scholarly journals The third-power moment of the Riesz mean error term of symmetric square $ L $-function

2021 ◽  
Vol 6 (9) ◽  
pp. 9436-9445
Author(s):  
Rui Zhang ◽  
◽  
Xiaofei Yan
Keyword(s):  
Error Term ◽  
Power Moment ◽  
The Third ◽  
Mean Error ◽  
Riesz Mean ◽  
Symmetric Square

scholarly journals Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2036
Author(s):  
Rui Zhang ◽  
Xue Han ◽  
Deyu Zhang
Keyword(s):  
Error Term ◽  
Asymptotic Formulas ◽  
Short Intervals ◽  
Mean Error ◽  
Riesz Mean ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Power Moments

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2,Z) and let L(s,sym2f)=∑n=1∞cnn−s,ℜs>1 denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form Dρ(x;sym2f)=L(0,sym2f)Γ(ρ+1)xρ+Δρ(x;sym2f) and derive the asymptotic formulas for ∫T−HT+HΔρk(x;sym2f)dx, when k≥3.

scholarly journals On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

2008 ◽  
Vol 51 (1) ◽  
pp. 148-160 ◽  
Author(s):  
Yoshio Tanigawa ◽  
WenGuang Zhai ◽  
DeYu Zhang
Keyword(s):  
Error Term ◽  
Mean Error ◽  
Riesz Mean ◽  
Higher Power ◽  
Power Moments

scholarly journals Tunable three-way topological energy-splitter

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Mehul P. Makwana ◽  
Gregory Chaplain
Keyword(s):  
Time Reversal ◽  
Rectangular Lattice ◽  
Geometrical Construction ◽  
Beam Splitters ◽  
The Third ◽  
Symmetric Square ◽  
Hall States

AbstractStrategically combining four structured domains creates the first ever three-way topological energy-splitter; remarkably, this is only possible using a square, or rectangular, lattice, and not the graphene-like structures more commonly used in valleytronics. To achieve this effect, the two mirror symmetries, present within all fully-symmetric square structures, are broken; this leads to two nondistinct interfaces upon which valley-Hall states reside. These interfaces are related to each other via the time-reversal operator and it is this subtlety that allows us to ignite the third outgoing lead. The geometrical construction of our structured medium allows for the three-way splitter to be adiabatically converted into a wave steerer around sharp bends. Due to the tunability of the energies directionality by geometry, our results have far-reaching implications for applications such as beam-splitters, switches and filters across wave physics.

scholarly journals On the periodic zeta-function with rational parameter

2011 ◽  
Vol 52 ◽  
Author(s):  
Sondra Černigova
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Error Term ◽  
Fourth Power ◽  
Power Moment ◽  
Rational Parameter ◽  
Estimated Error

We obtain an asymptotic formula with estimated error term for the fourth power moment of the periodic zeta-function with rational parameter.

A ZERO DENSITY ESTIMATE FOR THE SELBERG CLASS

2007 ◽  
Vol 03 (02) ◽  
pp. 263-273 ◽  
Author(s):  
ANIRBAN MUKHOPADHYAY ◽  
KOTYADA SRINIVAS
Keyword(s):  
Density Estimate ◽  
Selberg Class ◽  
Power Moment ◽  
Density Result ◽  
Zero Density ◽  
Symmetric Square

It is well known that bounds on moments of a specific L-function can lead to zero-density result for that L-function. In this paper, we generalize this argument to all L-functions in the Selberg class by assuming a certain second power moment. As an application, it is shown that in the case of symmetric-square L-function, this result improves the existing one.

scholarly journals The second moment of symmetric square L-functions over Gaussian integers

2021 ◽  
pp. 1-27
Author(s):  
Olga Balkanova ◽  
Dmitry Frolenkov
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Error Term ◽  
Maass Form ◽  
Gaussian Integers ◽  
Second Moment ◽  
Prime Geodesic Theorem ◽  
Symmetric Square

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog–Biro–Cherubini–Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

scholarly journals On the Riesz means of $\frac{n}{\phi(n)}$

2013 ◽  
Vol Volume 36 ◽  
Author(s):  
A Sankaranarayanan ◽  
Saurabh Kumar Singh
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Error Term ◽  
Arithmetical Function ◽  
Riesz Means ◽  
Riesz Mean ◽  
Euler Totient Function ◽  
International Audience

International audience Let $\phi(n)$ denote the Euler-totient function. We study the error term of the general $k$-th Riesz mean of the arithmetical function $\frac {n}{\phi(n)}$ for any positive integer $k \ge 1$, namely the error term $E_k(x)$ where \[ \frac{1}{k!}\sum_{n \leq x}\frac{n}{\phi(n)} \left( 1-\frac{n}{x} \right)^k = M_k(x) + E_k(x). \] The upper bound for $\left | E_k(x) \right |$ established here thus improves the earlier known upper bound when $k=1$.

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