Researcher:
Göral, Haydar

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Master Student

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Haydar

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Göral

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Göral, Haydar

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2013 - 20197

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Researcher Of Publications: search.filters.isAuthorOfPublication.0abbb80b-6985-485d-abba-a9b5b1e54ea4

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Now showing 1 - 7 of 7
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    On sums over the mobius function and discrepancy of fractions
    (Academic Press Inc Elsevier Science, 2013) Department of Mathematics; N/A; Alkan, Emre; Göral, Haydar; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 32803; 252019
    We obtain quantitative upper bounds on partial sums of the Mobius function over semigroups of integers in an arithmetic progression. Exploiting, the cancellation of such sums, we deduce upper bounds for the discrepancy of fractions in the unit interval [0, 1] whose denominators satisfy the same restrictions. In particular, the uniform distribution and approximation of discrete weighted averages over such fractions are established as a consequence.
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    Trigonometric series and special values of L-functions
    (Academic Press Inc Elsevier Science, 2017) Department of Mathematics; N/A; Alkan, Emre; Göral, Haydar; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 32803; 252019
    Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L-functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L-functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions.
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    Structural properties of dirichlet series with harmonic coefficients
    (Springer, 2018) Department of Mathematics; N/A; Alkan, Emre; Göral, Haydar; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 32803; 252019
    An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan's constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan's constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.
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    Height bounds, nullstellensatz and primality
    (Taylor & Francis, 2018) N/A; Göral, Haydar; Master Student; Graduate School of Sciences and Engineering; 252019
    In this study, we find height bounds in the polynomial ring over the field of algebraic numbers to test the primality of an ideal. We also obtain height bounds in the arithmetic Nullstellensatz. We apply nonstandard analysis and hence our constants will be ineffective.
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    Almost all hyperharmonic numbers are not integers
    (Elsevier, 2017) Sertbaş, Doğa Can; N/A; Göral, Haydar; Master Student; Graduate School of Sciences and Engineering; 252019
    It is an open question asked by Mezo that there is no hyperharmonic integer except 1. So far it has been proved that all hyperharmonic numbers are not integers up to order r = 25. In this paper, we extend the current results for large orders. Our method will be based on three different approaches, namely analytic, combinatorial and algebraic. From analytic point of view, by exploiting primes in short intervals we prove that almost all hyperharmonic numbers are not integers. Then using combinatorial techniques, we show that if n is even or a prime power, or r is odd then the corresponding hyperharmonic number is not integer. Finally as algebraic methods, we relate the integerness property of hyperharmonic numbers with solutions of some polynomials in finite fields. (C) 2016 Elsevier Inc. All rights reserved.
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    Interpretable groups in Mann pairs
    (Springer Heidelberg, 2018) N/A; Göral, Haydar; Master Student; Graduate School of Sciences and Engineering; 252019
    In this paper, we study an algebraically closed field expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup . This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple . This enables us to characterize the interpretable groups when is divisible. Every interpretable group H in is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in by an interpretable group N, which is the quotient of an algebraic group by a subgroup , which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in .
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    Divisor function and bounds in domains with enough primes
    (Hacettepe Üniversitesi, 2019) Department of Mathematics; Göral, Haydar; Master Student; Department of Mathematics; College of Sciences
    In this note, first we show that there is no uniform divisor bound for the Bezout identity using Dirichlet's theorem on arithmetic progressions. Then, we discuss for which rings the absolute value bound for the Bezout identity is not trivial and the answer depends on the number of small primes in the ring.