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  • Some exceptional sets of Borel–Bernstein theorem in continued fractions
    Ramanujan J. (IF 0.79) Pub Date : 2020-10-20
    Lulu Fang, Jihua Ma, Kunkun Song

    Let \([a_1(x),a_2(x), a_3(x),\ldots ]\) denote the continued fraction expansion of a real number \(x \in [0,1)\). This paper is concerned with certain exceptional sets of the Borel–Bernstein Theorem on the growth rate of \(\{a_n(x)\}_{n\geqslant 1}\). As a main result, the Hausdorff dimension of the set $$\begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log

    更新日期:2020-10-20
  • Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums
    Ramanujan J. (IF 0.79) Pub Date : 2020-10-20
    Hamed Mousavi, Maxie D. Schmidt

    We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of \(q^n\) as \(\sum _{d|n}

    更新日期:2020-10-20
  • Completions and algebraic formulas for the coefficients of Ramanujan’s mock theta functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-10-20
    David Klein, Jennifer Kupka

    We present completions of mock theta functions to harmonic weak Maass forms of weight \(\nicefrac {1}{2}\) and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight \(\nicefrac {1}{2}\) that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic

    更新日期:2020-10-20
  • The reciprocal of $$(q;q)_n$$ ( q ; q ) n as sums over partitions
    Ramanujan J. (IF 0.79) Pub Date : 2020-10-19
    Mircea Merca

    In this paper we show that the reciprocal of the finite product \((q;q)_n\) can be expressed as a sum over all the partitions of n in two different ways. We derive similar results for the reciprocal of \(1-q^n\). Product identities involving sums over partitions are derived as consequences of these results.

    更新日期:2020-10-20
  • A continuity of cycle integrals of modular functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-27
    Yuya Murakami

    In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean

    更新日期:2020-09-28
  • A general q -expansion formula based on matrix inversions and its applications
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-22
    Jin Wang

    In this paper, by the technique of matrix inversions, we establish a general q-expansion formula of arbitrary formal power series F(z) with respect to the base $$\begin{aligned} \left\{ z^n\frac{(az)_{n}}{(bz)_{n}}\bigg |n=0,1,2,\ldots \right\} . \end{aligned}$$ Some concrete expansion formulas and their applications to q-series identities are presented, including Carlitz’s q-expansion formula and

    更新日期:2020-09-23
  • Certain eta-quotients and $$\ell $$ ℓ -regular overpartitions
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-19
    Chiranjit Ray, Kalyan Chakraborty

    Let \({\overline{A}}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In this paper, we prove that \({\overline{A}}_{\ell }(n)\) is almost always divisible by \(p_i^j\) if \(p_i^{2a_i}\ge \ell \), where j is a fixed positive integer and \(\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}\) with primes \(p_i>3\). We obtain a Ramanujan-type congruence for \({\overline{A}}_{7}\)

    更新日期:2020-09-20
  • Dilogarithm identities for solutions to Pell’s equation in terms of continued fraction convergents
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-18
    Martin Bridgeman

    We describe a new connection between the dilogarithm function and the solutions of Pell’s equation \(x^2-ny^2 = \pm 1\). For each solution x, y to Pell’s equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit \(x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]\). We further show that Ramanujan’s dilogarithm value-identities correspond to an

    更新日期:2020-09-20
  • Integrals of inverse trigonometric and polylogarithmic functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-18
    Anthony Sofo

    In this paper we study the representation of integrals whose integrand involves the product of a polylogarithm and an inverse or inverse hyperbolic trigonometric function. We further demonstrate many connections between these integrals and Euler sums. We develop recurrence relations and give some examples of these integrals in terms of Riemann zeta values, Dirichlet values and other special functions

    更新日期:2020-09-20
  • On the pseudorandom properties of subsets constructed by using primitive roots
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-18
    Huaning Liu, Mengyao Jing

    Dartyge and Sárközy (partly with other coauthors) introduced pseudorandom measures of subsets, and studied the pseudorandomness of certain subsets related to primitive roots. In this paper we answer a conjecture of Dartyge, Sárközy and Szalay, and study the pseudorandom properties of some subsets constructed by using primitive roots.

