ON A NEW CIRCLE PROBLEM
| Autor: | Yoshio Tanigawa, Jun Furuya, Makoto Minamide |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
General Mathematics
010102 general mathematics Function (mathematics) 01 natural sciences Prime (order theory) Dirichlet distribution Dirichlet character 010101 applied mathematics Combinatorics Riemann hypothesis symbols.namesake symbols Arithmetic function Asymptotic formula 0101 mathematics Dirichlet series Mathematics |
| Zdroj: | Journal of the Australian Mathematical Society. 103:231-249 |
| ISSN: | 1446-8107 1446-7887 |
| DOI: | 10.1017/s1446788716000525 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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