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Generating functions of the Hurwitz class numbers associated with certain mock theta functions Ramanujan J. (IF 0.7) Pub Date : 2024-03-12 Dandan Chen, Rong Chen
We find Hecke–Rogers type series representations of generating functions of the Hurwitz class numbers which are similar to certain mock theta functions. We also prove two combinatorial interpretations of Hurwitz class numbers which appeared on OEIS (see A238872 and A321440).
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Sums of squares of integral multiples of an integral element of real bi-quadratic fields Ramanujan J. (IF 0.7) Pub Date : 2024-03-04 Srijonee Shabnam Chaudhury
For any given positive integer m we construct certain totally positive algebraic integers \(\alpha \) of a real bi-quadratic field K and obtain some necessary conditions for which \(m\alpha \) cannot be represented as sum of integral squares. We show this for integers that lie in quadratic subfields of K and for integers which are in K but not in any quadratic subfield of K. We provide examples in
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Lower bounds for negative moments of quadratic twist of modular L-functions Ramanujan J. (IF 0.7) Pub Date : 2024-02-24 Huaqing Bian, Xiaofei Yan, Ruiyang Yue
We establish sharp lower bonds for the 2k-th moment of families of quadratic twists of modular L-functions at the central point for all real \(k < 0\), assuming a conjecture of S. Chowla on the non-vanishing of these L-values.
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Sufficient conditions for the weighted integrability of Jacobi–Dunkl transforms Ramanujan J. (IF 0.7) Pub Date : 2024-02-22 Salah El Ouadih
In this paper, we obtain sufficient conditions for the weighted integrability of Jacobi–Dunkl transforms in terms of the moduli of smoothness connected with Jacobi–Dunkl translation operators. These results generalize a famous Titchmarsh’s theorem and Younis’ theorem for functions from Lipschitz classes.
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Fourier coefficients twisted with exponential functions Ramanujan J. (IF 0.7) Pub Date : 2024-02-22 Lijuan Cao, Yanxue Yu
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On some determinants involving the tangent function Ramanujan J. (IF 0.7) Pub Date : 2024-02-21 Zhi-Wei Sun
Let p be an odd prime and let \(a,b\in {\mathbb {Z}}\) with \(p\not \mid ab\). In this paper,we mainly evaluate $$\begin{aligned} T_p^{(\delta )}(a,b,x):=\det \left[ x+\tan \pi \frac{aj^2+bk^2}{p}\right] _{\delta \leqslant j,k\leqslant (p-1)/2}\ \ (\delta =0,1). \end{aligned}$$ For example, in the case \(p\equiv 3\ ({\textrm{mod}}\ 4)\), we show that \(T_p^{(1)}(a,b,0)=0\) and $$\begin{aligned} T_p^{(0)}(a
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New contiguous relations for a balanced q-series and their applications Ramanujan J. (IF 0.7) Pub Date : 2024-02-17 Chenying Wang, Jianan Xu
The present paper is mainly concerned with a balanced partial q-series. By the classical Abel lemma on summation by parts, we establish the q-contiguous relations for the partial sum whose iterating cases embrace many new summation and transformation formulae as well as some known results. Of particular interest are one identity of Rogers–Ramanujan type and two identities of partial theta functions
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Factoring numbers with elliptic curves Ramanujan J. (IF 0.7) Pub Date : 2024-02-16 Jorge Jiménez Urroz, Jacek Pomykała
In the present paper, we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer n to counting points on elliptic curves modulo n, succeeding with probability \(1-\varepsilon \), for any\(\varepsilon >0\).
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Some new Ramanujan-type modular equations of degree 15 Ramanujan J. (IF 0.7) Pub Date : 2024-02-01 Zhang Chuan-Ding, Yang Li
Ramanujan in his notebook recorded two modular equations involving multipliers with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan-type modular equations involving multipliers with moduli of degrees (3,5) and (1,15), and give concise proofs by employing Ramanujan’s multiplier functional equation.