    更新日期:2020-09-20
  • A variation of the Andrews–Stanley partition function and two interesting q -series identities
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-18
    Bernard L. S. Lin, Lin Peng, Pee Choon Toh

    Stanley introduced a partition statistic \(srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')\), where \(\mathcal {O}(\pi )\) denote the number of odd parts of the partition \(\pi \), and \(\pi '\) is the conjugate of \(\pi \). Let \(p_i(n)\) denote the number of partitions of n with srank \(\equiv i\pmod 4\). Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5: $$\begin{aligned}

    更新日期:2020-09-20
  • Discrete Kontorovich–Lebedev transforms
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-09
    Semyon Yakubovich

    Discrete analogs of the classical Kontorovich–Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function \(K_{in}(x), x >0, n \in {\mathbb {N}}, i \) is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application

    更新日期:2020-09-10
  • On a class of elliptic functions associated with even Dirichlet characters
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-09
    Dandan Chen, Rong Chen

    We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function \(\wp (z|\tau )\) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular

    更新日期:2020-09-10
  • k -Regular partitions and overpartitions with bounded part differences
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-08
    Bernard L. S. Lin, Saisai Zheng

    Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions

    更新日期:2020-09-08
  • Multivariate holomorphic Hermite polynomials
    Ramanujan J. (IF 0.79) Pub Date : 2020-09-08
    Mourad E. H. Ismail, Plamen Simeonov

    We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Itô polynomials. We show how these

    更新日期:2020-09-08
  • Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms
    Ramanujan J. (IF 0.79) Pub Date : 2020-08-29
    Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim

    We construct bases of the space of harmonic weak Maass forms of weight \(\kappa \in \frac{1}{2}{\mathbb {Z}}\). Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms.

    更新日期:2020-08-29
  • A note on the zeros of approximations of the Ramanujan $$\Xi $$ Ξ -function
    Ramanujan J. (IF 0.79) Pub Date : 2020-08-28
    Andrés Chirre, Oswaldo Velásquez Castañón

    In this paper we review the study of the distribution of the zeros of certain approximations for the Ramanujan \(\Xi \)-function given by Ki (Ramanujan J 17(1):123–143, 2008), and we provide new proofs of his results. Our approach is motivated by the ideas of Velásquez (J Anal Math 110:67–127, 2010) in the study of the zeros of certain sums of entire functions with some condition of stability related

    更新日期:2020-08-29
  • Note on a theorem of Farkas and Kra
    Ramanujan J. (IF 0.79) Pub Date : 2020-08-04
    Kazuhide Matsuda

    In this paper, we focus on applications of high-level versions of Jacobi’s derivative formula to number theory, such as quaternary quadratic forms and convolution sums of some arithmetical functions.

    更新日期:2020-08-05
  • Some results on divisor problems related to cusp forms
    Ramanujan J. (IF 0.79) Pub Date : 2020-08-04
    Wei Zhang

    Let \(\lambda _{f}(n)\) be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with \(\lambda _{f}(n)\) over some special sequences. More precisely, for any fixed integer \(k\ge 2\) and \(j\in \{1,2,3,4\},\) we are interested in the following sums$$\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum

    更新日期:2020-08-04
  • Highly composite polynomials and the maximum order of the divisor function in $$\pmb {\mathbb {F}}_q[t]$$ F q [ t ]
    Ramanujan J. (IF 0.79) Pub Date : 2020-08-03
    Ardavan Afshar

    We investigate the analogues, in \(\mathbb {F}_q[t]\), of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller

    更新日期:2020-08-03
  • An inequality for the modified Selberg zeta-function
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-30
    Igoris Belovas

    We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg zeta-function in a different way, reproving and extending the result of Garunkštis and Grigutis and completing the extension of a result of Belovas and Sakalauskas.

