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Abstract  

For the number of integer solutions of the title equation, withW≤;x (x a large parameter), an asymptotics of the form Ax log x + Bx + O(x 1/2 (log x)3 (loglog x)2) is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.

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Abstract  

We disprove some power sum conjectures of Tur�n that would have implied the density hypothesis of the Riemann zeta-function if true.

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Abstract  

By using p-adic q-deformed fermionic integral on ℤp, we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.

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] Montgomery , H. L. , Vaughan , R. C. 2007 Multiplicative Number Theory I. Classical Theory Cambridge University Press . [4] Titchmarsh , E. C. , The Riemann Zeta-Function , 2nd ed

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References [1] Chandrasekharan , K. , Narasimhan , R. 1963 The approximate functional equation for a class of zeta-functions Math. Ann. 152 30 – 64 10.1007/BF01343729

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Summary  

A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of those by trigonometric polynomials. We consider spectral sets of unbounded functions in a certain class which contains the square of the Riemann zeta-function as a typical example.

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] Edwards , H. M. 2001 Riemann's Zeta Function Dover New York . [4] Groenevelt , W. , The Wilson function transform

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