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 Arithmetic Progressions in Certain Subsets of Finite Fields(Elsevier, 2023) Eyidoğan, Sadık; Göral, Haydar; Kutlu, Mustafa KutayIn this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp = {t2 : t & ISIN; Fp} and Cp = {t3 : t & ISIN; Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k & GE; 3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given by p2 2k + R, where and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length in the set Sp when & ISIN; {3,4, 5} and p is an odd prime number. For = 4, 5, our formulas are based on the number of points onArticle Some Remarks on Harmonic Type Matrices(Colgate University, 2022) Göral, HaydarIn 1915, Theisinger proved that all harmonic numbers are not integers except for the first one. In 1862, Wolstenholme proved that the numerator of the reduced form of the harmonic number Hp−1 is divisible by p2 and the numerator of the reduced form of the generalized harmonic number (Formula presented) is divisible by p for all primes p ≥ 5. In this note, we define harmonic type matrices and our goal is to extend Theisinger’s and Wolstenholme’s results to harmonic type matrices. © 2022, Colgate University. All rights reserved.Article Class Number and the Special Values of L-Functions(Romanian Academy, 2022) Göral, HaydarWe give infinitely many explicit new representations of the class number of imag inary quadratic fields in terms of certain trigonometric series. Our result relies on a hybrid between power series and trigonometric series. Furthermore, in some cases we prove that the special values of Dirichlet L-functions can be evaluated as certain finite sums.Article Citation - WoS: 2Citation - Scopus: 3The Green-Tao Theorem and the Infinitude of Primes in Domains(Taylor & Francis, 2022) Göral, Haydar; Özcan, Hikmet Burak; Sertbaş, Doğa CanWe first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.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 Citation - WoS: 1Citation - Scopus: 1Applications of Class Numbers and Bernoulli Numbers To Harmonic Type Sums(Korean Mathematical Society, 2021) Göral, Haydar; Sertbaş, Doğa CanDivisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.Article Citation - WoS: 1Dedekind Harmonic Numbers(Indian Academy of Sciences, 2021) Altuntaş, Çağatay; Göral, HaydarFor any number field, we define Dedekind harmonic numbers with respect to this number field. First, we show that they are not integers except finitely many of them. Then, we present a uniform and an explicit version of this result for quadratic number fields. Moreover, by assuming the Riemann hypothesis for Dedekind zeta functions, we prove that the difference of two Dedekind harmonic numbers are not integers after a while if we have enough terms, and we prove the non-integrality of Dedekind harmonic numbers for quadratic number fields in another uniform way together with an asymptotic result.Article 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.