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Connections between binomial coefficients and binary quadratic forms Ramanujan J. (IF 0.7) Pub Date : 2024-01-31
Abstract In this paper, we mainly prove some congruences involving binomial coefficients and binary quadratic forms. One such example is the following: Let p b be a prime such that \(p=x^2+2y^2\equiv 1\ ({\textrm{mod}}\ 8)\) . Then, $$\begin{aligned} p\sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2}{(8k+1)16^k}\equiv 3p\sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\
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Rooted partitions and number-theoretic functions Ramanujan J. (IF 0.7) Pub Date : 2024-01-30 Bruce E. Sagan
Recently, Merca and Schmidt proved a number of identities relating partitions of an integer with two classic number-theoretic functions, namely the Möbius function and Euler’s totient function. Their demonstrations were mainly algebraic. We give bijective proofs of some of these results. Our main tools are the concept of a rooted partition and an operation which we call the direct sum, which combines
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Wave packet transform and wavelet convolution product involving the index Whittaker transform Ramanujan J. (IF 0.7) Pub Date : 2024-01-25 Jeetendrasingh Maan, Akhilesh Prasad
The main goal of this paper is to study the wave packet transform (WPT) and wavelet convolution product involving the index Whittaker transform (IWT). Some estimates for the Whittaker wavelet (W-Wavelet) and W-wavelet transform are obtained, and Plancherel’s relation for the WPT-transform is also deduced. Calderón’s formula associated with IWT is obtained. Furthermore, using the convolution property
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Primes of higher degree and annihilators of class groups Ramanujan J. (IF 0.7) Pub Date : 2024-02-01 Nimish Kumar Mahapatra, Prem Prakash Pandey, Mahesh Kumar Ram
Abstract Let L/K be a Galois extension of number fields with Galois group G. We discuss a new method, by studying primes of higher residue degree, to obtain elements in \({\mathbb {Z}}[G]\) which annihilate the class group of L. We illustrate the method by obtaining annihilators of class groups for some cyclotomic fields. We mention some results on factors of class numbers which can be obtained from
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A shifted convolution sum for $$GL(3) \times GL(2)$$ with weighted average Ramanujan J. (IF 0.7) Pub Date : 2024-01-25 Mohd Harun, Saurabh Kumar Singh
In this paper, we will prove a non-trivial bound for the weighted average version of a shifted convolution sum for \(GL(3) \times GL(2)\), i.e. for arbitrary small \(\epsilon >0\) and \(X^{1/4+\delta } \le H \le X\) with \(\delta >0\), we prove $$\begin{aligned} \frac{1}{H}\sum _{h=1}^\infty \lambda _f(h) V \left( \frac{h}{H}\right) \sum _{n=1}^\infty \lambda _{\pi }(1,n) \lambda _g (n+h) W \left(
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On the evaluation of the alternating multiple $$ t $$ value $$ t({\overline{1}},\ldots ,{\overline{1}}, 1, {\overline{1}},\ldots ,{\overline{1}}) $$ Ramanujan J. (IF 0.7) Pub Date : 2024-01-24
Abstract We prove an evaluation for the stuffle-regularised alternating multiple \( t \) value \( t^{*,V}({\overline{1}},\ldots ,{\overline{1}}, 1, {\overline{1}},\ldots ,{\overline{1}}) \) in terms of \( V \) , the regularisation parameter, \(\log (2), \zeta (k) \) and \( \beta (k) \) . This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a
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On the exponential diophantine equation $$U_{n}^x+U_{n+1}^x=U_m$$ Ramanujan J. (IF 0.7) Pub Date : 2024-01-23 Herbert Batte, Mahadi Ddamulira, Juma Kasozi, Florian Luca
Let \( \{U_n\}_{n\ge 0} \) be the Lucas sequence. For integers x, n and m, we find all solutions to \(U_{n}^x+U_{n+1}^x=U_m\). The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms
