scholarly journals Dirichlet series expansions of p-adic L-functions

2021 ◽  
Author(s):  
Heiko Knospe ◽  
Lawrence C. Washington
Keyword(s):  
Dirichlet Series ◽  
Series Expansion ◽  
Regularization Parameter ◽  
Complex Case ◽  
Alternative Proof ◽  
Series Expansions ◽  
Bernoulli Distribution ◽  
Natural Manner ◽  
Limiting Behavior ◽  
Dirichlet Characters

AbstractWe study p-adic L-functions $$L_p(s,\chi )$$ L p ( s , χ ) for Dirichlet characters $$\chi $$ χ . We show that $$L_p(s,\chi )$$ L p ( s , χ ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of $$\chi $$ χ . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $$c=2$$ c = 2 , where we obtain a Dirichlet series expansion that is similar to the complex case.

scholarly journals An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
Simone Di Marino ◽  
Augusto Gerolin
Keyword(s):  
Optimal Transport ◽  
A Priori Estimates ◽  
Regularization Parameter ◽  
Alternative Proof ◽  
Revealed Preferences ◽  
Discussion Paper ◽  
Existence Of A Solution ◽  
Trade Offs ◽  
Bridge Problem ◽  
Schrödinger Bridge

AbstractThis paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.

scholarly journals Nested derivatives: a simple method for computing series expansions of inverse functions

2003 ◽  
Vol 2003 (58) ◽  
pp. 3699-3715 ◽  
Author(s):  
Diego Dominici
Keyword(s):  
Series Expansion ◽  
Special Functions ◽  
Series Expansions ◽  
Simple Method ◽  
Inverse Functions

We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the firstNterms of the series. We show several examples of its application in calculating the inverses of some special functions.

scholarly journals A Dirichlet series expansion for the p-adic zeta-function

2006 ◽  
Vol 81 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Daniel Delbourgo
Keyword(s):  
Fractional Derivative ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Series Expansion ◽  
Generating Function

AbstractWe prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion Where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of p(–s, ω1+σ) for s ∈ Zp, with a branch of the ‘sth-fractional derivative’, of a suitable generating function.

scholarly journals ON THE q-DERIVATIVE AND q-SERIES EXPANSIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 2069-2089 ◽  
Author(s):  
ZHI-GUO LIU
Keyword(s):  
Series Expansion ◽  
Generating Function ◽  
Series Expansions ◽  
Infinite Products ◽  
Type Series ◽  
Famous Result ◽  
Triangular Numbers

Using a general q-series expansion, we derive some nontrivial q-formulas involving many infinite products. A multitude of Hecke-type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.

SERIES EXPANSIONS OF THE GENERALIZED K-BESSEL FUNCTIONS ON SYMMETRIC CONES

2008 ◽  
Vol 06 (01) ◽  
pp. 1-10 ◽  
Author(s):  
HONGMING DING ◽  
WEI HE
Keyword(s):  
Series Expansion ◽  
Bessel Functions ◽  
Series Expansions ◽  
Symmetric Cones ◽  
Expansion Formula

In this paper, we generalize the series expansion formula of classical K-Bessel functions to symmetric cones.

scholarly journals Subconvexity for a double Dirichlet series and non-vanishing of L-functions

2018 ◽  
Vol 14 (06) ◽  
pp. 1573-1604
Author(s):  
Alexander Dahl
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Dirichlet Series ◽  
Upper Bound ◽  
Dihedral Group ◽  
Central Point ◽  
Double Dirichlet Series ◽  
Dirichlet Characters ◽  
Equation Group

We study a double Dirichlet series of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are quadratic Dirichlet characters with prime conductors [Formula: see text] and [Formula: see text] respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to [Formula: see text]. The developed theory is used to prove an upper bound for the smallest positive integer [Formula: see text] such that [Formula: see text] does not vanish. Additionally, a convexity bound at the central point is established to be [Formula: see text] and a subconvexity bound of [Formula: see text] is proven. An application of bounds at the central point to the non-vanishing problem is also discussed.

scholarly journals Some Algebraic Techniques for obtaining Low-temperature Series Expansions

1978 ◽  
Vol 31 (6) ◽  
pp. 515 ◽  
Author(s):  
IG Enting
Keyword(s):  
Low Temperature ◽  
Statistical Mechanics ◽  
Series Expansion ◽  
General Result ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Models ◽  
Temperature Series ◽  
Series Expansions ◽  
Lattice Method

It is shown that low-temperature series expansions for lattice models in statistical mechanics can be obtained from a consideration of only connected strong subgraphs of the lattice. This general result is used as the basis of a linked-cluster form of the method of partial generating functions and also as the basis for extending the finite lattice method of series expansion to low-temperature series.

On a class of interpolation series

1979 ◽  
Vol 84 (3-4) ◽  
pp. 283-307 ◽  
Author(s):  
Einar Hille
Keyword(s):  
Dirichlet Series ◽  
Legendre Polynomial ◽  
Rational Functions ◽  
Series Expansions ◽  
Finite Rate ◽  
Converse Problem ◽  
Rate Of Growth ◽  
Abscissa Of Convergence ◽  
Interpolation Series ◽  
Image Position

SynopsisThis paper deals with a class of interpolation series of the formcalled R-series. It is equiconvergent with the Dirichlet seriesIf the nth Legendre polynomial for the interval (0,1) is denoted by (−1)nLn(t), then the bilinear formulaserves as generating function for the Rn(z). It also leads in easy steps to R-series expansions for rational functions.Lagrange [7] has shown that a function holomorphic and of finite rate of growth in a right half-plane can be expanded in an R-series whose abscissa of convergence is limited by the rate of growth of f(z). The converse problem is attacked in Theorem 2 below where it is shown that

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