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 Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Explicit upper bound on the least primitive root modulo p2

2021 ◽  
pp. 1-7
Author(s):  
Bo Chen
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Primitive Root ◽  
Character Sums ◽  
Primitive Root Modulo ◽  
Primitive Roots

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

scholarly journals Upper bounds for geodesic periods over hyperbolic manifolds

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009
Author(s):  
Feng Su
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Maass Forms ◽  
Maass Form ◽  
Ambient Manifold

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.

scholarly journals Sums of Kloosterman sums in the prime geodesic theorem

2018 ◽  
Vol 70 (2) ◽  
pp. 649-674 ◽  
Author(s):  
Olga Balkanova ◽  
Dmitry Frolenkov
Keyword(s):  
Error Term ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
New Method ◽  
Prime Geodesic Theorem

Abstract We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.

Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes

2019 ◽  
Vol 56 (4) ◽  
pp. 1033-1043 ◽  
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Hazard Rate ◽  
Rate Function ◽  
Semiparametric Models ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Parametric And Semiparametric Models ◽  
Reversed Hazard Rate Function

AbstractAn upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

scholarly journals An $\Omega$-result related to $r_4(n)$.

1989 ◽  
Vol Volume 12 ◽  
Author(s):  
Sukumar Das Adhikari ◽  
R Balasubramanian ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Error Term ◽  
Elementary Proof ◽  
Arithmetical Function ◽  
Average Order ◽  
Elementary Method ◽  
International Audience

International audience Let $r_4(n)$ be the number of ways of writing $n$ as the sum of four squares. Set $P_4(x)= \sum \limits_{n\le x} r_4(n)-\frac {1}{2}\pi^2 x^2$, the error term for the average order of this arithmetical function. In this paper, following the ideas of Erd\"os and Shapiro, a new elementary method is developed which yields the slightly stronger result $P_4(x)= \Omega_{+}(x \log \log x)$. We also apply our method to give an upper bound for a quantity involving the Euler $\varphi$-function. This second result gives an elementary proof of a theorem of H. L. Montgomery

scholarly journals Nonvanishing for cubic L-functions

2021 ◽  
Vol 9 ◽  
Author(s):  
Chantal David ◽  
Alexandra Florea ◽  
Matilde Lalin
Keyword(s):  
Critical Point ◽  
Number Field ◽  
Upper Bound ◽  
Riemann Hypothesis ◽  
Implied Constant ◽  
Field Case ◽  
Second Moment ◽  
Positive Proportion ◽  
First Moment

Abstract We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

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.

Sign in / Sign up

Export Citation Format

Share Document