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A bound for twists of $${\textrm{GL}}_3\times GL_2$$ L-functions with composite modulus Ramanujan J. (IF 0.7) Pub Date : 2024-01-19 Qingfeng Sun, Yanxue Yu
Let \(\pi \) be a Hecke-Maass cusp form for \(\textrm{SL}_3({\textbf{Z}})\) and let g be a holomorphic or Maass cusp form for \(\textrm{SL}_2({\textbf{Z}})\). Let \(\chi \) be a primitive Dirichlet character of modulus \(M=M_1M_2\) with \(M_i\) prime, \(i=1,2\). Suppose that \(M^{1/2+2\eta }
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From continuous to discrete: weak limit of normalized Askey–Wilson measure Ramanujan J. (IF 0.7) Pub Date : 2024-01-18 Dan Dai, Mourad E. H. Ismail, Xiang-Sheng Wang
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Andrews-Beck type congruences modulo powers of 5 Ramanujan J. (IF 0.7) Pub Date : 2024-01-10 Nankun Hong, Renrong Mao
Let NT(m, k, n) denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck’s conjecture on congruences for NT(m, k, n) modulo 5 and 7. Generalizing Andrews’ results, Chern obtained congruences for NT(m, k, n) modulo 11 and 13. More recently, the second author used the theory of Hecke operators to establish congruences for such partition statistics
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An elementary proof of a theorem of Hardy and Ramanujan Ramanujan J. (IF 0.7) Pub Date : 2024-01-08 Asaf Cohen Antonir, Asaf Shapira
Let Q(n) denote the number of integers \(1 \le q \le n\) whose prime factorization \(q= \prod ^{t}_{i=1}p^{a_i}_i\) satisfies \(a_1\ge a_2\ge \cdots \ge a_t\). Hardy and Ramanujan proved that $$\begin{aligned} \log Q(n) \sim \frac{2\pi }{\sqrt{3}} \sqrt{\frac{\log (n)}{\log \log (n)}}\;. \end{aligned}$$ Before proving the above precise asymptotic formula, they studied in great detail what can be obtained
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Deterministic comparison of cusp form coefficients over certain sequences Ramanujan J. (IF 0.7) Pub Date : 2024-01-04 Guodong Hua
Let f and g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1}\) and \(k_{2}\) for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\), respectively. Denote by \(\lambda _{f}(n)\) and \(\lambda _{g}(n)\) the nth normalized Fourier coefficients of f and g, respectively. And set \(Q(\textbf{x})\) a primitive integral positive-definite binary quadratic form of fixed discriminant
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On a problem on restricted k-colored partitions, II Ramanujan J. (IF 0.7) Pub Date : 2024-01-04 Wu-Xia Ma, Yong-Gao Chen
For two integers \(1\le j\le k\), we define (k, j)-colored partitions to be those partitions in which parts may appear in k different types and at most j types can appear for a given part size. Let \(c_{k,j}(n)\) be the number of (k, j)-colored partitions of n. Recently, Keith studied (k, j)-colored partitions and proved the following results: For \(j\in \{2,5,8,9\}\), we have \(c_{9,j}(3n+2)\equiv
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A simple proof of Slater’s transformations for bilateral series $${}_{r}\psi _{r},\, {}_{2r}\psi _{2r}$$ and $${}_{2r-1}\psi _{2r-1}$$ Ramanujan J. (IF 0.7) Pub Date : 2024-01-03 Katsuhisa Mimachi
We give a simple proof of Slater’s transformations for bilateral series \({}_r\psi _r\), \({}_{2r}\psi _{2r}\), and \({}_{2r-1}\psi _{2r-1}\), using only the residue theorem only, without technical manipulation of the series.
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A Voronoi summation formula for the shifted triple divisor function Ramanujan J. (IF 0.7) Pub Date : 2024-01-03 Alessandro Fazzari
In this paper, we prove a Voronoi summation formula for the shifted threefold divisor function twisted by additive characters. As the main tool, we provide the functional equation for the shifted GL(3) Estermann function.
