MATHEMATICS

M. M. Tyan

REMAINDER TERMS IN THE PROBLEM OF THE DISTRIBUTION OF VALUES OF TWO ARITHMETIC FUNCTIONS

(Presented by Academician I. M. Vinogradov on 26 II 1963)

Let \(f(n)\) denote some real-valued function of a natural argument, and let \(\lambda\) be a real number. By \(P_N(f(n)<\lambda)\) we shall denote the number of integers among \(1, 2, \ldots, N\) for which \(f(n)<\lambda\).

1. Let \(f(n)=\varphi(n)/n\), where \(\varphi(n)\) is Euler’s function. Schoenberg \((^{1})\) proved that for every real \(\lambda\) there exists the limit

\[ \lim_{N\to\infty}\frac{P_N(f(n)<\lambda)}{N}=v(\lambda), \tag{1} \]

and the limiting function is continuous. The properties of the function \(v(\lambda)\) are the subject of a paper by B. A. Venkov \((^{2})\).

Since the values of \(\varphi(n)/n\) are contained in the interval \([0,1]\), it follows, by the second theorem of Helly \(((^{3}), \text{ p. }220)\), that for every function \(F(x)\) continuous on the interval \([0,1]\),

\[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}F\!\left(\frac{\varphi(n)}{n}\right) = \int_{0}^{1}F(x)\,dv(x). \tag{2} \]

We shall formulate theorems concerning the remainder terms in equalities (1) and (2).

Theorem 1. Let \(\lambda\) belong to the interval \([0,1]\). Then

\[ \frac{P_N(\varphi(n)/n<\lambda)}{N} = v(\lambda)+O\!\left(\frac{1}{\ln\ln\ln N}\right). \]

Let \(F(x)\) be defined on \([0,1]\). Denote \(\max_{0\le x\le 1}|F(x)|=M\).

Theorem 2. Given an integer function of order \(\rho\), \(F(x)\). For every \(\varepsilon>0\),

\[ \frac{1}{N}\sum_{n\le N}F\!\left(\frac{\varphi(n)}{n}\right) = \int_{0}^{1}F(x)\,dv(x) + O\!\left(\frac{M+1}{N^{\frac{1}{1+\rho}-\varepsilon}}\right). \]

Theorem 3. Let \(F(x)\) be a function given on the interval \([0,1]\). Suppose that the function is analytic inside the ellipse with foci at the points \(0\) and \(1\) and with sum of semiaxes equal to \(R\) \((R>1/2)\). Then

\[ \frac{1}{N}\sum_{n\le N}F\!\left(\frac{\varphi(n)}{n}\right) = \int_{0}^{1}F(x)\,dv(x) + O\!\left( e^{-\ln(2R-\varepsilon)\left( \frac{\ln \frac{N}{M}}{\ln\ln N+\ln 12(2R-\varepsilon)}-1 \right)} \right). \]

Theorem 4. Let \(F(x)\) be a function given on the interval \([0,1]\). Suppose that \(F(x)\) has on the interval \([0,1]\) an \(r\)-th derivative \((r\ge 0)\) satisfying a Lipschitz condition of first order,

\[ \left|F^{(r)}(x_1)-F^{(r)}(x_2)\right|<C|x_1-x_2|. \]

Then

\[ \frac{1}{N}\sum_{n\le N}F\!\left(\frac{\varphi(n)}{n}\right) = \int_{0}^{1}F(x)\,dv(x) + O\!\left( \frac{C(\ln\ln N)^{r+1}}{\left(\ln \frac{CN}{M}\right)^{r+1}} \right). \]

1. Consider the function introduced by E. V. Novoselov \((^{4})\):

\[ \|n\|=1-\sum_{d\mid n}\frac{1}{2^d}. \]

Let us denote by \(t(n)\)

\[ t(n)=\sum_{d\mid n}\frac{1}{2^d}. \]

It can be proved that for every real \(\lambda\) there exists the limit *

\[ \lim_{N\to\infty}\frac{P_N(t(n)<\lambda)}{N}=v(\lambda). \tag{3} \]

