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Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product
Estimates of the form L (j) (s,A) ,j, DA R A in the range |s 1| 1/logRA for general L-functions, where RA is a parameter related to the functional equation of L(s,A), can be quite easily obtained ifExpand
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Representation of a 2-power as sum of k 2-powers: A recursive formula
Abstract For every integer k, a k-representation of 2 k − 1 is a string n = ( n 1 , … , n k ) of nonnegative integers such that ∑ j = 1 k 2 n j = 2 k − 1 , and W ( 1 , k ) is their number. We presentExpand
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Linear independence of L-functions
Abstract We prove the linear independence of the L-functions, and of their derivatives of any order, in a large class 𝒞 defined axiomatically. Such a class contains in particular the Selberg classExpand
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Inequalities for the beta function
Let g(x):= (e/x)xΓ(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y (k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) ∈ R+ 2 one has (-1)k-1Dx,y (k) (X,Y) > 0 forExpand
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Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH
TLDR
We have recently proved several explicit versions of the prime ideal theorem under GRH. Expand
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REPRESENTATION OF A 2-POWER AS SUM OF k 2-POWERS: THE ASYMPTOTIC BEHAVIOR
A k-representation of an integer l is a representation of l as sum of k powers of 2, where representations differing by the order are considered as distinct. Let be the maximum number of suchExpand
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On the algebraic independence in the selberg class
We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=Hb for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentFp. This implies that theExpand
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Extremal values for the sum ∑r=1τe(a2r/q)
Abstract Let q be an odd integer, let τ be the order of 2 modulo q , and let a be coprime with q . Finally, let s ( a / q ) : = ∑ r = 1 τ e ( a 2 r / q ) . We prove that | s ( a / q ) | can be asExpand
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Some remarks on the unique factorization in certain semigroups of classical $L$-functions
A well known result by Ram Murty [8] states that the Selberg orthonormality conjecture (SOC for short) for the Selberg class S implies the Artin conjecture on the holomorphy of the Artin L-functionsExpand
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Some Arithmetical Properties of the Generating Power Series for the Sequence {ζ(2k+1)}k=1∞
Let fodd(z):= ∑∞k=1ζ(2k + 1)z2k be the power series with the values of the Riemann ζ function at odd integers as coefficients. This function can be analytically continued to a meromorphic functionExpand
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