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On the Northcott property of Dedekind zeta functions Ramanujan J. (IF 0.7) Pub Date : 2024-01-03 Xavier Généreux, Matilde Lalín
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On the index of appearance of a Lucas sequence Ramanujan J. (IF 0.7) Pub Date : 2024-01-03 Carlo Sanna
Let \(\varvec{u} = (u_n)_{n \ge 0}\) be a Lucas sequence, that is, a sequence of integers satisfying \(u_0 = 0\), \(u_1 = 1\), and \(u_n = a_1 u_{n - 1} + a_2 u_{n - 2}\) for every integer \(n \ge 2\), where \(a_1\) and \(a_2\) are fixed nonzero integers. For each prime number p with \(p \not \mid 2a_2D_{\varvec{u}}\), where \(D_{\varvec{u}}:= a_1^2 + 4a_2\), let \(\rho _{\varvec{u}}(p)\) be the rank
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Some identities for Lin–Peng–Toh’s k-colored partition statistic Ramanujan J. (IF 0.7) Pub Date : 2024-01-03 Yang Lin, Ernest X. W. Xia, Xuan Yu
Recently, Andrews proved two conjectures for a partition statistic introduced by Beck. Very recently, Chern established some results for weighted rank and crank moments and proved many Andrews–Beck type congruences. Motivated by Andrews and Chern’s work, Lin, Peng, and Toh introduced a partition statistic of k-colored partitions \(NB_k(r,m,n)\) which counts the total number of parts of \(\pi ^{(1)}\)
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Congruences for the q-Fibonacci sequence related to its transcendence Ramanujan J. (IF 0.7) Pub Date : 2024-01-02 Takumi Anzawa, Hidetaka Funakura
By using Andrews’ explicit formulae of the q-Fibonacci sequence introduced by Schur, we prove certain congruences of the q-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that it yields a transcendental element in the \(\mathbb {Q}\)-algebra \(\mathscr {A}\) of integers modulo arbitrarily large primes under the generalized Riemann hypothesis
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A new Andrews–Crandall-type identity and the number of integer solutions to $$x^2+2y^2+2z^2=n$$ Ramanujan J. (IF 0.7) Pub Date : 2023-12-30 Mariia Dospolova, Ekaterina Kochetkova, Eric T. Mortenson
Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews–Crandall-type identity and use it to count the number of integer solutions to \(x^2+2y^2+2z^2=n\).
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Congruences modulo powers of 2 for restricted partition triples due to Lin and Wang Ramanujan J. (IF 0.7) Pub Date : 2023-12-29
Abstract In 2018, Lin and Wang introduced two restricted partition triples, whose generating functions are related to the reciprocals of Ramanujan–Gordon identities. Let \(RG_2(n)\) denote the number of partitions triples of n, where odd parts in the first two components are distinct and the last component only contains even parts. Lin and Wang proved some congruences modulo 5 and 7 satisfied by \(RG_2(n)\)
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On the Dotsenko–Fateev complex twin of the Selberg integral and its extensions Ramanujan J. (IF 0.7) Pub Date : 2023-12-29 Yury A. Neretin
The Selberg integral has a twin (‘the Dotsenko–Fateev integral’) of the following form. We replace real variables \(x_k\) in the integrand \(\prod |x_k|^{\sigma -1}\,|1-x_k|^{\tau -1} \prod |x_k-x_l|^{2\theta }\) of the Selberg integral by complex variables \(z_k\), integration over a cube we replace by an integration over the whole complex space \({\mathbb {C}}^n\). According to Dotsenko, Fateev,
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A generalization of Markov Numbers Ramanujan J. (IF 0.7) Pub Date : 2023-12-28 Esther Banaian, Archan Sen
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Positive differences of overpartitions with separated parts Ramanujan J. (IF 0.7) Pub Date : 2023-12-28 Mohamed El Bachraoui
In this note, we investigate partition differences of overpartitions where overlined and non-overlined part are separated. As a consequence of our main results, we deduce a variety of positivity results for overpartition differences.
