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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]

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. [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 consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on S L 2 (Z) by Kaneko and Koike as orthogonal polynomials and clarify their properties. Using them, we show that the normalized extremal quasimodular forms have a certain expression by the linear functional corresponding to the Atkin inner product and prove an unexpected connection between generalized Faber polynomials, which are closely related to certain bases of the vector space of weakly holomorphic modular forms, and normalized extremal quasimodular forms. In particular, we reveal that the orthogonal polynomial expansion coefficients of the generalized Faber polynomials by the Atkin-like polynomials appear in the Fourier coefficients of normalized extremal quasimodular forms multiplied by certain (weakly) holomorphic modular forms. [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]

Let g ∈ S k (Γ 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 α g (f) . When g has complex multiplication by an imaginary quadratic field K and p is split in O K , it is known that α g (f) is zero. On the other hand, we do not know much about α g (f) for an inert prime p. In this paper, we prove that α g (f) is a p-adic unit when p is inert in O K and dim C S k (Γ 0 (N)) = 1 . [ABSTRACT FROM AUTHOR]

Let f be any normalized Hecke eigen forms with even integral weight k ≥ 2 for the full modular group S L (2 , Z) , and χ be a primitive Dirichlet character modulo q. Let L f (s , χ) be the automorphic L-function attached to f and χ . In this paper, we study the mean-square estimate of L f (s , χ) weighted by incomplete character sums, via using properties of Fourier coefficients and analytic methods, and establish an asymptotic formula which refines the previous results. [ABSTRACT FROM AUTHOR]

Recently, Mangerel extended the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions in typical short intervals. In this paper, we combine Mangerel's result with Halász-type result recently established by Granville, Harper and Soundararajan to consider the distribution of a class of multiplicative functions in short intervals. First, we prove cancellation in the sum of the coefficients of the standard L-function of an automorphic irreducible cuspidal representation of GL m over Q with unitary central character in typical intervals of length h (log X) c with h = h (X) → ∞ and some constant c > 0 (under Vinogradov–Korobov zero-free region and GRC). Then we also establish a non-trivial bound for the product of divisor-bounded multiplicative functions with the Liouville function in arithmetic progressions over typical short intervals. [ABSTRACT FROM AUTHOR]

Jeon, Kang, and Kim defined the Zagier lifts between harmonic weak Maass forms of negative integral weights and half integral weights. These lifts were defined by establishing that traces related to cycle integrals of harmonic weak Maass forms of integral weights appear as Fourier coefficients of harmonic weak Maass forms of half integral weights. For fundamental discriminants d and δ , they studied δ -th Fourier coefficients of the d-th Zagier lift with respect to the condition that d δ is not a perfect square. For d δ being a perfect square, the interpretation of coefficients in terms of traces is not possible due to the divergence of cycle integrals. In this paper, we provide an alternate definition of traces called modified trace in the condition that d δ is a perfect square and interpret such coefficients in terms of the modified trace. [ABSTRACT FROM AUTHOR]

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O (b + , b -) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π , which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp 2 (R) cusp forms by using the Borcherds' method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions. [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]

We obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng. [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]