• 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
• 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 • 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 • 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 • 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 • 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 • 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}$$更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Sheng-Chi Liu, Shenhui Liu Let $$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}}).$$ We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F). 更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Lenny Jones Let $$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$$. 更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Su-Ping Cui, Nancy S. S. Gu, Chen-Yang Su Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function$$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^

更新日期：2020-07-28
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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
• 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
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-23

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
• 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
• 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-21 Stavros Garoufalidis, Don Zagier We give a formula for the radial asymptotics to all orders of the special q-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in Calegari et al. (Bloch groups, algebraic K-theory, units and Nahm’s conjecture. arXiv:1712.04887, 2017) to prove Nahm’s conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in K-theory. The power series 更新日期：2020-07-21 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-20 Mohamed El Bachraoui We combine an extended version of Bailey’s transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions. 更新日期：2020-07-21 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-20 Khang Tran, Andres Zumba This paper investigates the zero distribution of a sequence of polynomials \(\left\{ P_{m}(z)\right\} _{m=0}^{\infty }$$ which satisfy a four-term recurrence whose coefficients are linear polynomials in z. In particular, we study necessary and sufficient conditions for the reality of the zeros of $$P_{m}(z)$$. Under these conditions, we find an explicit interval containing these zeros, whose union 更新日期：2020-07-21 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-20 Terence Coelho, Jongwon Kim, Matthew C. Russell In the spirit of Göllnitz’s “big” partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Göllnitz’s big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews 更新日期：2020-07-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-20 N. A. Rather, Ishfaq Dar, Suhail Gulzar If all the zeros of nth degree polynomials f(z) and $$g(z) = \sum _{k=0}^{n}\lambda _k\left( {\begin{array}{c}n\\ k\end{array}}\right) z^k$$ respectively lie in the cricular regions $$|z|\le r$$ and $$|z| \le s|z-\sigma |$$, $$s>0$$, then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial 更新日期：2020-07-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-19 Ratnadeep Acharya, Saurabh Kumar Singh Let $$\lambda _f (n)$$ denote the normalized n-th Fourier coefficient of a holomorphic Hecke eigencuspform or a Hecke–Maass cusp form for the full modular group. In this paper we shall exhibit cancellations in the following sum:\begin{aligned} \sum _{N

更新日期：2020-07-20
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-19
Maciej Ulas, Błażej Żmija

Let $$k\in \mathbb {N}_{\ge 2}$$ and for given $$m\in \mathbb {Z}{\setminus }\{0\}$$ consider the sequence $$(S_{k,m}(n))_{n\in \mathbb {N}}$$ defined by the power series expansion \begin{aligned} \frac{1}{(1-x)^{m}}\prod _{i=0}^{\infty } \frac{1}{(1-x^{k^{i}})^{m(k-1)}}=\sum _{n=0}^{\infty }S_{k,m}(n)x^{n}. \end{aligned} The number $$S_{k,m}(n)$$ for $$m\in \mathbb {N}_{+}$$ has a natural combinatorial

更新日期：2020-07-20
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-18
Min Zhang, Jinjiang Li

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{7/18+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$$, where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers, which constitutes an improvement over some previous work.

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-18
Tessa Cotron, Anya Michaelsen, Emily Stamm, Weitao Zhu

An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, Gordon and Ono studied the generating functions for t-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-18
Keshav Aggarwal, Roman Holowinsky, Yongxiao Lin, Qingfeng Sun

Let g be a fixed Hecke cusp form for $$\mathrm SL(2,{\mathbb {Z}})$$ and $$\chi$$ be a primitive Dirichlet character of conductor M. The best known subconvex bound for $$L(1/2,g\otimes \chi )$$ is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-18
Wenchang Chu

For the very well-poised $$\Omega$$-series, a universal iteration pattern is established that yields numerous infinite series identities including several important ones discovered by Ramanujan (in Q J Pure Appl Math 45:350–372, 1914) and recently by Guillera.

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-17
Xia Wu

In this paper, we apply Qin’s theorem for the 4-rank of $$K_2O_F$$ to establish the relation between the 4-rank of the ideal class group of $$F=\mathbb {Q}(\sqrt{d})$$ and the 4-rank of $$K_2O_F$$ provided that all odd prime factors of d are congruent to 1 mod 8. As an application, we give a concise and unified proof of two conjectures proposed by Conner and Hurrelbrink (Acta Arith 73:59–65, 1995)

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-17
Sanoli Gun, Winfried Kohnen, Biplab Paul

Let F and G be Siegel cusp forms for $${\mathrm{Sp}}_4({{\mathbb {Z}}})$$ and weights $$k_1, k_2$$, respectively. Also let F and G be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither F nor G is a Saito–Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.

更新日期：2020-07-18
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-16
Marc Munsch

In the present note, we prove new lower bounds on large values of character sums $$\varDelta (x,q):=\max _{\chi \ne \chi _0} \big \vert \sum _{n\le x} \chi (n)\big \vert$$ in certain ranges of x. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of L-functions, we improve some results of Hough in the range $$\log x = o\big 更新日期：2020-07-17 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-16 R. M. Green, Ilia D. Mishev, Eric Stade In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised \(_9F_8$$ hypergeometric series of unit argument. We consider a known, transitive

更新日期：2020-07-17
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-16
Min Bian, Dazhao Tang, Ernest X. W. Xia, Fanggang Xue

Recently, Lin and Wang introduced two special partition functions $$RG_1(n)$$ and $$RG_2(n)$$, the generating functions of which are the reciprocals of two identities due to Ramanujan and Gordon. They established several congruences modulo 5 and 7 for $$RG_1(n)$$ and $$RG_2(n)$$ and posed four conjectures on congruences modulo 25 for $$RG_1(n)$$ and $$RG_2(n)$$ at the end of their paper. In this paper

更新日期：2020-07-16
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-16
Chuanan Wei

In terms of Dougall’s $$_2H_2$$ series identity and the series rearrangement method, we establish a symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschütz’s theorem. Similarly, we also show that Bailey’s $$_6\psi _6$$ series identity implies the nonterminating form of Jackson’s $$_8\phi _7$$ summation formula. Considering the reversibility

更新日期：2020-07-16
Contents have been reproduced by permission of the publishers.

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