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Factors of certain basic hypergeometric sums Ramanujan J. (IF 0.7) Pub Date : 2023-12-28 Jian Cao, Victor J. W. Guo, Xiao Yu
We prove that certain truncated basic hypergeometric series contain the factor \(\Phi _n(q)^2\), where \(\Phi _n(q)\) is the nth cyclotomic polynomial. This result may be regarded as a generalization of Theorem 1.1 in Guo (J Math Anal Appl 476:851–859, 2019).
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On integer partitions and continued fraction type algorithms Ramanujan J. (IF 0.7) Pub Date : 2023-11-27 Wael Baalbaki, Claudio Bonanno, Alessio Del Vigna, Thomas Garrity, Stefano Isola
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Modular equations of degrees 13, 29, and 61 Ramanujan J. (IF 0.7) Pub Date : 2023-11-21 Ahmet M. Güloğlu, Hamza Yesilyurt
Schröter-type theta function identities were very instrumental in proving modular equations. In this paper, by employing a generalization of this identity, we prove for the first time a modular equation of degree 61. Furthermore, new modular equations of degrees 13 and 29 are obtained.
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On the sum and the regularized products of some Dirichlet series Ramanujan J. (IF 0.7) Pub Date : 2023-11-22 Mounir Hajli
In this paper, using zeta functions, we evaluate the regularized products of some Dirichlet series generalizing Lerch’s formula and we show that they are multiplicative. As an application, we determine the sum of some Dirichlet series generalizing Euler’s formula on the sum of the reciprocal of squares.
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Modularity of a certain continued fraction of Ramanujan Ramanujan J. (IF 0.7) Pub Date : 2023-11-21 Russelle Guadalupe
We apply the methods of Lee and Park to study a certain continued fraction \(H(\tau )\) of Ramanujan, which is a particular case of his identity written in one of his notebooks. We prove that \(H(\tau )\) can be expressed in terms of an \(\eta \)-quotient \(s(\tau )\), which is a generator for the field of all modular functions on the congruence subgroup \(\Gamma _0(12)\). We also show that there is
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The p-adic constant for mock modular forms associated to CM forms Ramanujan J. (IF 0.7) Pub Date : 2023-11-20 Ryota Tajima
Let \(g \in S_{k}(\Gamma _{0}(N))\) be a normalized newform and f be a harmonic Maass form that is good for g. The holomorphic part of f is called a mock modular form and denoted by \(f^{+}\). For odd prime p, Bringmann et al. (Trans Am Math Soc 364(5):2393–2410, 2012) obtained a p-adic modular form of level pN from \(f^{+}\) and a certain p-adic constant \(\alpha _{g}(f)\). When g has complex multiplication
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Inverses of r-primitive k-normal elements over finite fields Ramanujan J. (IF 0.7) Pub Date : 2023-10-24 Mamta Rani, Avnish K. Sharma, Sharwan K. Tiwari, Anupama Panigrahi
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Seesaw identities and theta contractions with generalized theta functions, and restrictions of theta lifts Ramanujan J. (IF 0.7) Pub Date : 2023-10-24 Shaul Zemel
We prove a seesaw identity for theta functions with polynomials, and establish a formula for the corresponding theta contractions. We then use it to determine the restrictions of theta lifts to sub-Grassmannians, with some applications.
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On a variant of Pillai’s problem with factorials and S-units Ramanujan J. (IF 0.7) Pub Date : 2023-10-24 Bernadette Faye, Florian Luca, Volker Ziegler
Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S-unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S-unit associated with the set of primes \(\{2,3,5,7\}\).
