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In this paper, we build on the recent results of Frank Garvan and Rishabh Sarma as well as classical results of Bruce Berndt in order to establish the 11-dissection of the deviations of the rank and crank modulo 11. Using our new dissections, we re-derive the results of Garvan, Atkin, Swinnerton-Dyer, Hussain, Ekin and Chern. By developing and exploiting positivity conditions for quotients of theta functions, we will also prove new rank–crank inequalities and make several conjectures, one of which was recently solved by Kathrin Bringmann and Badri Vishal Pandey. For other applications of our methods, in this paper, we will also prove new congruences for rank moments as well as the Andrews' smallest parts function and Eisenstein series. [ABSTRACT FROM AUTHOR]

Let f(z) be a holomorphic or Maass cusp form for the full modular group S L (2 , Z) . In this paper, we consider the asymptotic behavior of the sums U w (x) = ∑ n ≤ x λ w , f × f (n) , which can be regarded as the general divisor problems considered by Kanemitsu et al. (Monatsh Math 136(1):17–34, 2002), and get some new results which improve previous results. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct normalized primitive Hecke cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f ⊗ f ⊗ f ⊗ g (n) and λ sym 2 f ⊗ f ⊗ g (n) the nth normalized coefficients of the automorphic L-functions L (f ⊗ f ⊗ f ⊗ g , s) and L (sym 2 f ⊗ f ⊗ g , s) , respectively. In this paper, we are interested in the average behavior of the coefficients λ f ⊗ f ⊗ f ⊗ g (n) and λ sym 2 f ⊗ f ⊗ g (n) on a primitive integral binary quadratic form with negative discriminant whose class number is 1, and we also provide the asymptotic formulae of these summatory functions. As an application, we also consider the number of sign changes of the sequences { λ f ⊗ f ⊗ f ⊗ g (n) } n ⩾ 1 and { λ sym 2 f ⊗ f ⊗ g (n) } n ⩾ 1 on the same binary quadratic form in short intervals. [ABSTRACT FROM AUTHOR]

Let K 3 be a non-normal cubic extension over Q . And let τ k K 3 (n) denote the k-dimensional divisor function in the number field K 3 / Q . In this paper, we investigate the higher moments of the coefficients attached to the Dedekind zeta function over sum of two squares of the form ∑ n 1 2 + n 2 2 ≤ x (τ k K 3 (n 1 2 + n 2 2)) l , where n 1 , n 2 ∈ Z , and k ≥ 2 , l ≥ 2 are any fixed integers. [ABSTRACT FROM AUTHOR]

Lehmer's conjecture that τ (n) never vanishes remains open. In the spirit of this conjecture, it is natrual to consider for a given integer α , whether there exists n ∈ Z + such that τ (n) = α . Several recent papers exclude many odd α involving one prime factor or small values of the τ -function using the theory of congruences of τ (n) , Lucas sequences, integer points of elliptic curves, and the Thue equation. In this paper, we generalize the results and methods to infinitely many odd α involving two prime factors, showing that τ (n) ≠ ± p 1 p 2 b for arbitrary b ∈ Z , where 3 ≤ p 1 , p 2 ≤ 31 are primes such that p 1 p 2 ≤ 247 or p 1 = p 2 . Moreover, together with the previous results, we get τ (n) ≠ 3 b 1 5 b 2 7 b 3 11 b 4 13 b for b ∈ Z , and b i ∈ { 0 , 1 } , i = 1 , 2 , 3 , 4 . Besides, we show that | τ (n) | > | τ (3) | for all odd τ (n) , n ≠ 1 . [ABSTRACT FROM AUTHOR]

Let f and g be two distinct normalized primitive Hecke–Maass cusp forms of weight zero with Laplacian eigenvalues 1 4 + u 2 and 1 4 + v 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the non-trivial upper bounds for the sum ∑ n ∈ S | λ f (n) λ g (n) | , where S is a suitable subset of Z + ∩ [ 1 , x ] with certain properties. [ABSTRACT FROM AUTHOR]

