We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
