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In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems. ==Definition== The divisor summatory function is defined as : where : is the divisor function. The divisor function counts the number of ways that the integer ''n'' can be written as a product of two integers. More generally, one defines : where ''d''''k''(''n'') counts the number of ways that ''n'' can be written as a product of ''k'' numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in ''k'' dimensions. Thus, for ''k''=2, ''D''(''x'') = ''D''2(''x'') counts the number of points on a square lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis, and to the upper-right by the hyperbola ''jk'' = ''x''. Roughly, this shape may be envisioned as a hyperbolic simplex. This allows us to provide an alternative expression for ''D''(''x''), and a simple way to compute it in time: :, where If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is known as the Gauss circle problem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「divisor summatory function」の詳細全文を読む スポンサード リンク
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