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Semi-ordinary p-stabilization of the Siegel Eisenstein series of arbitrary genus
Siegel modular forms Eisenstein series p-adic analytic family
2012/7/11
For a given odd prime p, we define a certain p-stabilization of the Siegel Eisenstein series of arbitrary genus so that the resulting eigenvalue of a generalized Atkin operator at p is a p-adic unit (...
On higher congruences between cusp forms and Eisenstein series
higher congruences cusp forms Eisenstein series Number Theory
2012/6/30
In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properti...
New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series
Congruence finite central binomial sum multiple harmonic sum Bernoulli number
2012/6/15
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congru...
On an incomplete argument of Erdos on the irrationality of Lambert series
Lambert series divisor function q-logarithm Number Theory
2012/6/15
We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.
Computing power series expansions of modular forms
power series expansions of modular forms Number Theory
2012/5/9
We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra.
Abstract: The calculation, by L.\ Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of $\pi$ and rational numbers, was a watershed event in the history of...
An Elliptic Analogue Of Generalized Cotangent Dirichlet Series And Its Transformation Formulae At Some Integer Arguments
Elliptic Analogue Cotangent Dirichlet Series Integer Arguments Number Theory
2011/9/20
Abstract: B.C. Berndt evaluated special values of the cotangent Dirichlet series. T. Arakawa studied a generalization of the series, or generalized cotangent Dirichlet series, and gave its transformat...
Values of certain L-series in positive characteristic
L-functions in positive characteristic Drinfeld modular forms function fields of positive characteristic
2011/9/19
Abstract: We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[...
Abstract: Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This ...
Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups; the CM method
Eisenstein series unitary groups the CM method Number Theory
2011/8/26
Abstract: In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences...
Exponential power series, Galois module structure and differential modules
Exponential power series Galois module structure Number Theory
2011/8/26
Abstract: We use new over-convergent p-adic exponential power series, inspired by work of Pulita, to build self-dual normal basis generators for the square root of the inverse different of certain abe...
Extending Landau's Theorem on Dirichlet Series with Non-Negative Coefficients
Extending Landau's Theorem Non-Negative Coefficients
2010/11/26
A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissae of absolute convergence.
A new series for $\pi^3$ and related congruences
Series for 3 central binomial coefficients congruences modulo prime powers
2010/12/14
Let H(2)n denote the second-order harmonic numberP0 for n = 0, 1, 2, . . . . In this paper we obtain the following new identity for 3: 1 X k=1 2kH(2) k−1 k2k k = 3 48 .We e...