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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]

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 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]

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 establish a doubly-weighted vertical Sato-Tate law for GL(4) with explicit error terms. The main ingredient is an extension of the orthogonality relation for Maass cusp forms on GL(4) of Goldfeld, Stade, and Woodbury from spherical to general forms, and without their assumption of the Ramanujan conjecture for the error term. [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]

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]

In this article, we investigate a non-Archimedean analogue of a question of Atkin and Serre. More precisely, we derive lower bounds for the largest prime factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms of even weight k ≥ 2 , level N with integer Fourier coefficients. In particular, we show that for such a form f and for any real number ϵ > 0 , the largest prime factor of the p-th Fourier coefficient a f (p) of f, denoted by P (a f (p)) , satisfies P (a f (p)) > (log p) 1 / 8 (log log p) 3 / 8 - ϵ for almost all primes p. This improves on earlier bounds. We also investigate a number field analogue of a recent result of Bennett, Gherga, Patel and Siksek about the largest prime factor of a f (p m) for m ≥ 2. [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]

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length x (log x) A for any A > 0 , modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length x 1 / 2 + ϵ for any ϵ > 0 . [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 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]

Fix an integer κ ≥ 2 . Let P ≥ 2 be a prime, and F be the symmetric-square lift of a Hecke newform f ∈ S κ * (P) . We study the exponential sum L F (α) = ∑ n ∼ N A F (n , 1) e (n α) by implementing an average over a family in such a way to investigate the best possible magnitude of the level aspect bound for L F (α) . We prove a uniform bound with respect to any α ∈ R and the level parameter P , and present that there exist certain forms with fairly strong oscillations in L F (α) , if the associated level of f is allowed to vary. As applications, we consider the shifted convolution sums for GL (3) × GL (d) , for any d ≥ 2 , in a family as well as theWaring-Goldbach problem associated to Fourier coefficients of SL (3 , Z) -Maa β forms. [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]

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]

Let Γ be a subgroup of finite index of SL 2 ⁡ (퐙) . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ (2) , where we can consider convenient generalized Jacobians instead of J Γ . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves. [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]

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]

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]

We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms. [ABSTRACT FROM AUTHOR]

Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ (n) is odd if and only if n is an odd square, we only need to consider τ ⁢ (p 2 ⁢ n) for primes p and natural numbers n ≥ 1 . This is a rather delicate question. In this direction, we show that for any ϵ > 0 and integer n ≥ 1 , the largest prime factor of τ ⁢ (p 2 ⁢ n) , denoted by P ⁢ (τ ⁢ (p 2 ⁢ n)) , satisfies P ⁢ (τ ⁢ (p 2 ⁢ n)) > (log ⁡ p) 1 8 ⁢ (log ⁡ log ⁡ p) 3 8 - ϵ for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. [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, g be two distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 for the full modular group Γ = S L (2 , Z) , respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving λ f 2 i 1 (n) , λ f 2 (n j 1) and λ f 2 (n i 2) λ g 2 (n j 2) on average in terms of f, g respectively. Here i 1 ≥ 2 , j 1 ≥ 3 and i 2 , j 2 ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Let 2 ≤ j ≤ 8 be any fixed positive integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . Denote by λ sym 2 f (n) the nth normalized coefficient of the Dirichlet expansion of the symmetric square L-function L (sym 2 f , s) attached to f. In this paper, we are interested in the average behaviour of the following summatory function ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym 2 f j (a 2 + b 2 + c 2 + d 2) for x ≥ x 0 (sufficiently large), which improves and generalizes the recent works of Sharma and Sankaranarayanan (Res Number Theory 8:19, 2022, Rend Circ Mat Palermo II Ser 72:1399–1416, 2023). [ABSTRACT FROM AUTHOR]

In this paper, we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3) L-functions for all k ≥ 1, based on our recent work on the twisted first moment of central values in this family of L-functions. [ABSTRACT FROM AUTHOR]

Let H k ∗ be the set of all normalized primitive holomorphic cusp forms of even weight k ≥ 2 for S L 2 (Z) . For f ∈ H k ∗ , let λ sym j f (n) be the nth normalized Fourier coefficient of the jth symmetric power L-function attached to f. In this paper, we establish asymptotic formulas for the power sums of λ sym j f (n) and improve previous results. [ABSTRACT FROM AUTHOR]

In this paper, we study shells of the D 4 lattice with a slight generalization of spherical t-designs due to Delsarte–Goethals–Seidel, namely, the spherical design of harmonic index T (spherical T-design for short) introduced by Delsarte-Seidel. We first observe that, for any positive integer m, the 2m-shell of D 4 is an antipodal spherical { 10 , 4 , 2 } -design on the three dimensional sphere. We then prove that the 2-shell, which is the D 4 root system, is a tight { 10 , 4 , 2 } -design, using the linear programming method. The uniqueness of the D 4 root system as an antipodal spherical { 10 , 4 , 2 } -design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the D 4 lattice in terms of orthogonal transformations of the D 4 root system, and the uniqueness of the D 4 lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the D 4 lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed. [ABSTRACT FROM AUTHOR]

In monstrous moonshine, the replication formula and the Hecke operator played a central role. We generalize the replication formula and the Hecke operator to higher genus modular curves, with an eye toward extending moonshine to these cases. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera and obtain uniform proofs for numerous arithmetic properties of Fourier coefficients of modular functions of arbitrary level, which have been proved only for special cases of curves of genus zero or small prime levels. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for S L (2 , Z) with normalized Fourier coefficients λ f (n) . Let χ be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form S f , χ (N) : = ∑ n ∈ Z λ f (n) χ (n) W (n N). We show that, for 0 < θ < 1 10 , S f , χ (N) ≪ f , ϵ N 3 / 4 + θ / 2 p 1 / 6 (p N) ϵ + N 1 - θ (p N) ϵ , which is non-trivial if N ≥ p 2 / 3 + α + ϵ where α = = 4 θ 1 - 6 θ . Previously, such a bound was known for N ≥ p 3 / 4 + ϵ. [ABSTRACT FROM AUTHOR]

We investigate the average behavior of the nth normalized Fourier coefficients of the jth (j ≽ 2 be any fixed integer) symmetric power L-function (i.e., L(s,symjf)), attached to a primitive holomorphic cusp form f of weight k for the full modular group S L (2 , ℤ) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum S j ∗ : = ∑ 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 2 (a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2) , where x is sufficiently large, and L (s , sym j f) : = ∑ n = 1 ∞ λ sym j f (n) n s . When j = 2, the error term which we obtain improves the earlier known result. [ABSTRACT FROM AUTHOR]

Let f be a fixed Maass form for SL3 (ℤ) with Fourier coefficients Af(m, n). Let g be a Maass cusp form for SL2 (ℤ) with Laplace eigenvalue 1 4 + k 2 and Fourier coefficient λg(n), or a holomorphic cusp form of even weight k. Denote by SX(f × g, α, β) a smoothly weighted sum of Af(1, n)λg(n)e(αnβ) for X < n < 2X, where α ≠ 0 and β > 0 are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of SX(f × g, α, β) over g as k tends to ∞ with X. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak. [ABSTRACT FROM AUTHOR]