scholarly journals Bounds for GL3L-functions in depth aspect

2019 ◽  
Vol 31 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Qingfeng Sun ◽  
Rui Zhao
Keyword(s):  
Cusp Form ◽  
Prime Power ◽  
Dirichlet Character ◽  
Implied Constant ◽  
Maass Cusp Form ◽  
Primitive Dirichlet Character

AbstractLet f be a Hecke–Maass cusp form for {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor {\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity boundL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon}for any {\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

scholarly journals Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

2019 ◽  
Author(s):  
Keshav Aggarwal ◽  
Yeongseong Jo ◽  
Kevin Nowland
Keyword(s):  
Positive Integer ◽  
Cusp Form ◽  
Moment Method ◽  
Dirichlet Character ◽  
Delta Method ◽  
Primitive Dirichlet Character ◽  
International Audience ◽  
Subconvexity Bounds ◽  
Second Moment Method ◽  
Level Aspect

International audience Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P \sim M^{\eta}$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1]{2}, f \otimes \chi)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\chi$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

Cancellation of Cusp Forms Coefficients over Beatty Sequences on GL(m)

2011 ◽  
Vol 54 (4) ◽  
pp. 757-762
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form ◽  
Beatty Sequences

AbstractLet A(n1, n2, … , nm–1) be the normalized Fourier coefficients of a Maass cusp form on GL(m). In this paper, we study the cancellation of A(n1, n2, … , nm–1) over Beatty sequences.

scholarly journals An explicit upper bound for |L(1,χ)| when χ(2) = 1 and χ is even

2016 ◽  
Vol 12 (08) ◽  
pp. 2299-2315 ◽  
Author(s):  
Sumaia Saad Eddin
Keyword(s):  
Upper Bound ◽  
Dirichlet Character ◽  
Primitive Dirichlet Character

Let [Formula: see text] be a primitive Dirichlet character of conductor [Formula: see text] and let us denote by [Formula: see text] the associated [Formula: see text]-series. In this paper, we provide an explicit upper bound for [Formula: see text] when [Formula: see text] is a primitive even Dirichlet character with [Formula: see text].

scholarly journals Rapid computation of L-functions attached to Maass forms

2018 ◽  
Vol 14 (05) ◽  
pp. 1459-1485 ◽  
Author(s):  
Andrew R. Booker ◽  
Holger Then
Keyword(s):  
Cusp Form ◽  
Critical Line ◽  
Maass Forms ◽  
Maass Cusp Form

Let [Formula: see text] be a degree-[Formula: see text] [Formula: see text]-function associated to a Maass cusp form. We explore an algorithm that evaluates [Formula: see text] values of [Formula: see text] on the critical line in time [Formula: see text]. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.

On a Congruence Related to Character Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 431-439 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Prime Power ◽  
Dirichlet Character ◽  
Character Sums ◽  
Arithmetic Progressions ◽  
Solution Set ◽  
Character Sum ◽  
Equal Degree

AbstractIf χ is a Dirichlet character to a prime-power modulus pα, then the problem of estimating an incomplete character sum of the form ∑1≤x≤h χ (x) by the method of D. A. Burgess leads to a consideration of congruences of the typef(x)g'(x) - f'(x)g(x) ≡ 0(pα),where fg(x) ≢ 0(p) and f, g are monic polynomials of equal degree with coefficients in Ζ. Here, a characterization of the solution-set for cubics is given in terms of explicit arithmetic progressions.

scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

scholarly journals Complex numbers similar to the generalized Bernoulli numbers and their applications

2021 ◽  
Vol 50 ◽  
pp. 15-26
Author(s):  
Brahim Mittou ◽  
Abdallah Derbal
Keyword(s):  
Dirichlet Character ◽  
Bernoulli Numbers ◽  
Complex Numbers ◽  
Primitive Dirichlet Character

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

On sums of Fourier coefficients of Maass cusp forms

2017 ◽  
Vol 13 (05) ◽  
pp. 1233-1243 ◽  
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficient ◽  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form

Let [Formula: see text] be a Hecke–Maass cusp form, and [Formula: see text] be its [Formula: see text]th Fourier coefficient at the cusp infinity. In this paper, we are interested in the estimation on sums [Formula: see text] for [Formula: see text] We are able to improve previous results by introducing some inequalities concerning Fourier coefficients and other techniques.

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