Researcher:
Alkan, Emre

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Emre

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Alkan

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Researcher Of Publications: search.filters.isAuthorOfPublication.51385f62-155d-4cd8-ae59-c4e403ba6e48

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Now showing 1 - 10 of 42
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    Addendum to “On the mean square average of special values of L-functions” [J. Number Theory 131 (8) (2011) 1470–1485]
    (Elsevier, 2011) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
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    Asymptotic behavior of the irrational factor
    (Springer, 2008) Ledoan, A. H.; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    We study the irrational factor function I(n) introduced by Atanassov and defined by I(n) = Pi(k)(k=1)p(v)(1/alpha v), where n = Pi(k)(v=1) p(v)(alpha v) is the prime factorization of n. We show that the sequence {G(n)/n}(n >= 1), where G(n) = Pi(n)(v=1) I(v)(1/n), is covergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).
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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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    Biased behavior of weighted mertens sums
    (World Scientific Publ Co Pte Ltd, 2020) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at s = 1.
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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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    Ramanujan sums and the burgess zeta function
    (World Scientific Publ Co Pte Ltd, 2012) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.
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    Average size of gaps in the fourier expansion of modular forms
    (World Scientific Publ Co Pte Ltd, 2007) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. in particular we obtain quantitative results on the number of large values of the gap function.
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    Approximation by special values of harmonic zeta function and log-sine integrals
    (Int Press Boston, Inc, 2013) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    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, 2 pi] and [0, pi].
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    Inequalities between sums over prime numbers in progressions
    (Springer International Publishing Ag, 2020) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the correspondingL-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain DirichletL-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series.
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    Applications of bombieri-vinogradov type theorems to power-free integers
    (Ars Polona-Ruch, 2021) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803
    Studying a variant of a classical result of Walfisz on the number of representations of an integer as the sum of a prime number and a square-free integer with an extra additive constraint on the prime summand, we obtain an asymptotic formula for the number of representations of an integer N such that N - 1 is a prime number in the form p + N - p, where p is a prime number, N - p is square-free and p - 1 is cube-free. We improve the error term for the number of representations of an integer as the sum of a prime number and a k-free integer conditionally by assuming weaker forms of the Riemann hypothesis for Dirichlet L-functions. As a further application of our method, we find an asymptotic formula for the number of prime numbers p <= x such that p + 2y, 1 <= y <= 7, are all square-free. Our formula shows that a positive proportion of prime numbers leads to a longest possible progression of eight consecutive odd, square-free integers. A key ingredient in our approach is the Bombieri-Vinogradov theorem and its variant for sparse moduli due to Baier and Zhao which regulates the uniform distribution of prime numbers along certain short arithmetic progressions.