Holomorphic Cusp Forms Research Articles

Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Eigenspaces of newforms with nontrivial character

Let [Formula: see text] denote the space of holomorphic cuspforms with Dirichlet character [Formula: see text] and modular subgroup [Formula: see text]. We will characterize the space of newforms [Formula: see text] as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the [Formula: see text]-adic case which we then de-adelize into relations of classical Hecke operators.

Quadratic base change and resonance sums for holomorphic cusp forms on Γ0(N)

Let D,k be integers with D square free and k even. Let N be a positive integer so that (N,D)=1 when D has residue one modulo four and (N,4D)=1 when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum SX(α,β;π) attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by D is computed. First a Voronoi summation formula is derived that expresses SX(α,β;π) in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of SX(α,β;π) as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.

On the sign changes of the coefficients attached to triple product L-functions

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights [Formula: see text] and [Formula: see text] for the full modular group SL[Formula: see text], respectively. In this paper, we investigate the sign changes of the sequence [Formula: see text] in short intervals, where [Formula: see text] denote the normalized coefficients of triple product L-functions [Formula: see text] attached to f and [Formula: see text], respectively. We provide quantitative results for the number of sign changes of the sequences mentioned above in short intervals.

Structure of the space of polyharmonic Maass forms with an application to L-values

A polyharmonic Maass form introduced by Lagarias and Rhoades is a generalization of a harmonic Maass form in a sense. In this paper, we construct some bases for the spaces of polyharmonic Maass forms from the structure of the spaces of weakly holomorphic modular forms for prime levels or square-free levels. Furthermore, we relate the polyharmonic Maass forms of depth 1 or depth 3/2 to the L-series of holomorphic cusp forms for square-free levels.

On Möbius Functions from Automorphic Forms and a Generalized Sarnak’s Conjecture

Abstract In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of L -functions associated with cusp Hecke eigenforms f of even integral weight, and prove that N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon} holds for \sigma \geq 1407/1601 . This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range \sigma\geq 53/60 . Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums.

𝐿-values of harmonic Maass forms

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the “central L L -value” of the modular j j -invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this “ L L -value” as the value of an actual L L -series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L L -series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L L -values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L L -functions using test functions.

ESTIMATE FOR HIGHER MOMENTS OF CUSP FORM COEFFICIENTS OVER SUM OF TWO SQUARES COEFFICIENTS OVER SUM OF TWO SQUARES

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group Γ = SL(2, Z), respectively. Denote by λf (n) and λg(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we consider the summatory function X n=a2+b2≤x λf (n)iλg(n)j , for x ≥ 2, where a, b ∈ Z and i, j ≥ 1 are positive integers.

Shifted convolution sums of divisor functions with Fourier coefficients

Let f(z) be a holomorphic cusp form of weight κ for the full modular group SL2(Z). Denote its n-th normalized Fourier coefficient by λf(n). Let τk(n) denote that k-th divisor function with k≥4. In this paper, we consider the shifted convolution sum∑n≤Xτk(n)λf(n+h). We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h.

High power sums of Fourier coefficients of holomorphic cusp forms and their applications

<p>Let $ \lambda_f(n) $ be the $ n $th normalized Fourier coefficient of a holomorphic cusp form $ f $ for the full modular group. In this paper, we will established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $ \{\lambda_f(n)\} $ and $ \{\lambda_f(n)\lambda_g(n)\} $ in short intervals, and presented some quantitative results for the number of sign changes for $ n\leq x $.</p>

Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights

In this paper, we prove that the Hecke eigenvalue square for a holomorphic cusp form and the Piltz divisor functions are good weighting functions for the pointwise ergodic theorem. Additionally, we prove similar results for various other arithmetical functions in the last section.

On the average behavior of coefficients related to triple product $L$-functions

In this paper, we study the average behaviour of the coefficients of triple product $L$-functions and some related $L$-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\mathbbZ).$ Here we call $f(z)$ a primitive cusp form if it is an eigenfunction of all Hecke operators simultaneously.

The average of the exponential sums related to cusp forms

In this paper, we investigate the average [Formula: see text] where [Formula: see text] for any [Formula: see text] and [Formula: see text], with [Formula: see text] running over primitive holomorphic cusp forms of weight [Formula: see text] and prime level P. As a result, we prove pointwise uniform bounds with respect to [Formula: see text], [Formula: see text] for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum [Formula: see text], so long as the level of f varies. Concerning the applications of the main result, we consider the quadratic Waring problem associated to the coefficients of cusp forms exhibiting considerable cancellations.

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

AbstractLet f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$ . More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$ , where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$ .For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$ .

On the Local Maxima Behaviour of Hecke Eigenvalues and Its Applications

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 $${\lambda _{{\rm{sy}}{{\rm{m}}^m}f}}(n)$$ 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.

Divisor-bounded multiplicative functions in short intervals

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) rightarrow infty and where c = c_f ge 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 |lambda _f(n) |^2, where {lambda _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 hlog X, if h = h(X) rightarrow infty . We also generalize this result to sequences {|lambda _{pi }(n) |^2}_n, where lambda _{pi }(n) is the nth coefficient of the standard L-function of an automorphic representation pi with unitary central character for GL_m, m ge 2, provided pi 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 {|lambda _f(n) |^{alpha }}_n over intervals of length h(log X)^{c_{alpha }}, with c_{alpha } > 0 explicit, for any alpha > 0, as h = h(X) rightarrow infty . Finally, we show that the (non-multiplicative) Hooley Delta -function has average value gg log log X in typical short intervals of length (log X)^{1/2+eta }, where eta >0 is fixed.

The average behaviour of a hybrid arithmetic function associated to cusp form coefficients over certain sparse sequence

Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group ? = SL(2,Z), and let ?f (n), ?(n) and ?(n) be the nth normalized Fourier coefficient of the cusp form f , the sum-of-divisors function and the Euler totient function, respectively. In this paper, we investigate the asymptotic behaviour of the following summatory function Sj,b,c(x) := X n=a21 +a22 +a23 +a24 ?x (a1,a2,a3,a4)?Z4 ?j f (n)?b(n)?c(n), where j ? 2 is any given integer. In a similar manner, we also establish other similar results related to normalized coefficients of the symmetric power L-functions associated to holomorphic cusp form f.

Double square moments and bounds for resonance sums of cusp forms

Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 and k2 with Fourier coefficients λf(n) and λg(n), respectively. For real α≠0 and 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) of λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments of SX(f,g;α,β) over both f and g are nontrivially bounded when their weights k1 and k2 tend to infinity together. By allowing both f and g to move, these double moments are indeed square moments associated with automorphic forms for GL(4). By taking out a small exceptional set of f and g, bounds for individual SX(f,g;α,β) shall then be proved. These individual bounds break the resonance barrier of X5/8 for 1/6<β<1 and achieve a square-root cancellation for 1/3<β<1 for almost all f and g as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.