Primitive Cusp Form Research Articles

On the Fourier Coefficients for General Product 𝐿-Functions

Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

An approximation formula for the shifted cubic moment of automorphic L-functions in the weight aspect

Abstract Consider the family of automorphic L-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted L-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.

Averages of long Dirichlet polynomials with modular coefficients

AbstractWe study the moments ofL‐functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of alongDirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

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.

Weyl-type bounds for twisted GL(2) short character sums

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for \(SL(2,{\mathbb {Z}})\) with normalized Fourier coefficients \(\lambda _{f}(n)\). Let \(\chi \) 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 $$\begin{aligned} S_{f,\chi }(N):= \mathop \sum _{n \in {\mathbb {Z}}}\lambda _{f}(n)\chi (n)W\Big (\frac{n}{N}\Big ). \end{aligned}$$We show that, for \(0<\theta < \frac{1}{10}\), $$\begin{aligned} S_{f,\chi }(N) \ll _{f,\epsilon }N^{3/4 + \theta /2}p^{1/6}(pN)^{\epsilon } + N^{1-\theta }(pN)^{\epsilon }, \end{aligned}$$which is non-trivial if \(N \ge p^{2/3 + \alpha + \epsilon }\) where\(\alpha = = \frac{4\theta }{1-6\theta }\). Previously, such a bound was known for \(N \ge p^{3/4 + \epsilon }.\)

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ &gt; 0 \epsilon&gt;0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Nondiagonal terms in the second moment of automorphic -functions

We investigate the exact formula for the second moment of central -values associated to primitive cusp forms of level one and weight , , which was proved by Kuznetsov in 1994. Nondiagonal terms in this formula are expressed in terms of several convolution sums of divisor functions. We investigate the main terms of the convolution sums and show that they cancel out. Bibliography: 8 titles.

Mixed moment of GL(2) and GL(3)L‐functions

Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the symmetric square $L$-function $ L(sym^2\mathfrak{f}, 1/2)$, relating it to the dual mixed moment of the double Dirichlet series and the Riemann zeta function weighted by the ${}_3F_{2}$ hypergeometric function. Analysing the corresponding special functions by the means of the Liouville-Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded by $\log^3k$.

The cubic moment of automorphic L-functions in the weight aspect

We prove an explicit formula for the cubic moment of central values of automorphic L-functions associated to primitive cusp forms of level one and large weight. The resulting explicit formula contains the main term predicted by the random matrix theory conjectures, while the error term is expressed as the fourth moment of the Riemann zeta function weighted by the F23 hypergeometric function. As a corollary, we derive a new upper bound for the cubic moment improving the previous result of Peng. Furthermore, we obtain a new subconvexity estimate for automorphic L-functions in the weight aspect.

Level aspect subconvexity for twisted L-functions

Let M be a square-free integer and P be a prime such that (M,P)=1. We prove a new level aspect hybrid subconvexity bound for L(1/2,f⊗χ) where f is a primitive (either holomorphic or Maass) cusp form of level P and χ a primitive Dirichlet character modulo M satisfying P∼Mη with 0<η<3/2−3ϑ, where ϑ is the current known approximation towards the Ramanujan-Petersson conjecture. Particularly we obtain a stronger subconvexity for max⁡{6ϑ,1/2}<η<(3−6ϑ)/2 which has not been covered by the work of Blomer-Harcos [0].

On mean values of mollifiers and L-functions associated to primitive cusp forms

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.

On M-Functions Associated with Modular Forms

Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or $(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $\chi$ or $f$ vary. First, for a quasi-character $\psi\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit for the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi, s)),$ when $f$ is fixed and $\chi$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes \chi,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h_f \psi(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimes\chi, s).$

Determining Hilbert modular forms by central values of Rankin-Selberg convolutions: The level aspect

In this paper, we prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of level $\mathfrak{q}$ for infinitely many prime ideals $\mathfrak{q}$. This result is a generalization of a theorem of Luo to the setting of totally real number fields.

On the coefficients of triple product $L$-functions

In this paper, we investigate the average behavior of coefficients of the triple product $L$-function $L(f \otimes f \otimes f,s)$ attached to a primitive holomorphic cusp form $f(z)$ of weight~$k$ for the full modular group $SL(2, \Z )$. Here we call $f(z)$ a primitive cusp form if it is an eigenfunction of all Hecke operators simultaneously.

Hybrid bounds for Rankin–Selberg L-functions

Let M be a square-free integer and P be a prime such that (P,M)=1. We prove a new hybrid bound for L(12,f⊗g) where f is a primitive holomorphic cusp form of level M and g a primitive (either holomorphic or Maass) cusp form of level P satisfying P∼Mη with 0<η<2/15. Particularly in the range β<η<(2−32β)/15 with β=11/4875 we present a strengthened level aspect hybrid subconvexity bound for L(12,f⊗g) relative to the current bounds obtained by Holowinsky–Munshi [11] and Ye [27].

Values of automorphic L-functions with prime power level at the point $$s=1$$ s = 1

We discuss in this work the distributions of values of L(1, f), where f is a primitive cusp form whose level is a prime power. We prove the upper bound part of the Montgomery–Vaughan’s first conjecture and give a weaker version of the lower bound part for automorphic L-functions in this case. We establish an unweighted trace formula in aspect of prime power level in our proof.

On special values of certain L-functions, II

We prove an algebraicity result concerning special values at critical points, in the sense of Deligne, of tensor product $L$-functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite, and, cuspidal representations of ${\rm GL}(2)$ corresponding to primitive cusp forms, over totally real number fields. We also prove the reciprocity law, i.e., the equivariance under the action of ${\rm Gal}({\overline{\Bbb Q}}/{\Bbb Q})$, for the special values. In the appendix, the second author calculates the Deligne periods for such $L$-functions, assuming the existence of corresponding motives and the automorphic transfer to a quasi-split form of the special orthogonal group. Our result conforms with the celebrated conjecture of Deligne on special values of motivic $L$-functions.

Higher Moments of Fourier Coefficients of Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

Gaps between zeros of [formula omitted]L-functions

Let L(s,f) be an L-function associated to a primitive (holomorphic or Maass) cusp form f on GL(2) over Q. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of L(1/2+it,f) and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of L(s,f) along the critical line that are at least 3=1.732… times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive L-function of the Selberg class. In particular, when f is a primitive holomorphic cusp form on GL(2) over Q, we prove that there are infinitely many gaps between consecutive zeros of L(s,f) along the critical line that are at most 0.823 times the average spacing.