    更新日期:2020-07-30
  • On minimal complements in groups
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-29
    Arindam Biswas, Jyoti Prakash Saha

    Let \(W,W'\subseteq G\) be non-empty subsets in an arbitrary group G. The set \(W'\) is said to be a complement to W if \(W\cdot W'=G\) and it is minimal if no proper subset of \(W'\) is a complement to W. We show that, if W is finite then every complement of W has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal r-nets for every \(r\geqslant 0\) in finitely

    更新日期:2020-07-30
  • Notes on theta series for Niemeier lattices II
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-29
    Shoyu Nagaoka

    Following (Nagaoka and Takemori in Ramanujan J 42: 385–400, 2017), we study some congruence properties satisfied by the theta series associated with Niemeier lattices.

    更新日期:2020-07-29
  • 1-Shell totally symmetric plane partitions (TSPPs) modulo powers of 5
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    Shane Chern

    Let s(n) be the number of 1-shell totally symmetric plane partitions (TSPPs) of n. In this paper, an infinite family of congruences modulo powers of 5 for s(n) will be deduced through an elementary approach. Namely, $$\begin{aligned} s\left( 2\cdot 5^{2\alpha -1}n+5^{2\alpha -1}\right) \equiv 0 \pmod {5^{\alpha }}. \end{aligned}$$

    更新日期:2020-07-29
  • A $$\mathrm {GL}_3$$ GL 3 analog of Selberg’s result on S ( t )
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    Sheng-Chi Liu, Shenhui Liu

    Let \(S(t,F):=\pi ^{-1}\arg L\big (\frac{1}{2}+it,F\big ),\) where F is a Hecke–Maass cusp form for \(\mathrm {SL}_3({\mathbb {Z}}).\) We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F).

    更新日期:2020-07-29
  • Sextic reciprocal monogenic dihedral polynomials
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    Lenny Jones

    Let \(D_n\) denote the dihedral group of order 2n. We find infinite \(D_6\)-families and an infinite \(D_3\)-family of monic irreducible reciprocal sixth-degree polynomials \(f(x)\in \mathbb {Z}[x]\), such that \(\{1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^5\}\) is a basis for the ring of integers of \(L=\mathbb {Q}(\theta )\), where \(f(\theta )=0\).

    更新日期:2020-07-29
  • Four identities related to third-order mock theta functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    Su-Ping Cui, Nancy S. S. Gu, Chen-Yang Su

    Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function $$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^

    更新日期:2020-07-28
  • Farkas’ identities with quartic characters
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    P. Guerzhoy, Ka Lun Wong

    Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) introduced an arithmetic function \(\delta \) and found an identity involving \(\delta \) and a sum of divisor function \(\sigma '\). The first-named author and Raji in (Ramanujan J 19(1):19–27, 2009) discussed a natural generalization of the

    更新日期:2020-07-28
  • Goldbach–Linnik type problems on cubes of primes
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
    Xiaodong Zhao

    In this paper, we will investigate three Goldbach–Linnik type problems on cubes of primes. For example, it was proved that, for \(k=1364,\) every pair of sufficiently large even integers can be represented by a pair of eight prime cubes and k powers of 2. In this paper, we give the detailed proof for the first time and sharpen the value of k to 658.

    更新日期:2020-07-28
  • 1-Universal binary and ternary Hermitian lattices over imaginary quadratic fields
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
    Byeong Moon Kim, Ji Young Kim

    A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices, including both free and non-free Hermitian lattices. This paper is more concerned with the representations of unary non-free Hermitian lattices by Hermitian lattices. We estimate the minimal rank\(u_m^1\) of 1-universal Hermitian lattices and we classify all 1-universal binary

    更新日期:2020-07-28
  • On special values at integers of L -functions of Jacobi theta products of weight 3
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
    Ryojun Ito

    In this paper, we consider L-functions of two modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at \(s=3\), 4 in terms of special values of Kampé de Fériet hypergeometric functions. Moreover, via L-values, we give some relations between special values of Kampé de Fériet hypergeometric functions and generalized hypergeometric functions.