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Parity of the 8-regular partition function Ramanujan J. (IF 0.7) Pub Date : 2023-10-16 Giacomo Cherubini, Pietro Mercuri
We give a complete characterisation of the parity of \(b_8(n)\), the number of 8-regular partitions of n. Namely, we prove that \(b_8(n)\) is odd precisely when \(24n+7\) has the form \(p^{4a+1}m^2\) with p prime and \(p\not \mid m\).
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On algebraic twists with composite moduli Ramanujan J. (IF 0.7) Pub Date : 2023-10-17 Yongxiao Lin, Philippe Michel
We study bounds for algebraic twists sums of automorphic coefficients by trace functions of composite moduli.
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Infinite product representations of some q-series Ramanujan J. (IF 0.7) Pub Date : 2023-10-16 Florian Münkel, Lerna Pehlivan, Kenneth S. Williams
For integers a and b (not both 0) we define the integers \(c(a,b;n)\ \ (n=0,1,2,\ldots )\) by $$\begin{aligned} \sum _{n=0}^\infty c(a,b;n)q^n = \prod _{n=1}^\infty \left( 1-q^n\right) ^a (1-q^{2n})^b \quad (|q|<1). \end{aligned}$$ These integers include the numbers \(t_k(n) = c(-k,2k;n)\), which count the number of representations of n as a sum of k triangular numbers, and the numbers \((-1)^n r_k(n)
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Some identities on Beck’s partition statistics Ramanujan J. (IF 0.7) Pub Date : 2023-09-26 Yongqiang Chen, Jing Jin, Olivia X. M. Yao
Recently, Chern proved a number of congruences modulo 5, 7, 11, and 13 on Beck’s partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\), which enumerate the total number of parts in the partitions of n with rank congruent to r modulo m and the total number of ones in the partitions of n with crank congruent to r modulo m, respectively. In this paper, we prove some identities on NT(r, 5, n) and
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Identities on mex-related partitions Ramanujan J. (IF 0.7) Pub Date : 2023-09-27 Jane Y. X. Yang, Li Zhou
The minimal excludant, or mex-function, on a set of positive integers is the smallest positive integer not in it. Andrews and Newman defined the mex-function \(\text{ mex}_{A,a}(\lambda )\) to be the smallest positive integer congruent to a modulo A that is not part of partition \(\lambda \), and denote by \(p_{A,a}(n)\) (reps. \(\overline{p}_{A,a}(n)\)) the number of partitions \(\lambda \) of n satisfying
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Alternating multiple T-values: weighted sums, duality, and dimension conjecture Ramanujan J. (IF 0.7) Pub Date : 2023-09-26 Ce Xu, Jianqiang Zhao
In this paper, we define some weighted sums of the alternating multiple T-values (AMTVs) and study several duality formulas for them. Then we introduce the alternating version of the convoluted T-values and Kaneko–Tsumura \(\psi \)-function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the \(\mathbb {Q}\)-vector space generated by the AMTVs of any fixed weight
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Composition-theoretic series in partition theory Ramanujan J. (IF 0.7) Pub Date : 2023-09-15 Robert Schneider, Andrew V. Sills
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are k-gonal numbers; our proofs employ Ramanujan’s theta functions. We explore applications to lacunary q-series, and to a new class of composition-theoretic Dirichlet series.
