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    EULER-ZAGIER SUMS VIA TRIGONOMETRIC SERIES
    (Editura Acad Romane, 2023) Celik, Sermin Cam; Goral, Haydar
    In 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.
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    On classification of sequences containing arbitrarily long arithmetic progressions
    (World Scientific Publ Co Pte Ltd, 2023) Celik, Sermin Cam; Eyidogan, Sadik; Goral, Haydar; Sertbas, Doga Can
    In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map n(s) can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.