In order to formulate a theorem on the rate of convergence to the limit in relation (3), it is necessary to introduce several definitions. On the half-interval \([1/2,1)\) we consider dyadic-rational points, i.e. numbers of the form

\[ s=\frac{1}{2}+\frac{1}{2^{n_2}}+\frac{1}{2^{n_3}}+\cdots+\frac{1}{2^{n_k}}, \]

where \(n_1=1<n_2<\cdots<n_k\) (\(n_1,n_2,\ldots,n_k\) are natural numbers). Denote by \(D(s)\) the least common multiple of the numbers \(n_1,n_2,\ldots,n_k\). A dyadic-rational point \(s\) lying on the half-interval \([1/2,1)\) will be called a point of work if the system of exponents \(1,n_2,\ldots,n_k\) corresponding to this point has the following property: if \(d\mid D(s)\) and \(d\le n_k\), then \(d\) is among the numbers \(1,n_2,\ldots,n_k\). Any dyadic-rational point lying on \([1/2,1)\) and not being a point of work will be called a point of rest. Near each dyadic-rational point

\[ s=\frac{1}{2}+\frac{1}{2^{n_2}}+\frac{1}{2^{n_3}}+\cdots+\frac{1}{2^{n_k}} \]

we consider, to the right, the interval \(\left(s,s+\frac{1}{2^{n_k}}\right)\). If \(s\) is a point of work, then this interval will be called an interval of work; if \(s\) is a point of rest, then the interval will be called an interval of rest. We divide the points \(\lambda\) of the half-interval \([1/2,1)\) into three categories.

1) Dyadic-rational points \(\lambda\). It is easy to prove that every dyadic-rational point is the right endpoint of an interval of rest. Denote by \(\delta(\lambda)\) the length of the longest interval of rest adjacent to \(\lambda\) on the left.

2) Points \(\lambda\) that are not dyadic-rational, but lie on some interval of rest \((\alpha,\beta)\). Denote

\[ \delta(\lambda)=\max(\beta-\lambda,\lambda-\alpha). \]

3) Points \(\lambda\) that are not dyadic-rational and do not lie on any interval of rest. Let

\[ \lambda=\frac{1}{2}+\frac{1}{2^{n_2}}+\frac{1}{2^{n_3}}+\cdots \]

be the dyadic expansion of \(\lambda\). Denote, for a natural \(m\),

\[ s_m=\frac{1}{2}+\frac{1}{2^{n_2}}+\cdots+\frac{1}{2^{n_m}}, \]

\[ \delta_m(\lambda)=\min\left(\lambda-s_m,\;s_m+\frac{1}{2^{n_m}}-\lambda\right). \]

* The theorem was communicated to me by E. V. Novoselov.

As \(m \to \infty\),

\[ \frac{\ln n_m}{\delta_m(\lambda)} \to \infty . \]

For a given sufficiently large natural number \(N\), denote by \(m(N)\) the greatest number \(m\) for which still

\[ \frac{\ln n_m}{\delta_m(\lambda)} \leqslant 2^{\frac{\ln N}{\ln(2\ln N)}} . \]

Theorem 5. Let \(\lambda\) belong to the half-open interval \([\,1/2,1)\).

1) If \(\lambda\) is a dyadic-rational number, then

\[ \frac{P_N(t(n)<\lambda)}{N} = \vartheta(\lambda) + O\!\left( \frac{1}{\delta(\lambda)} \frac{1}{ \displaystyle \frac{\ln \frac{N}{\delta(\lambda)}}{ \ln\!\left(2\ln \frac{N}{\delta(\lambda)}\right)} } \right). \]

2) If \(\lambda\) belongs to an interval of rest, then

\[ \frac{P_N(t(n)<\lambda)}{N} = \vartheta(\lambda) + O\!\left( \frac{1}{\delta(\lambda)} \frac{1}{ \displaystyle \frac{\ln \frac{N}{\delta(\lambda)}}{ \ln\!\left(2\ln \frac{N}{\delta(\lambda)}\right)} } \right). \]

3) If \(\lambda\) is not a dyadic-rational number and does not belong to any interval of rest, then

\[ \frac{P_N(t(n)<\lambda)}{N} = \vartheta(\lambda) + O\!\left(\frac{1}{\ln n_{m(N)}}\right). \]

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
20 II 1963

REFERENCES

¹ J. Schoenberg, Math. Zs., 28, 171 (1928).
² B. A. Venkov, Scientific Notes of Leningrad State University, issue 16, 3 (1949).
³ B. V. Gnedenko, Course in Probability Theory, Moscow, 1954.
⁴ E. V. Novoselov, Scientific Notes of the Elabuga Pedagogical Institute, Physico-Mathematical Sciences, 8, 3 (1960).