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Inequalities for the $$M_2$$ -rank modulo 12 of partitions without repeated odd parts Ramanujan J. (IF 0.7) Pub Date : 2023-09-15 Yan Fan, Eric H. Liu, Ernest X. W. Xia
Let \(N_2(a, M; n)\) denote the number of partitions of n without repeated odd parts whose \(M_2\)-rank is congruent to a modulo M. Lovejoy, Osburn and Mao have found formulas for \(M_2 \)-rank differences modulo 3, 5, 6, and 10. Recently, Xia and Zhao established generating functions for \(N_2(a, 8; n)\) with \(0 \le a\le 7\). Motivated by their works, we establish generating functions for \(N_2(a
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Additive evaluations of the number of divisors Ramanujan J. (IF 0.7) Pub Date : 2023-09-11 Mircea Merca
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On restricted approximation measures of Jacobi’s triple product Ramanujan J. (IF 0.7) Pub Date : 2023-09-11 Leena Leinonen, Marko Leinonen
We obtain rational approximations for Jacobi’s triple product $$\begin{aligned} \Pi _q(t):= \prod _{m=1}^\infty (1-q^{2m})(1+q^{2m-1}t)(1+q^{2m-1}t^{-1}), \end{aligned}$$ when \(t=a/b\in {\mathbb {Q}}\) is non-zero and \(q=1/d\) with \(d\in {\mathbb {Z}}{\setminus }\{0, \pm 1 \}\). Especially we give effective and restricted approximation for the values of Jacobi’s triple product and for the values
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Hypergeometric functions over finite fields Ramanujan J. (IF 0.7) Pub Date : 2023-09-11 Noriyuki Otsubo
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix,
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The first negative Fourier coefficient of an Eisenstein series newform Ramanujan J. (IF 0.7) Pub Date : 2023-09-11 Sebastián Carrillo Santana
There have been a number of papers on statistical questions concerning the sign changes of Fourier coefficients of newforms. In one such paper, Linowitz and Thompson gave a conjecture describing when, on average, the first negative sign of the Fourier coefficients of an Eisenstein series newform occurs. In this paper, we correct their conjecture and prove the corrected version.
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Finite trigonometric sums arising from Ramanujan’s theta functions Ramanujan J. (IF 0.7) Pub Date : 2023-09-12 Bruce C. Berndt, Sun Kim, Alexandru Zaharescu
Two classes of finite trigonometric sums, each involving only sines, are evaluated in closed form. The previous and original proofs arise from Ramanujan’s theta functions and modular equations.
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A closed-form expression for the Euler–Kronecker constant of a quadratic field Ramanujan J. (IF 0.7) Pub Date : 2023-09-04 Suraj Singh Khurana
Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point \(s=1\). In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid
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Some problems about friable numbers in Piatetski-Shapiro sequence Ramanujan J. (IF 0.7) Pub Date : 2023-09-05 Wenbin Zhu
In this article, we consider several problems about friable numbers in Piatetski-Shapiro sequences, such as the ternary Goldbach type problem, Diophantine approximation, almost primes, intersections of Piatetski-Shapiro sequences and Beatty sequences, intersections of Piatetski-Shapiro sequences and so on. By using well known properties, we establish two exponential sums involving fractional powers
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Parity bias for the partitions with bounded number of appearances of the part of size 1 Ramanujan J. (IF 0.7) Pub Date : 2023-09-01 Byungchan Kim, Eunmi Kim
While there are more partitions of n having more odd parts than even parts compared to partitions having more even parts than odd parts, we show that if there is a limit on the number of appearances of the part of size 1, there will eventually be more partitions of n with more even parts than odd parts compared to partitions with more odd parts than even parts. To demonstrate this, we obtain the asymptotic
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On index divisors and monogenity of certain number fields defined by $$x^{12}+ax^m+b$$ Ramanujan J. (IF 0.7) Pub Date : 2023-09-01 Lhoussain El Fadil, Omar Kchit
In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial \(F(x)=x^{12}+ax^m+b\in \mathbb {Z}[x]\) with \(1\le m\le 11\) an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if \(i(K)\ne 1\), then K is not monogenic. For \(m=1\), we give necessary and sufficient conditions on a
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Euler sums of generalized alternating hyperharmonic numbers II Ramanujan J. (IF 0.7) Pub Date : 2023-09-02 Rusen Li
In this paper, we introduce a new type of generalized alternating hyperharmonic number \(H_n^{(p,r,s_{1},s_{2})}\), and show that the Euler sums of the generalized alternating hyperharmonic numbers \(H_n^{(p,r,s_{1},s_{2})}\) can be expressed in terms of linear combinations of the classical (alternating) Euler sums.