AI 中文总结
该研究围绕莫比乌斯函数部分和M(x)的振荡，证明其平均阶随Y趋向无穷大，且对大Y与黎曼-冯·曼戈尔特素数公式的最大误差项高度吻合。
AI 中文摘要
自1885年斯蒂尔杰斯在给埃尔米特的信中以较弱形式提出默滕斯猜想后，1905年默滕斯的著名猜想使莫比乌斯函数的部分和M(x)的振荡成为素数理论研究者的关注焦点。区间[0,Y]上|M(x)|的平均阶显然与素数分布相关。作者在20世纪80年代初证明，该平均阶无条件地随Y趋向无穷大。本研究表明，与素数定理误差项的平均阶类似，对于足够大的Y，该平均阶与黎曼-冯·曼戈尔特素数公式对应Y值的最大误差项高度吻合。
英文摘要
The oscillation of M(x), the partial sum of the Möbius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in connection with the distribution of primes. The author proved at the beginning of 1980's that this average tends unconditionally to infinity with Y. The present work shows that (similarly to the average order of the error term of the Prime Number Theorem) this average agrees with great accuracy for large values of Y with the largest error term of the Riemann-von Mangoldt prime number formula for the value Y.
