1,721,241 research outputs found
Counting sets with exceptions
No full textLet be integers such that and . The author proves the inequality where the motivation is that the left-hand side counts those -element subsets of an -element set which intersect a fixed -element subset in at least elements. The inequality is compared with another one of the same sort that was obtained by Brüdern and Perelli
Factorization in the extended Selberg class of L-functions associated to modular forms
No full textLet and let be a Dirichlet character modulo . Let be a modular form with respect to the group , multiplier and weight . Let be the -function associated with and normalized in such a way that satisfies a functional equation where reflects in . The modular forms for which belongs to the extended Selberg class \Selberg^\sharp are characterized. For these forms the factorization of in primitive elements of \Selberg^\sharp is enquired. In particular, it is proved that if is a cusp form and F\in\Selberg^\sharp then is almost primitive (i.e., that if is a factorization with P,G\in\Selberg^\sharp and the degree of is then is a Dirichlet polynomial). It is also proved that the conductor of the polynomial factor is bounded by . If belongs to the space generated by newforms and then is actually primitive (i.e., is a constant)
Upper and lower bounds at s=1 for certain Dirichlet series with Euler product
No full textEstimates of the form L(j)(s, script A sign) ≪ε,j,script D sign script A signRεscript A sign in the range |s - 1| ≪ 1/log Rscript A sign for general L-functions, where Rscript A sign, is a parameter related to the functional equation of L(s, script A sign), can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the L-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every L(s, π), where π is an automorphic cusp form on GL(d, double-struck A signK). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form
Thermal behavior of dipolarophile-containing 2-azidocarbonylpyrroles and -indoles
No full textThe thermal reaction of N-allyl- or N-propargyl-2-azidocarbonylpyrroles (2) or -indoles (9) involves competition between intramolecular azide cycloaddition and Curtius rearrangement
Multiplicity results for the functional equation of the Dirichlet -functions: case
No full textGiven an integer q and a primitive character χ modulo q, the functionalequation of the Dirichlet L-function L(s, χ) is determined by the signature of χ, i.e. byχ(−1) (the parity) and τ(χ) (the Gauss sum). In this paper we prove several resultsabout the cardinalities of the sets T(χ) := {ψ : τ(ψ) = τ(χ)} and W(χ) := {ψ : τ(ψ) =τ(χ), ψ(−1) = χ(−1)}, mainly an algorithm for their computation and optimal upperand lower bounds for their values, when q is either an odd prime power or a compositenumber of special form. For the same q we compute also the number of distinct Gausssums and of distinct signatures: the latter number deserves a special attention because itcoincides with the number of non-trivial functional equations of degree 1 and conductorq in the Selberg class
Multiplicity results for the functional equation of the Dirichlet -functions
No full textGiven an integer and a primitive character modulo , the functional equation of the
Dirichlet -function is determined by the \emph{signature} of , i.e. by (the parity) and (the Gauss sum). In this paper we prove several results about the cardinalities of the sets and , mainly an algorithm for their computation and optimal upper and lower bounds for their values, when is either an odd prime power or a composite number of special form. For the same we compute also the number of distinct Gauss sums and of distinct signatures: the latter number deserves a special attention because it coincides with the number of non-trivial functional equations of degree and conductor in the Selberg class
Explicit bounds for even moments of Bernstein's polynomials
Full text linkWe prove explicit and optimal lower and upper bounds for even-order moments of Bernstein polynomials
Some arithmetical properties of the generating power series for the sequence {ζ(2k+1)}k=1∞
Full text linkThe first case of diastereoselective cycloadditions of enantiopure nitrilimines in aqueous media
No full textThe diastereoselective cycloadditions of enantiopure nitrilimines 4 with ethyl acrylate were exploited in dry toluene and in aqueous sodium hydrogencarbonate as reaction media. Shorter reaction times and improved diastereoisomeric ratios of the resulting 5-ethoxycarbonyl-4,5-dihydropyrazoles 5 and 6 were observed in aqueous media
An explicit bound for the error term of the development at s=1 of a set of lacunary series
No full textAn explicit bound for the error term of the expansion of as is given
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