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Displaying similar documents to “On discrete mean values of Dirichlet L -functions”

The equation - Δ 𝑢 - λ 𝑢 | 𝑥 | 2 = | 𝑢 | 𝑝 + 𝑐 𝑓 ( 𝑥 ) : The optimal power

Boumediene Abdellaoui, Ireneo Peral (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We will consider the following problem - Δ u - λ u | x | 2 = | u | p + c f , u > 0 in Ω , where Ω N is a domain such that 0 Ω , N 3 , c > 0 and λ > 0 . The main objective of this note is to study the precise threshold p + = p + ( λ ) for which there is novery weak supersolutionif p p + ( λ ) . The optimality of p + ( λ ) is also proved by showing the solvability of the Dirichlet problem when 1 p < p + ( λ ) , for c > 0 small enough and f 0 under some hypotheses that we will prescribe.

Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura (2013)

Acta Arithmetica

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Let p = 1 + 2 e + 1 q be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order 2 e + 1 (resp. order d φ dividing q), and let ψₙ be an even character of conductor p n + 1 and order pⁿ. We put χ = δφψₙ, whose value is contained in K = ( ζ ( p - 1 ) p ) . It is well known that the Bernoulli number B 1 , χ is not zero, which is shown in an analytic way. In the extreme cases d φ = 1 and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: T r n / 1 ( ξ B 1 , χ ) 0 for any...

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

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A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .

Correct solvability of a general differential equation of the first order in the space L p ( )

Nina A. Chernyavskaya, Leonid A. Shuster (2015)

Archivum Mathematicum

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We consider the equation - r ( x ) y ' ( x ) + q ( x ) y ( x ) = f ( x ) , x where f L p ( ) , p [ 1 , ] ( L ( ) : = C ( ) ) and 0 < r C ( ) , 0 q L 1 ( ) . We obtain minimal requirements to the functions r and q , in addition to (), under which equation () is correctly solvable in L p ( ) , p [ 1 , ] .

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N , both eigenfunctions of the Hecke operators, and both normalized so that a 1 ( f ) = a 1 ( E ) = 1 . The main result we prove is that when E and f are congruent mod a prime 𝔭 (which we take in this paper to be a prime of ¯ lying over a rational prime p &gt; 2 ), the algebraic parts of the special values L ( E , χ , j ) and L ( f , χ , j ) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

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We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

Spectral condition, hitting times and Nash inequality

Eva Löcherbach, Oleg Loukianov, Dasha Loukianova (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let X be a μ -symmetric Hunt process on a LCCB space 𝙴 . For an open set 𝙶 𝙴 , let τ 𝙶 be the exit time of X from 𝙶 and A 𝙶 be the generator of the process killed when it leaves 𝙶 . Let r : [ 0 , [ [ 0 , [ and R ( t ) = 0 t r ( s ) d s . We give necessary and sufficient conditions for 𝔼 μ R ( τ 𝙶 ) l t ; in terms of the behavior near the origin of the spectral measure of - A 𝙶 . When r ( t ) = t l , l 0 , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order l + 1 for τ 𝙶 ...