An algebraic independence result for Euler products of finite degree

Zaharescu, Alexandru; Zaki, Mohammad

doi:10.1090/s0002-9939-08-09622-6

Cited by 4 publications

(2 citation statements)

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“…For a detailed account of recent results on this topic, we refer to [23]. Let us also mention that functional independence of general families of L-functions (including the ones that are associated to cusp forms) was shown by Zaharescu and Zaki [28] using a purely algebraic method. Let us also mention that functional independence of general families of L-functions (including the ones that are associated to cusp forms) was shown by Zaharescu and Zaki [28] using a purely algebraic method.…”

Section: Introductionmentioning

confidence: 99%

Special values of the Riemann zeta function capture all real numbers

Alkan

2015

Proc. Amer. Math. Soc.

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It is shown that the set of odd values {ζ(3), ζ(5), . . . , ζ(2k + 1), . . . }of the Riemann zeta function is rich enough to capture real numbers in an approximation aspect. Precisely, we prove that any real number can be strongly approximated by certain linear combinations of these odd values, where the coefficients belonging to these combinations are universal in the sense of being independent of ζ(n) for all integers n ≥ 2. This approximation property is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals.

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Section: Introductionmentioning

confidence: 99%

Special values of the Riemann zeta function capture all real numbers

Alkan

2015

Proc. Amer. Math. Soc.

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“…We should mention that approximation type results in some nonstandard settings have been the subject of important research. [54] with the help of a purely algebraic method. The author [5] showed that weighted averages of Gauss sums can be well approximated by Q-linear combinations of special values of L-functions in which coefficients belonging to the combinations only depend on the parity of the character.…”

Section: Introductionmentioning

confidence: 99%

Approximation by special values of harmonic zeta function and log-sine integrals

Alkan

2013

Communications in Number Theory and Physics

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Motivated by applications of log-sine integrals to a wide range of mathematical and physical problems, it is shown that real numbers and certain types of log-sine integrals can be strongly approximated by linear combinations of special values of the harmonic zeta function with the property that the coefficients belonging to these combinations turn out to be universal in the sense of being independent of special values. The approximation of real numbers by combinations of special values is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals. Moreover, explicit representations of some specific log-sine integrals are obtained in terms of special values of the harmonic zeta function and the Riemann zeta function through a study of Fourier series involving harmonic numbers. In particular, special values of the harmonic zeta function and the less studied odd harmonic zeta function are expressed in terms of log-sine integrals over [0, 2π] and [0, π].

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On linear combinations of special values of L-functions

Alkan

2012

manuscripta math.

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An algebraic independence result for Euler products of finite degree

Abstract: Abstract. We investigate the algebraic independence of some derivatives of certain multiplicative arithmetical functions over the field C of complex numbers.

Cited by 4 publications

References 10 publications

Special values of the Riemann zeta function capture all real numbers

Special values of the Riemann zeta function capture all real numbers

Approximation by special values of harmonic zeta function and log-sine integrals

On linear combinations of special values of L-functions

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