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ON A NEW CIRCLE PROBLEM

  • JUN FURUYA (a1), MAKOTO MINAMIDE (a2) and YOSHIO TANIGAWA (a3)

Abstract

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ( $\text{Re}\,s>1$ ) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$ -function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ( $\text{Re}\,s>1$ ) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$ $(\text{Re}\,s>1)$ . This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$ . In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$ . As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$ . As a direct application, we show the mean square for the error term in our new problem.

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This work is supported by JSPS KAKENHI: 26400030, 15K17512, and 15K04778.

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[AC] Ayoub, R. and Chowla, S., ‘On a theorem of Müller and Carlits’, J. Number Theory 2 (1970), 342344.
[FMT] Furuya, J., Minamide, M. and Tanigawa, Y., ‘Representations and evaluations of the error term in a certain divisor problem’, Math. Slovaca 66 (2016), 575582.
[FT] Furuya, J. and Tanigawa, Y., ‘On integrals and Dirichlet series obtained from the error term in the circle problem’, Funct. Approx. Comment. Math. 51 (2014), 303333.
[G] Gonek, S. M., ‘Mean values of the Riemann zeta-function and its derivatives’, Invent. Math. 75 (1984), 123141.
[I] Ivić, A., The Riemann Zeta-Function (Dover, Mineola, 2003).
[J] Jutila, M., Lectures on a Method in the Theory of Exponential Sums (Springer, Bombay, 1987).
[K] Krätzel, E., Lattice Points (Kluwer Academic, Dordrecht, 1988).
[M] Minamide, M., ‘The truncated Voronoï formula for the derivative of the Riemann zeta function’, Indian J. Math. 55 (2013), 325352.
[T] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd edn (ed. Heath-Brown, D. R.) (Oxford University Press, Oxford, 1986).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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