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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]

Let a( n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of $$SL_2({\mathbb R})$$ . This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of $$a(n)a(n+r)$$ for a fixed positive integer r. [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 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]

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]

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 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]

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]

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ 0 (N) with N odd square-free. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level. [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]

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]

Let f and g be 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 n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function λ f (n i) λ f (n j) and λ f (n i) λ g (n j) for any integers i , j ≥ 1 . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 2 + ε for arbitrarily small ε > 0 . In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients λ f × f × f (n) and λ f × f × g (n) of triple product L-functions L (f × f × f , s) and L (f × f × g , s) , respectively. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the n-th normalized Fourier coefficient of primitive holomorphic cusp form of even integral weight k ≥ 2 for the full modular group S L (2 , Z) . In this paper, we first refine related previous results on the l-th power sums of the Fourier coefficient λ f (n) . Then we establish asymptotic formulas of the variance of λ f l (n) . [ABSTRACT FROM AUTHOR]

Let f, g be two distinct normalized primitive holomorphic cusp forms of weights k 1 , k 2 and levels N 1 , N 2 , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the first change of sequences { λ f (n i) } n ∈ N and { λ f (n j) λ g (n l) } n ∈ N with positive integers i ≥ 2 and j , l ≥ 1 . And we also derive the lower bounds for the sequences { λ f (n i) } n ∈ N , { λ f (n j) λ f (n l) } n ∈ N of the same signs in the interval [1, x], which remains the best possible results in terms of order of magnitude, where i ≥ 2 , j , l ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Suppose that H ∗ is the set of all primitive cusp forms f of even integral weight k ≥ 2 for the full modular group S L 2 (Z) . In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product L-function L (s , f ⊗ f ⊗ f) and the related L-function L (s , sym 2 f ⊗ f) attached to f on average, which improves previous results. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of the coefficients appearing in the q-series expansion of ∏ n ≥ 1 [ (1 - q n) / (1 - q pn) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p = 3 case. [ABSTRACT FROM AUTHOR]

A natural variant of Lehmer's conjecture that the Ramanujan τ -function never vanishes asks whether, for any given integer α , there exist any n ∈ Z + such that τ (n) = α . A series of recent papers excludes many integers as possible values of the τ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for τ (n) . We synthesize these results and methods to prove that if 0 < α < 100 and α ∉ T : = { 2 k , - 24 , - 48 , - 70 , - 90 , 92 , - 96 } , then τ (n) ≠ α for all n > 1 . Moreover, if α ∈ T and τ (n) = α , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that τ (n) > 100 for all n > 2 . [ABSTRACT FROM AUTHOR]

Let f be a primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . And let λ f (n) be the nth normalized Fourier coefficient of f. Let a(n) be the function with squarefull kernel. Let j ≥ 2 be any fixed positive integer. In this paper, we establish the following result: ∑ n ≤ x a (n) λ f (n + 1) 2 j = C f , j x P A j - 1 (log x) + O (x 1 - 1 2 (3 · 2 2 j - A j) + ε ) , where P A j - 1 (t) is polynomial of t with degree A j - 1 , and C f , j > 0 is some suitable constant that can be explicitly evaluated. [ABSTRACT FROM AUTHOR]

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively. [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]

The purpose of this paper is to study the masses over CM points on Drinfeld–Stuhler modular curves and definite Shimura curves over global function fields. After connecting the CM masses with the modified Hurwitz class numbers, we show that the generating function of these masses is a particular metaplectic theta series. Applying the Poisson summation formula, we are able to describe explicitly the meromorphic continuation of the Mellin transform of the generating theta series. This enables us to derive not only a Gauss-type mean value formula of the average masses in question, but also provides “new” class number relations in the positive characteristic world. [ABSTRACT FROM AUTHOR]

In this paper, we will first establish a power saving result for the shifted convolution sums of k-th powers and the normalized Fourier coefficients of SL 2 (Z) cusp forms. Later we will generalize the result to higher rank cases. [ABSTRACT FROM AUTHOR]

In [4], the first and second authors studied the signatures of Dirichlet characters and elliptic curves. In this paper we improve their lower bound estimates in the case of elliptic curves. We also introduce and investigate integral and interval d-constant signatures for cusp forms, and prove lower bound results for the number of twisted cusp forms having positive or negative signatures. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to prove an asymptotic estimate for sums of Fourier coefficients of holomorphic cusp forms over integers without large prime factors by using a Chebyshev-type functional equation. [ABSTRACT FROM AUTHOR]