    更新日期:2020-07-28
  • Visible lattice points along curves
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
    Kui Liu, Xianchang Meng

    This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the “visible” points and the “invisible” points in the definition of previous research. We prove asymptotic formulas for the number of lattice

    更新日期:2020-07-28
  • Adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
    Jinfa Cheng, Weizhong Dai

    In this article, we obtain the adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint difference equation are then obtained. As an application of these particular solutions, we use them to obtain

    更新日期:2020-07-28
  • Some extensions for Ramanujan’s circular summation formulas and applications
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
    Ji-Ke Ge, Qiu-Ming Luo

    In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we also obtain many interesting identities of Jacobi’s theta functions.

    更新日期:2020-07-27
  • An explicit Waldspurger formula for Hilbert modular forms II
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-26
    Nicolás Sirolli, Gonzalo Tornaría

    We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.

    更新日期:2020-07-26
  • Shift-plethystic trees and Rogers–Ramanujan identities
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-26
    Miguel A. Méndez

    By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers–Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers–Ramanujan continued fraction. By specializing the non-commutative series

    更新日期:2020-07-26
  • The divisor function on residue classes III
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-25
    Prapanpong Pongsriiam, Robert C. Vaughan

    Let \(n, q, a\in \mathbb N\), d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over \(n\le x\) and \(n\equiv a\pmod q\). Previously we obtained an asymptotic formula for the second moment $$\begin{aligned} V(x,Q) = \sum _{q\le Q}\sum _{a=1}^q \left| A(x,q,a)-M(x,q,a)\right| ^2, \end{aligned}$$ where \(Q\le x\) and M(x, q, a) is a usual expected value for A(x, q, a). In this

    更新日期:2020-07-25
  • On two conjectural supercongruences of Z.-W. Sun
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Chen Wang

    In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity: $$\begin{aligned} \sum _{k=0}^n\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) ^2=16^n\sum _{k=0}^n \frac{\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right)

    更新日期:2020-07-25
  • A new approach to hypergeometric transformation formulas
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Noriyuki Otsubo

    We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for q-hypergeometric functions is also studied.

    更新日期:2020-07-25
  • Simultaneous core partitions with nontrivial common divisor
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Jean-Baptiste Gramain, Rishi Nath, James A. Sellers

    A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are relatively prime integers. Here we change perspective slightly and explore properties of (s, t)-core and \((\bar{s},\bar{t})\)-core partitions for s and t with a nontrivial common divisor g. We begin by recovering, using the g-core and g-quotient construction, the generating function for

    更新日期:2020-07-25
  • Counting pattern-avoiding integer partitions
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Jonathan Bloom, Nathan McNew

    A partition \(\alpha \) is said to contain another partition (or pattern) \(\mu \) if the Ferrers board for \(\mu \) is attainable from \(\alpha \) under removal of rows and columns. We say \(\alpha \) avoids \(\mu \) if it does not contain \(\mu \). In this paper we count the number of partitions of n avoiding a fixed pattern \(\mu \), in terms of generating functions and their asymptotic growth rates

    更新日期:2020-07-25
  • Hyperelliptic parametrizations of $$\pmb {\mathbb {Q}}$$ Q -curves
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Francesc Bars, Josep González, Xavier Xarles

    For a square-free integer N, we present a procedure to compute \(\mathbb {Q}\)-curves parametrized by rational points of the modular curve \(X_0^*(N)\) when this is hyperelliptic.

