Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 1Citation - Scopus: 1Euler-Zagier Sums Via Trigonometric Series(Publishing House of the Romanian Academy, 2023) Çam Çelik, Şermin; Göral, HaydarIn this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series.Article Citation - WoS: 2Citation - Scopus: 1The Difference of Hyperharmonic Numbers Via Geometric and Analytic Methods(Korean Mathematical Society, 2022) Altuntaş, Çağatay; Göral, Haydar; Sertbaş, Doğa CanOur motivation in this note is to find equal hyperharmonic numbers of different orders. In particular, we deal with the integerness property of the difference of hyperharmonic numbers. Inspired by finite-ness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyper-harmonic numbers of fixed indices to have a certain rational difference. Moreover, using analytic techniques, we get that almost all differences are not integers. On the contrary, we also obtain that there are infinitely many order values where the corresponding differences are integers.Article Lehmer’s Conjecture Via Model Theory(Japan Academy, 2022) Göral, HaydarIn this short note, we study Lehmer's conjecture in terms of stability theory. We state Bounded Lehmer's conjecture, and we prove that if a certain formula is uniformly stable in a class of structures, then Bounded Lehmer's conjecture holds. Our proof is based on Van der Waerden's theorem from additive combinatoricsArticle A Note on Points on Algebraic Sets(Hacettepe Üniversitesi, 2021) Çam Çelik, Şermin; Göral, HaydarIn this short note, we count the points on algebraic sets which lie in a subset of a domain. It is proved that the set of points on algebraic sets coming from certain subsets of a domain has the full asymptotic. This generalizes the first theorem of [E. Alkan and E.S. Yoruk, Statistics and characterization of matrices by determinant and trace, Ramanujan J., 2019] and also anwers some questions from the same article.