Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level \(\Gamma _0(5)\) is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15. PubDate: 2022-08-04

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract We investigate some basic analytic properties of the \(L^2\) -holomorphic automorphic functions on a g-complex vector space associated with isotropic discrete subgroups \(\Gamma _r\) of rank \(r\le g\) . We show that each of these spaces form an infinite dimensional reproducing kernel Hilbert space which looks like a tensor product of a theta Bargmann–Fock space on \(Span_{{\mathbb {C}}}(\Gamma _r)\) and the classical Bargmann–Fock space on a \((g-r)\) -complex space. An explicit orthonormal basis using Fourier series is constructed and the explicit expression of its reproducing kernel function is given in terms of several variables Riemann theta function of particular characteristics. PubDate: 2022-08-03

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Let \(d_k(n)\) denote the kth divisor function. In this paper, we proved that the sum $$\begin{aligned} \sum _{1\le p_1,p_2,p_3\le X}d_k(p_1^2+p_2^2+p_3^2) \end{aligned}$$ has an asymptotic formula, where \(p_1,p_2,p_3\) are primes and \(k\ge 2\) is an integer. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Using the relationship between Siegel cusp forms of degree 2 and cuspidal automorphic representations of \(\mathrm{GSp}(4,{\mathbb A}_{{\mathbb Q}})\) , we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree 2 and the class number of \({\mathbb Q}(\sqrt{-p})\) . We also obtain some congruences between the 4-core partition function \(c_4(n)\) and dimensions of spaces of Siegel cusp forms of degree 2. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In 2002, Berkovich and Garvan introduced the \(M_2\) -rank of partitions without repeated odd parts. Let \(N_2(a, M, n)\) denote the number of partitions of n without repeated odd parts in which \(M_2\) -rank is congruent to a mod M. Lovejoy and Osburn, and Mao found a number of nice results for \(M_2 \) -rank differences modulo 3, 5, 6, and 10. In this paper, by using some properties for Appell–Lerch sums, we establish the generating functions for \(N_2(a,8,n) \) with \(0\le a \le 7\) . With these generating functions, we obtain some equalities and inequalities on \(M_2\) -rank modulo 8 of partitions without repeated odd parts. We also relate some differences of the \(M_2\) -rank to eighth-order mock theta functions. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of \(A \subseteq \{1, \dots , n\}\) in k-sets such that the first \(k-1\) symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial \(f_A(x) := \prod _{x \in A} (x-a)\) . Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation \(f_A(x)=P(y)\) where \(P(y)\in \mathbb {Q}[y]\) and on shifted power values of \(f_A(x)\) . PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Let N be a sufficiently large real number. In this paper, it is proved that, for \(1<c<\frac{2173}{1930}\) , the Diophantine equation $$\begin{aligned} \left[ p_1^c\right] +\left[ p_2^c\right] +\left[ p_3^c\right] =N \end{aligned}$$ is solvable in prime variables \(p_1,p_2,p_3\) such that each of the numbers \(p_i+2\,(i=1,2,3)\) has at most \([\frac{11387}{4346-3860c}]\) prime factors, counted according to multiplicity. This result constitutes a large improvement upon the previous result of Petrov (God Sofiĭ Univ “Sv Kliment Okhridski” Fac Mat Inform 104:171–183, 2017). PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In 1984, Kairies proved that for all positive real numbers x the geometric mean of \(\Gamma _q(x)\) and \(\Gamma _q(1/x)\) is greater than or equal to 1, that is, $$\begin{aligned} 1\le \sqrt{\Gamma _q(x)\Gamma _q(1/x)} \quad (0<q\ne 1), \end{aligned}$$ where \(\Gamma _q\) denotes the q-gamma function. This result can be improved if \(q\in (0,1)\) . We show that for all \(x>0\) the harmonic mean of \(\Gamma _q(x)\) and \(\Gamma _q(1/x)\) is greater than or equal to 1, that is, $$\begin{aligned} 1\le \frac{2}{1/\Gamma _q(x)+1/\Gamma _q(1/x)} \quad (0<q<1) \end{aligned}$$ with equality if and only if \(x=1\) . PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Let \(\sigma (n)=\sum _{d\mid n}d\) be the sum of divisors function and \(\gamma =0.577\ldots \) the Euler constant. In 1984, Robin proved that, under the Riemann hypothesis, \(\sigma (n)/n < e^\gamma \log \log n\) holds for \(n > 5040\) and that this inequality is equivalent to the Riemann hypothesis. Under the Riemann hypothesis, Ramanujan gave the asymptotic upper bound $$\begin{aligned} \frac{\sigma (n)}{n}\leqslant e^\gamma \Big (\log \log n- \frac{2(\sqrt{2}-1)}{\sqrt{\log n}}+S_1(\log n)+ \frac{\mathcal {O}(1)}{\sqrt{\log n}\log \log n} \Big ) \end{aligned}$$ with \(S_1(x)=\sum _\rho x^{\rho -1}/(\rho (1-\rho ))= \sum _\rho x^{\rho -1}/ \rho ^2\) where \(\rho \) runs over the non-trivial zeros of the Riemann \(\zeta \) function. In this paper, an effective form of the asymptotic upper bound of Ramanujan is given, which provides a slightly better upper bound for \(\sigma (n)/n\) than Robin’s inequality. