scholarly journals A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

2016 ◽  
Vol 294 (1) ◽  
pp. 13-46 ◽  
Author(s):  
Olga G. Balkanova ◽  
Dmitry A. Frolenkov
Keyword(s):  
Asymptotic Formula ◽  
Critical Line ◽  
Second Moment

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

scholarly journals THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS

2010 ◽  
Vol 06 (08) ◽  
pp. 1933-1944 ◽  
Author(s):  
SANDRO BETTIN
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Second Moment ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals On certain sums over ordinates of zeta-zeros II

2019 ◽  
Author(s):  
Andriy Bondarenko ◽  
Aleksandar Ivić ◽  
Eero Saksman ◽  
Kristian Seip
Keyword(s):  
Analytic Continuation ◽  
Critical Line ◽  
Laurent Expansion ◽  
Second Moment ◽  
International Audience ◽  
Complex Zeros ◽  
Gamma S

International audience Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function $G(s) :=\sum_{\gamma >0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].

scholarly journals Weighted value distributions of the Riemann zeta function on the critical line

2021 ◽  
Vol 33 (3) ◽  
pp. 579-592
Author(s):  
Alessandro Fazzari
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Formula ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Matrix Theory ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a central limit theorem for log ⁡ | ζ ⁢ ( 1 2 + i ⁢ t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ⁢ ( 1 2 + i ⁢ t ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ⁢ ( 1 2 + i ⁢ t + i ⁢ α ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.

scholarly journals The fourth moment of individual Dirichlet L-functions on the critical line

2020 ◽  
Author(s):  
Berke Topacogullari
Keyword(s):  
Critical Line ◽  
Power Saving ◽  
Zeta Functions ◽  
Number Fields ◽  
Fourth Moment ◽  
Quadratic Number ◽  
Second Moment ◽  
Dirichlet Characters ◽  
Asymptotic Formulae ◽  
Special Cases

Abstract We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

scholarly journals MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

2021 ◽  
pp. 1-45
Author(s):  
RIZWANUR KHAN ◽  
MATTHEW P. YOUNG
Keyword(s):  
Spectral Parameter ◽  
Critical Line ◽  
Short Interval ◽  
Central Point ◽  
Cusp Forms ◽  
Sharp Bounds ◽  
Second Moment ◽  
Symmetric Square

Abstract We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

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