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Abstract: Abstract 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 \(M_{\omega }(r,5,n)\) which are analogous to Ramanujan’s “most beautiful identity.” These identities along with some identities proved by Mao, and Jin, Liu, and Xia imply all congruences modulo 5 on NT(r, 5, n) and \(M_{\omega }(r,5,n)\) discovered by Chern. PubDate: 2024-03-01

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Abstract: Abstract 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 \(p\not \mid m\) . PubDate: 2024-03-01

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Abstract: Abstract We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell–Lerch functions—the building blocks of Ramanujan’s mock theta functions—can be used to give general string function formulas for the affine Lie algebra \(A_{1}^{(1)}\) for levels \(N=1,2,3,4\) . PubDate: 2024-03-01

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Abstract: Abstract We study bounds for algebraic twists sums of automorphic coefficients by trace functions of composite moduli. PubDate: 2024-03-01

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Abstract: Abstract Our goal is to show that the additive-slow-Farey version of the Triangle map (a type of multidimensional continued fraction algorithm) gives us a method for producing a map from the set of integer partitions of a positive number n into itself. We start by showing that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number n into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for p(2, n), the number of ways for partitioning n into two smaller positive integers, with multiplicity. We then turn to the general case, using the Triangle map to give a natural map from general integer partitions of a positive number n to integer partitions of n. This map will still be compatible with conjugation of the corresponding Young shapes. We will close by the observation that it appears few other multidimensional continued fraction algorithms can be used to study partitions. PubDate: 2024-03-01

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Abstract: Abstract 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 \(\{2,3,5,7\}\) . PubDate: 2024-03-01

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Abstract: Abstract 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 formulas for related partition functions via a Tauberian theorem, q-series manipulations, and partition combinatorics. PubDate: 2024-03-01

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Abstract: Abstract Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for these generalized sequentially congruent partitions. Finally, we investigate a question of Schneider and Schneider regarding how sequentially congruent partitions fit into Andrews’ theory of partition ideals. We prove that the maximal partition ideal of sequentially congruent partitions has infinite order and is therefore not linked, and we identify its order 1 subideals. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract This article studies the existence of elements \(\alpha \) in finite fields \(\mathbb {F}_{q^n}\) such that both \(\alpha \) and its inverse \(\alpha ^{-1}\) are r-primitive and k-normal over \(\mathbb {F}_q\) . We define a characteristic function for the set of k-normal elements and use it to establish a sufficient condition for the existence of the desired pair \((\alpha ,\alpha ^{-1})\) . Moreover, we find that for \(n\ge 7\) , there always exists a pair \((\alpha ,\alpha ^{-1})\) of 1-primitive and 1-normal elements in \(\mathbb {F}_{q^n}\) over \(\mathbb {F}_q\) . Additionally, we obtain that for \(n=5,6\) , if \(\textrm{gcd}(q,n)=1\) , there always exists such a pair in \(\mathbb {F}_{q^n}\) , except for the field \(\mathbb {F}_{4^5}\) . PubDate: 2024-03-01

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Abstract: Abstract If m and n are positive integers, then \(a_m(n)\) denotes the number of the parts congruent to 0 modulo m in all the partitions of n. On the strength of Euler’s pentagonal number theorem, this paper shows that the number of positive divisors of n can be expressed additively in terms of the partition function \(a_m(\cdot )\) . PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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 \((-1)^n r_k(n) = c(2k,-k;n)\) , where \(r_k(n)\) counts the number of representations of n as a sum of k squares. A computer search was carried out for integers a and b, satisfying \(-24\le a,b\le 24\) , such that at least one of the sums 0.1 $$\begin{aligned} \sum _{n=0}^\infty c(a,b;3n+j)q^n, \quad j=0,1,2, \end{aligned}$$ is either zero or can be expressed as a nonzero constant multiple of the product of a power of q and a single infinite product of factors involving powers of \(1-q^{rn}\) with \(r\in \{1,2,3,4,6,8,12,24\}\) for all powers of q up to \(q^{1000}\) . A total of 84 such candidate identities involving 56 pairs of integers (a, b) all satisfying \(a\equiv b\ (\textrm{mod}\ 3)\) were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums $$\begin{aligned} \sum _{n=0}^\infty t_k(3n+j)q^n, \quad \sum _{n=0}^\infty r_k(3n+j)q^n, \quad j=0,1,2. \end{aligned}$$ PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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 over friable numbers. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-02-24

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Abstract: Abstract 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. PubDate: 2024-02-22

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Abstract: Abstract Let f be a cusp form for \(\textrm{SL}(2,\mathbb {Z})\) , and \(\lambda _f(n)\) , the n-th Fourier coefficient of the L-function attached to f. Here a new bound is established for the exponential sum $$\begin{aligned} \sum _{{n} \le {x}}\Lambda (n)\lambda _{f}(n)e(n^{\theta }\alpha ), \end{aligned}$$ where \(0< \theta <1\) and \(\alpha \ne 0\) is a fixed parameter. This improves previous results when \(1/2\le \theta <1\) . PubDate: 2024-02-22

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Abstract: Abstract 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 $$\begin{aligned} T_p^{(0)}(a,b,x)={\left\{ \begin{array}{ll} 2^{(p-1)/2}p^{(p+1)/4}&{}\text {if}\ (\frac{ab}{p})=1, \\ p^{(p+1)/4}&{}\text {if}\ (\frac{ab}{p})=-1,\end{array}\right. } \end{aligned}$$ where \((\frac{\cdot }{p})\) is the Legendre symbol. When \((\frac{-ab}{p})=-1\) , we also evaluate the determinant \(\det [x+\cot \pi \frac{aj^2+bk^2}{p}]_{1\leqslant j,k\leqslant (p-1)/2}.\) In addition, we pose several conjectures one of which states that for any prime \(p\equiv 3\ ({\textrm{mod}}\ 4)\) , there is an integer \(x_p\equiv 1\ ({\textrm{mod}}\ p)\) such that $$\begin{aligned}\det \left[ \sec 2\pi \frac{(j-k)^2}{p}\right] _{0\leqslant j,k\leqslant p-1}=-p^{(p+3)/2}x_p^2.\end{aligned}$$ PubDate: 2024-02-21

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Abstract: Abstract 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. PubDate: 2024-02-17