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this article, we prove that a general version of Alladi’s formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom \(A^{\#}\) . As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry, and graph theory. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract The purpose of this note is to characterize all the sequences of orthogonal polynomials \((P_n)_{n\ge 0}\) such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$ where \(\,\mathrm {I}\) is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s)\) . The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In recent work, Miezaki introduced the notion of a spherical T-design in \(\mathbb {R}^2\) , where T is a potentially infinite set. As an example, he offered the \(\mathbb {Z}^2\) -lattice points with fixed integer norm (a.k.a. shells). These shells are maximal spherical T-designs, where \(T=\mathbb {Z}^+\setminus 4\mathbb {Z}^+\) . We generalize the notion of a spherical T-design to special ellipses, and extend Miezaki’s work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we obtain some Hecke-type triple sums for the third-order mock theta function \(\omega (q)\) and the fifth-order mock theta functions \(\chi _0(q)\) , \(\chi _1(q)\) . In addition, we extend this topic to the generating function of \(S^{*}(n)\) due to Andrews, Dyson, and Hickerson by investigating its new alternative representation. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Previous works have shown that certain weight 2 newforms are p-adic limits of weakly holomorphic modular forms under repeated application of the U-operator. The proofs of these theorems originally relied on the theory of harmonic Maass forms. Ahlgren and Samart obtained strengthened versions of these results using the theory of holomorphic modular forms. Here, we use such techniques to express all weight 2 CM newforms which are eta quotients as p-adic limits. In particular, we show that these forms are p-adic limits of the derivatives of the Weierstrass mock modular forms associated to their elliptic curves. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Recently, Andrews proved congruences for the total number of parts functions associated with ranks of partitions modulo 5 and 7. In this paper, applying the method of Atkin and Swinnerton-Dyer, we establish identities for these functions from which Andrews’ congruences modulo 5 follow immediately. In particular, the generating functions for total number parts functions associated with ranks of partitions modulo 7 turn out to be theta-functions. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Let \(\mathbb {Z}_{n}\) be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group \(\mathbb {Z}_{m} \times \mathbb {Z}_{n}\) , where m and n are arbitrary positive integers. Let \(k\ge 2\) be a fixed integer, we consider the k-th power sequence. For \(x\ge 1\) , define $$\begin{aligned} S_{k}(x):=\sum _{m,n \le x}s(m^{k},n^{k}). \end{aligned}$$ In this paper, we obtain an asymptotic formula of \(S_{k}(x)\) by using a multidimensional Perron formula and the complex integration method. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations. PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract The goal of the present paper is to show that all the zeros of the Eisenstein series for \(\Gamma _0^*(5)\) and \(\Gamma _0^*(7)\) on the standard fundamental domain lie on the lower boundary arcs. These are improvements of the results in Shigezumi (Kyushu J Math 61:527–549, 2007). PubDate: 2022-08-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Let \(d\ge 2\) be an integer which is not a square. We show that if \((L_n)_{n\ge 0}\) is the Lucas sequence and \((X_m,Y_m)_{m \ge 1}\) is the mth solution of the Pell equation \(X^2-dY^2=\pm 1\) , then the equation \(Y_m=L_n\) has at most two positive integer solutions (m, n) except for \(d=2\) when it has the three solutions \((m,n)=(1,1),~(2,0),~(5,7)\) . PubDate: 2022-07-31

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract We provide examples of multiplicative functions f supported on the k-free integers such that at primes \(f(p)=\pm 1\) and such that the partial sums of f up to x are \(o(x^{1/k})\) . Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1/k: There are examples such that the partial sums up to x are \(o(x^{1/(k+\frac{1}{2})+\epsilon })\) , for all \(\epsilon >0\) . This generalizes to the k-free integers the results of Aymone (J. Number Theory, 212:113-121, 2020). PubDate: 2022-07-31