    更新日期:2020-07-24
  • Klingen $${\mathfrak {p}}^2$$ p 2 vectors for $$\mathrm{GSp}(4)$$ GSp ( 4 )
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Shaoyun Yi

    We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level \({\mathfrak {p}}^2\) for all irreducible, admissible representations of \(\mathrm{GSp}(4, F)\) with trivial central character for F a \({\mathfrak {p}}\)-adic field.

    更新日期:2020-07-24
  • Regular ternary polygonal forms
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
    Zilong He, Ben Kane

    Inspired by Dickson’s classification of regular diagonal ternary quadratic forms, we prove that there are no primitive regular ternary m-gonal forms when m is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next such prime by a product of the previous primes, a question of

    更新日期:2020-07-24
  • Asymptotics for the Taylor coefficients of certain infinite products
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
    Shane Chern

    Let \((m_1,\ldots ,m_J)\) and \((r_1,\ldots ,r_J)\) be two sequences of J positive integers satisfying \(1\le r_j< m_j\) for all \(j=1,\ldots ,J\). Let \((\delta _1,\ldots ,\delta _J)\) be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\begin{aligned} \prod _{j=1}^J\Bigg (\prod _{k\ge 1}\big (1-q^{r_j+m_j(k-1)}\big

    更新日期:2020-07-24
  • On modular equations of degree 25
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
    K. R. Vasuki, M. V. Yathirajsharma

    On page 237–238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. Berndt proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.

    更新日期:2020-07-24
  • On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
    Mahadi Ddamulira, Florian Luca

    Let \(r\ge 1\) be an integer and \(\mathbf{U}:=(U_{n})_{n\ge 0} \) be the Lucas sequence given by \(U_0=0\), \(U_1=1, \) and \(U_{n+2}=rU_{n+1}+U_n\), for all \( n\ge 0 \). In this paper, we show that there are no positive integers \(r\ge 3,~x\ne 2,~n\ge 1\) such that \(U_n^x+U_{n+1}^x\) is a member of \(\mathbf{U}\).

    更新日期:2020-07-23
  • $$H_q$$ H q -Semiclassical orthogonal polynomials via polynomial mappings
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
    K. Castillo, M. N. de Jesus, F. Marcellán, J. Petronilho

    In this work we study orthogonal polynomials via polynomial mappings in the framework of the \(H_q\)-semiclassical class. We consider two monic orthogonal polynomial sequences \(\{p_n (x)\}_{n\ge 0}\) and \(\{q_n(x)\}_{n\ge 0}\) such that $$\begin{aligned} p_{kn}(x)=q_n(x^k),\quad n=0,1,2,\ldots , \end{aligned}$$ where \(k\ge 2\) is a fixed integer number, and we prove that if one of the sequences

    更新日期:2020-07-23
  • Some permutations over $$\pmb {\mathbb {F}}_p$$ F p concerning primitive roots
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
    Li-Yuan Wang, Hao Pan

    Let p be an odd prime and let \({\mathbb {F}}_p\) denote the finite field with p elements. Suppose that g is a primitive root of \({\mathbb {F}}_p\). Define the permutation \(\tau _g:\,{\mathcal {H}}_p\rightarrow {\mathcal {H}}_p\) by $$\begin{aligned} \tau _g(b):={\left\{ \begin{array}{ll} g^b&{}\text {if }g^b\in {\mathcal {H}}_p,\\ -g^b&{}\text {if }g^b\not \in {\mathcal {H}}_p,\\ \end{array}\right

    更新日期:2020-07-23
  • Maass relations for Saito–Kurokawa lifts of higher levels
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
    Jolanta Marzec

    It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for \(\mathrm {GSp}_4\) is predicted to fail only in case of CAP representations. It is not known

    更新日期:2020-07-23
  • New infinite q -product expansions with vanishing coefficients
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
    James Mc Laughlin

    Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared recently, we further examine such infinite products, and find that many such results on vanishing coefficients may be grouped into families. For example, one result proved in the present paper is that if \(b\in \{1,2,\dots , 9, 10\}\)

    更新日期:2020-07-23
  • Exact expansions of Hankel transforms and related integrals
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
    A. V. Kisselev

    The Hankel transform \(\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x\) is studied for integer \(n\geqslant -1\) and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f(x) and its derivatives are imposed. It is necessary to underline

    更新日期:2020-07-22
  • The Ramanujan sum and Chebotarev densities
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Biao Wang

    In this short note, we show an analogue of one of Alladi’s and Dawsey’s formulas with respect to the Ramanujan sum \(c_n(m)\) for \(m\geqslant 1\). Their formulas may be viewed as the case \(m=1\) in our result.

    更新日期:2020-07-22
  • Acyclotomy of torsion in the CM case
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Michael Chou, Pete L. Clark, Marko Milosevic

    Let F be a number field, and let \(F^{{\text {cyc}}}\) be obtained by adjoining to F all the roots of unity. We show that as E ranges over all elliptic curves defined over \(F^{{\text {cyc}}}\) with complex multiplication, the torsion subgroups \(E(F^{{\text {cyc}}})\) are finite and uniformly bounded in size.

    更新日期:2020-07-22
  • Combinatorial proofs and generalization of Bringmann, Lovejoy and Mahlburg’s overpartition theorems
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Doris D. M. Sang, Diane Y. H. Shi

    In 2013, Bringmann and Mahlburg defined a new type of partitions by adding a different restriction on the smallest parts in Gleissberg’s generalization of partitions considered by Schur. The generating function of this new Schur-type partitions is a mixed mock modular form, more precisely, it equals the product of the generating function of Gleissberg’s generalization and a specialization of a universal

    更新日期:2020-07-22
  • The first moment of Maaß form symmetric square L -functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Olga Balkanova

    We prove an asymptotic formula for the twisted first moment of Maaß form symmetric square L-functions on the critical line and at the central point. The error term is estimated uniformly with respect to all parameters.

    更新日期:2020-07-22
  • A note on additive twists, reciprocity laws and quantum modular forms
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Asbjørn Christian Nordentoft

    We prove that the central values of additive twists of a cuspidal L-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this, we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal L-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet L-functions

    更新日期:2020-07-22
  • Geometric and monotonic properties of Ramanujan type entire functions
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Erhan Deniz

    In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their Mittag–Leffler expansion in such a way that the resulting functions are analytic in the unit disk of the complex plane. A result of Zhang (Proc Am Math Soc 145:241–250, 2017) on the reality of the zeros of Ramanujan type entire functions

    更新日期:2020-07-21
  • On class numbers of pure quartic fields
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Jianing Li, Yue Xu

    Let p be a prime. The 2-primary part of the class group of the pure quartic field \({\mathbb {Q}}(\root 4 \of {p})\) has been determined by Parry and Lemmermeyer when \(p \not \equiv \pm \, 1\bmod 16\). In this paper, we improve the known results in the case \(p\equiv \pm \, 1\bmod 16\). In particular, we determine all primes p such that 4 does not divide the class number of \({\mathbb {Q}}(\root 4

    更新日期:2020-07-21
  • The first simultaneous sign change for Fourier coefficients of Hecke–Maass forms
    Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
    Moni Kumari, Jyoti Sengupta

    Let f and g be two Hecke–Maass cusp forms of weight zero for \(SL_2({\mathbb {Z}})\) with Laplacian eigenvalues \(\frac{1}{4}+u^2\) and \(\frac{1}{4}+v^2\), respectively. Then both have real Fourier coefficients say, \(\lambda _f(n)\) and \(\lambda _g(n)\), and we may normalize f and g so that \(\lambda _f(1)=1=\lambda _g(1)\). In this article, we first prove that the sequence \(\{\lambda _f(n)\lambda

    更新日期:2020-07-21
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