On linear combinations of special values of L-functions

Evaluations of a Weighted Average of Gauss Sums

“…We also refer to [8] for a further investigation for the high-order mean value of products of τ(χ) and the generalized Bernoulli numbers. More recently, Alkan [9] used a special evaluation method for Dirichlet L-functions to consider the following weighted average of Gauss sums:…”

“…where ω(•) is a real-valued function defined on the interval [0, 1] and depicted that the weighted averages of Gauss sums A χ (ω) and iA χ (ω) and the character values χ(m) at positive integers m can be well approximated by linear combinations of the algebraic parts of special values of Dirichlet L-functions under correct parity conditions (see, e.g., eorems 1, 2, and 3 in [9]). e special evaluation method for Dirichlet L-functions posed by Alkan (see eorem 4 in [9]) states that if ω(x) � x m for a positive integer m, then for a nonprincipal even Dirichlet character χ modulo q,…”

“…Motivated and inspired by the work of Alkan [9], in this paper, we use some properties of the cotangent function and the Bernoulli polynomials to study the above weighted average of Gauss sums and determine the coefficients appearing in (6) and ( 7) which can be explicitly evaluated by the binomial coefficients.…”

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

“…[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. Alkan et al [9,10] found some nonstandard Diophantine approximation results with values of arithmetic functions and Fourier coefficients of newforms.…”

Quadratic Form Gauss Sums

“…As mentioned in the previous section, the Gauss character sum has various connections with Dirichlet L-functions [4], [125]. As well, this Gauss sum can be used to estimate various character sums [33], [53], [58], [80], [108]; estimate the number of solutions for certain congruences [52], [109]; and has applications for the number of representations of an integer as a sum of primes [72].…”