Search

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 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]

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]

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 and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group S L (2 , Z) . We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑ n ≤ x λ sym i f × sym j g (n) , ∑ n ≤ x λ f (n i) λ g (n j) , ∑ n ≤ x | λ sym i f × sym j g (n) | , ∑ n ≤ x | λ f (n i) λ g (n j) | , and ∑ n ≤ x max { | λ sym i f × sym j g (n) | 2 φ , | λ sym i f × sym j g (n + h) | 2 φ } , where φ > 0 and h is a fixed positive integer. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform f and a matrix σ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of f | | k σ . [ABSTRACT FROM AUTHOR]

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that lim x → ∞ # { 1 ≤ n ≤ x ∣ τ (n) ≠ 0 } x > 1 - 1.15 × 10 - 12 , where τ (n) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ (n) ≠ 0 . [ABSTRACT FROM AUTHOR]

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the kth moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the p m th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak. [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]

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]

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 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]

We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form. [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]

We investigate the linear dependence among odd periods of modular forms. We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three odd periods are linearly independent. We also prove an asymptotic result for an arbitrary number of odd periods. The results are achieved by studying cuspidal projections of products of two Eisenstein series. [ABSTRACT FROM AUTHOR]

Let a E (n) be the normalized Frobenius traces. For any multiplicative function f satisfying some mild conditions, we investigate the sum ∑ n = 1 N f (n) a E (n) and obtain a nontrivial bound. We also prove cancelation in the sum of Frobenius traces twisted by the coefficients of automorphic L-function on GLm. [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]

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]

We formulate a precise conjecture about the size of the L ∞ -mass of the space of Jacobi forms on H n × C g × n of matrix index S of size g. This L ∞ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when n = 1 , g ≥ 1 . When n = 1 and g = 1 , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values L (1 / 2 , f ⊗ χ D) with f varying over newforms of level a prime p and even weight k as k , p → ∞ and D being (explicitly) polynomially bounded by k, p. Here χ D is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k 5 / 2 by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded H × H is k 2 . In an appendix, the same question is answered for the pullbacks of the whole space S k 2 , the size here being k 3 . [ABSTRACT FROM AUTHOR]

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h (log X) c , with h = h (X) → ∞ and where c = c f ≥ 0 is determined by the distribution of { | f (p) | } p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of | λ f (n) | 2 , where { λ f (n) } n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length h log X , if h = h (X) → ∞ . We also generalize this result to sequences { | λ π (n) | 2 } n , where λ π (n) is the nth coefficient of the standard L-function of an automorphic representation π with unitary central character for G L m , m ≥ 2 , provided π satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments { | λ f (n) | α } n over intervals of length h (log X) c α , with c α > 0 explicit, for any α > 0 , as h = h (X) → ∞ . Finally, we show that the (non-multiplicative) Hooley Δ -function has average value ≫ log log X in typical short intervals of length (log X) 1 / 2 + η , where η > 0 is fixed. [ABSTRACT FROM AUTHOR]