In this paper, we will prove a non-trivial bound for the weighted average version of a shifted convolution sum for G L (3) × G L (2) , i.e. for arbitrary small ϵ > 0 and X 1 / 4 + δ ≤ H ≤ X with δ > 0 , we prove 1 H ∑ h = 1 ∞ λ f (h) V h H ∑ n = 1 ∞ λ π (1 , n) λ g (n + h) W n X ≪ X 1 - δ + ϵ , where V, W are smooth and compactly supported functions, λ f (n) , λ g (n) and λ π (1 , n) are the normalized n-th Fourier coefficients of holomorphic or Hecke–Maass cusp forms f, g for S L (2 , Z) , and Hecke–Maass cusp form π for S L (3 , Z) . [ABSTRACT FROM AUTHOR]

Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λSymmf (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Symm f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λSymmf (n) and establish its asymptotic formula. [ABSTRACT FROM AUTHOR]

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 Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. And set Q (x) a primitive integral positive-definite binary quadratic form of fixed discriminant D < 0 with the class number h (D) = 1 . In this paper, we establish a lower bound for the analytic density of the set { p : p = Q (x) for some x ∈ Z 2 , λ f (p i) λ f (p j) < λ g (p i) λ g (p j) } , where j ⩾ 1 , 0 ⩽ i ⩽ j are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of λ f (p j) and λ g (p j) in a given interval supported on the binary quadratic form Q (x) . Similar problems for triple product L-functions are also studied. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

Let π be a cuspidal automorphic representation of GL 2 (A Q) associated to holomorphic forms with Fourier coefficients a π (n) . Consider an automorphic representation Π which is equivalent to sym m π or π × sym m π . We establish uniform upper bounds for ∑ n ⩽ X | a Π (| f (n) |) | , where f (x) ∈ Z [ x ] is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of determining formulas for the number of representations of a natural number n by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain conditions on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain the modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain the modularity property of the generating function of mixed forms involving figurate numbers (including the squares and triangular numbers) with coefficients and forms of the type m 2 + m n + n 2 with coefficients. In particular, we show the modularity of the generating function of odd number of squares and odd number of triangular numbers (with coefficients). As a consequence, explicit formulas for the number of representations of these mixed forms are obtained using a basis of the corresponding space of modular forms of integral weight. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in Ono et al. (Aequat Math 50:73–94, 1995). In 2016, Xia et al. (Int J Number Theory 12:945–954) considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the (p, k) parametrization method. We also derive these 21 formulas using our method and further obtain as a consequence, the (p, k) parametrization of the Eisenstein series E 4 (τ) and its duplications. It is to be noted that the (p, k) parametrization of E 4 and its duplications were derived by a different method in [3, 8]. We illustrate our method with several examples. [ABSTRACT FROM AUTHOR]

We consider LG(s) to be the L-function attached to a particular automorphic form G on GL(6). We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg L-function LG×G(s). As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of LG×G(s). [ABSTRACT FROM AUTHOR]

Let j ⩾ 2 be a given integer. Let Hk* be the set of all normalized primitive holomorphic cusp forms of even integral weight k ⩾ 2 for the full modulo group SL(2, ℤ). For f ∈ Hk*, denote by λ sym j f (n) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, symjf)) attached to f. We are interested in the average behaviour of the sum ∑ n = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 ⩽ x (a 1 , a 2 , a 3 , a 4 , a 5 , a 6) ∈ ℤ 6 λ sym j f (n) , 2 where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023). [ABSTRACT FROM AUTHOR]

We give a general conjecture concerning the existence of Eisenstein congruences between weight k ≥ 3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ of conductor N. The special cases M = 1 and M = p prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof. [ABSTRACT FROM AUTHOR]

Let f be a cusp form for SL (2 , Z) , and λ f (n) , the n-th Fourier coefficient of the L-function attached to f. Here a new bound is established for the exponential sum ∑ n ≤ x Λ (n) λ f (n) e (n θ α) , where 0 < θ < 1 and α ≠ 0 is a fixed parameter. This improves previous results when 1 / 2 ≤ θ < 1 . [ABSTRACT FROM AUTHOR]