Displaying similar documents to “Two problems related to the non-vanishing of L ( 1 , χ ) ”

On the power-series expansion of a rational function

D. V. Lee (1992)

Acta Arithmetica

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Introduction. The problem of determining the formula for P S ( n ) , the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, h s ₁ , . . . , h s k , of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of x ⁿ i n [(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part...

Nonconvolution transforms with oscillating kernels that map Ḃ 1 0 , 1 into itself

G. Sampson (1993)

Studia Mathematica

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We consider operators of the form ( Ω f ) ( y ) = ʃ - ∞ ∞ Ω ( y , u ) f ( u ) d u with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and h ∈ L ∞ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space Ḃ 1 0 , 1 (= B) into itself. In particular, all operators with h ( y ) = e i | y | a , a > 0, a ≠ 1, map B into itself.

Several q -series identities from the Euler expansions of ( a ; q ) ∞ and 1 ( a ; q ) ∞

Zhizheng Zhang, Yang, Jizhen (2009)

Archivum Mathematicum

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In this paper, we first give several operator identities which extend the results of Chen and Liu, then make use of them to two q -series identities obtained by the Euler expansions of ( a ; q ) ∞ and 1 ( a ; q ) ∞ . Several q -series identities are obtained involving a q -series identity in Ramanujan’s Lost Notebook.

Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson

Rolf Wallisser (2005)

Journal de Théorie des Nombres de Bordeaux

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Let Q be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function G ( x ) = ∑ n = 0 ∞ x n Q ( 1 ) Q ( 2 ) ⋯ Q ( n ) .

On some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the L p ( ℝ ) -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by T m f ( x ) = p . v . ∫ k ( x - y ) m ( x + y ) f ( y ) d y where k ( x ) = ∑ j ∈ ℤ 2 j φ j ( 2 j x ) for a family of functions φ j j ∈ ℤ satisfying conditions (1.1)-(1.3) given below.