Let f be an adelic Hilbert cusp form of weight k and level n over a totally real number field F. In this paper, we study the sign changes in the Fourier coefficients of f when restricted to square-free integral ideals and integral ideals in "arithmetic progression". In both cases we obtain qualitative results and in the former case we obtain a quantitative result as well. Our results are general in the sense that we do not impose any restriction to the totally real number field F, the weight k or the level n . [ABSTRACT FROM AUTHOR]

Let f be a Hecke cusp form of even integral weight k for the full modular group S L (2 , Z) . Denote by λ sym j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function associated with f. In this paper, we derive a upper bound for the first negative coefficient for sequence { λ sym j f (n) } n ≥ 1 in terms of the weight of f. Furthermore, we also investigate the length of the maximal string of terms with the same sign in an interval [1, x]. [ABSTRACT FROM AUTHOR]

Let j ≥ 2 be a given 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 j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L (sym j f , s) attached to f. In this paper, we are interested in the average behaviour of the following sum ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym j f 2 (a 2 + b 2 + c 2 + d 2) , where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence. [ABSTRACT FROM AUTHOR]

In this paper we estimate sums of the form ∑ n ≤ X | a Sym m π (| f (n) |) | , for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f (x) ∈ Z [ x ] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4 . Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves. [ABSTRACT FROM AUTHOR]

Let f(z) be a holomorphic Hecke eigenform of even weight k for SL(2, ℤ), and denote by L(s, symmf) the corresponding symmetric power L-function associated with f. Denote by ⋋symmf (n) the nth normalized coefficient of L(s, symmf). In this paper, we investigate the sum Σn≤x⋋symmf (n) for m ≥ 2 and get the uniform upper bound extending the previous results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide a complete and refined study of the standard L-functions L (π , Std , s) for certain non-generic cuspidal automorphic representations π of G 2 (A) . For a cuspidal automorphic representation π of G 2 (A) that corresponds to a modular form φ of level one and of even weight on G 2 , we explicitly define the completed standard L-function, Λ (π , Std , s) . Assuming that a certain Fourier coefficient of φ is nonzero, we prove the functional equation Λ (π , Std , s) = Λ (π , Std , 1 - s) . Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal. [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 Hk denote the space of holomorphic Hecke cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Let λf (n), λg(n), and λh(n) be the nth Hecke-normalized Fourier coefficients of three distinct holomorphic Hecke cusp forms f(z), g(z), h(z) ∈ H k i ′ , i ′ = 1 , 2 , 3 , respectively. In this paper, we establish an asymptotic formula for the summatory function (under suitable conditions)Σn≤x λ f 2 (ni) λ g 2 (nj), which generalizes the results of Lü [17]. As an application, we also consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms of the type Σn≤x |λf (n)λg(ni)| and Σn≤x|λf (n)λg(ni)λh(nj)|, which also generalizes the work of Lü [17]. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper we compute the Fourier coefficients of the Siegel Eisenstein series of degree 2, level p with the trivial or the quadratic character, associated to a certain cusp. For that we need to define the p-factor of the special type of Siegel series with character. [ABSTRACT FROM AUTHOR]

Duke and Imamoglu showed that the Saito–Kurokawa lift for Siegel modular forms can in an elegant manner be obtained from a converse theorem by Imai using spectral analysis of the hyperbolic Laplacian. In an earlier paper we gave a simplified approach without any appeal to spectral analysis. Here we want to show, that this generalizes to some orthogonal groups of signature (2 , m + 2) with m > 2 . [ABSTRACT FROM AUTHOR]

Let k ≥ 2 and N be positive integers and let χ be a Dirichlet character modulo N. Let f(z) be a modular form in M k (Γ 0 (N) , χ) . Then we have a unique decomposition f (z) = E f (z) + S f (z) , where E f (z) ∈ E k (Γ 0 (N) , χ) and S f (z) ∈ S k (Γ 0 (N) , χ) . In this paper, we give an explicit formula for E f (z) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities. [ABSTRACT FROM AUTHOR]

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group G (A K) , which we refer to as Fourier coefficients associated to the data of a 'Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are 'Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Rankin–Selberg problem. That is, we break the well known Rankin–Selberg's bound on the error term of the second moment of Fourier coefficients of a GL (2) cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions. [ABSTRACT FROM AUTHOR]

Let λ π (1 , ... , 1 , n) be the normalized Fourier coefficients of an even Hecke–Maass form π for S L (m , Z) with m ≥ 3 , and r 3 (n) = # { (n 1 , n 2 , n 3) ∈ Z 3 : n = n 1 2 + n 2 2 + n 3 2 } . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients λ π (1 , ... , 1 , n) and r 3 (n) , which improves previous results (even under the generalized Ramanujan conjecture). [ABSTRACT FROM AUTHOR]