilities for the Galois representation ρg attached to a weight-one newform g to be residually reducible, i.e. for the Hecke eigenvalues to be congruent to those of a weight-one Eisenstein series. A special role is played by Eisenstein series E11,ηK of level dK, where ηK is the quadratic character associated with an imaginary quadratic field K, of discriminant dK, with respect to

Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher-dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances with respect to any flat Riemannian metric. In dimension two they proved a five distance theorem, which is best possible. In this paper, we establish analogous bounds

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 Sα(x)=∑n≤x(−1)n−Ω(n)/nα and in Hα(x)=∑n≤x(−1)ω(n)/nα for α∈[0,1], and in W(x)=∑n≤x(−2)Ω(n). We show for example that each of the inequalities S0(x)<0, S0(x)>3.3x, S1(x)>0, and S1(x)x<−3

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

Let L(s,χ) be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined modq, where q is an odd prime. In this paper we introduce a fast method to compute |L(1,χ)| using the values of Euler's Γ function. We also introduce an alternative way of computing log⁡Γ(x) and ψ(x)=Γ′/Γ(x), x∈(0,1). Using such algorithms we numerically verify the classical Littlewood bounds

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight w(x;t,α):=e−x2|x−t|α(A+B⋅𝜃(x−t)),x∈(−∞,∞), where α>−1,A≥0,A+B≥0 and t∈ℝ. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation

We study the limiting distribution of a pair counting statistics of the form ∑1≤i≠j≤Nf(LN(𝜃i−𝜃j)) for the circular β-ensemble (CβE) of random matrices for sufficiently smooth test function f and LN=O(N). For β=2 and LN=N our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation 𝔼φ(z−X−f(X)Yf∗(X))−1|X for free random variables X,Y and a Borel function f is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form X+f(X)Yf∗(X)

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map 𝒮1) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) “small divisors” summatory functions σψsm(n). More

We prove that the function Pα(x)=exp(1−x−α) with α>0 is 1/α-mild. We apply this result to obtain a uniform 1/α-mild parametrization of the family of curves {xy=𝜖2|(x,y)∈(0,1)2} for 𝜖∈(0,1), which does not have a uniform 0-mild parametrization by work of Yomdin. More generally, we can parametrize families of power-subanalytic curves. This improves a result of Binyamini and Novikov that gives a 2-mild

In this paper, we study the order of the pole of the triple tensor product L-functions L(s,π1×π2×π3,⊗3) for cuspidal automorphic representations πi of GLni(𝔸F) in the setting where one of the πi is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’

Let q be a power of a prime number p, 𝔽q be a finite field with q elements and 𝒢 be a subgroup of (𝔽q,+) of order p. We give an existence criterion and an algorithm for computing maximally 𝒢-fixed c-Wieferich primes in 𝔽q[T]. Using the criterion, we study how c-Wieferich primes behave in 𝔽q[T] extensions.

In this paper, we complete the classification of transitive ovoids of finite Hermitian polar spaces.

We prove that for every integer t ⩾ 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having

A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the ‘correct number’ of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts, the global one and the local one. We first