Math Problems Vault

Problem 2.147

If $f(t)$ is differentiable and $f'(t)$ properly integrable, $t > 0$ , then

\sum _{r_{n} \leq r} f(r_{n}) = N(r) f(r) - \int _{0}^{r} N(t) f'(t) \, dt .

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Problem 2.148

Let $N(r)$ denote the counting function of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$ , which increases to infinity. Then

\limsup _{r \to \infty } \frac{N(r)}{r} = \limsup _{n \to \infty } \frac{n}{r_{n}}, \qquad \liminf _{r \to \infty } \frac{N(r)}{r} = \liminf _{n \to \infty } \frac{n}{r_{n}},

\limsup _{r \to \infty } \frac{\log N(r)}{\log r} = \limsup _{n \to \infty } \frac{\log n}{\log r_{n}}, \qquad \liminf _{r \to \infty } \frac{\log N(r)}{\log r} = \liminf _{n \to \infty } \frac{\log n}{\log r_{n}} .

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Problem 2.149

The counting function $N(r)$ and the convergence exponent $\lambda$ of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$ (see §2 of Chapter 3, Part I) are connected by the relation

\limsup _{r \to \infty } \frac{\log N(r)}{\log r} = \lambda .

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Problem 2.150

A function $L(r)$ defined and positive for $r > 0$ is termed slowly increasing if it is monotone increasing and satisfies the condition

\lim _{r \to \infty } \frac{L(2r)}{L(r)} = 1 .

Show that

\lim _{r \to \infty } \frac{L(cr)}{L(r)} = 1, \qquad c > 0 .

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Problem 2.151

Suppose that $L(r)$ is positive for $r > 0$ , monotone increasing and that for $r$ sufficiently large

L(r) = (\log r)^{\alpha _{1}} (\log _{2} r)^{\alpha _{2}} \cdots (\log _{k} r)^{\alpha _{k}}, \qquad \alpha _{1} > 0 .

Then $L(r)$ is slowly increasing.

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Remark.

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$\log_{k} x = \log_{k-1}(\log x)$ .

Problem 2.152

If $L(r)$ is slowly increasing then

\lim _{r \to \infty } \frac{\log L(r)}{\log r} = 0 .

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Problem 2.153

If $N(r)$ denotes the counting function of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$ and if

N(r) \sim r^{\lambda } L(r),

where $L(r)$ is slowly increasing, $0 < \lambda < \infty$ , then $\lambda$ is the convergence exponent of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$

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Problem 2.154

The counting function of a regular sequence with convergence exponent $\lambda$ satisfies the relation

\lim _{r \to \infty } \frac{N(cr)}{N(r)} = c^{\lambda }, \qquad c > 0 .

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Problem 2.155

Let $N(r)$ be the counting function of the regular sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$ with convergence exponent $\lambda$ and $f(x)$ be a piecewise constant function on the interval $(0, c]$ , $c > 0$ . Then

\lim _{r \to \infty } \frac{1}{N(r)} \sum _{r_{n} \leq cr} f\left(\frac{r_{n}}{r}\right) = \int _{0}^{c^{\lambda }} f\left(x^{\frac{1}{\lambda }}\right) \, dx .

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Problem 2.156

The limit relation in Problem 155 is also valid if $f(x)$ denotes a properly integrable function on $\left[0, c\right]$ .

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Problem 2.157

Let $N(r)$ denote the counting function of the regular sequence $r_{1}, r_{2}, \ldots , r_{n}, \ldots$ with convergence exponent $\lambda$ and let $\alpha > 0$ . Then

\lim _{r \to \infty } \frac{1}{N(r)} \sum _{r_{n} \leq r} \left(\frac{r_{n}}{r}\right)^{\alpha -\lambda } = \int _{0}^{1} x^{\frac{\alpha -\lambda }{\lambda }} \, dx = \frac{\lambda }{\alpha } .

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Problem 2.158

(continued).

\lim _{r \to \infty } \frac{1}{N(r)} \sum _{r_{n} > r} \left(\frac{r_{n}}{r}\right)^{-\alpha -\lambda } = \int _{1}^{\infty } x^{\frac{-\alpha -\lambda }{\lambda }} \, dx = \frac{\lambda }{\alpha } .

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Problem 2.159

Assume that $N(r)$ is the counting function of the regular sequence $r_{1}, r_{2}, \ldots , r_{n}, \ldots$ with convergence exponent $\lambda$ , that $f(x)$ is defined for $x > 0$ and properly integrable over every finite interval $\left[a, b\right]$ , $0 < a < b$ and that furthermore

\left|f(x)\right| < x^{\alpha -\lambda } \quad \text{in a neighbourhood of } x=0

and

\left|f(x)\right| < x^{-\alpha -\lambda } \quad \text{in a neighbourhood of } x=\infty , \qquad \alpha > 0 .

Show that

\lim _{r \to \infty } \frac{1}{N(r)} \sum _{n=1}^{\infty } f\left(\frac{r_{n}}{r}\right) = \int _{0}^{\infty } f\left(x^{\frac{1}{\lambda }}\right) \, dx .

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Problem 2.160

Suppose that the function $f(x)$ is defined and monotone on the interval $(0, 1]$ and that it satisfies in the neighbourhood of $x=0$ the inequality

\left|f(x)\right| < x^{\alpha -\lambda }, \qquad \alpha > 0 .

The counting function and the convergence exponent of the positive sequence $r_{1}, r_{2}, \ldots , r_{n}, \ldots$ are called $N(r)$ and $\lambda$ resp.; let $0 < \lambda < \infty$ . Then

\liminf _{r \to \infty } \frac{1}{N(r)} \sum _{r_{n} \leq r} f\left(\frac{r_{n}}{r}\right) \leq \int _{0}^{1} f\left(x^{\frac{1}{\lambda }}\right) \, dx \leq \limsup _{r \to \infty } \frac{1}{N(r)} \sum _{r_{n} \leq r} f\left(\frac{r_{n}}{r}\right) .

The sequence $r_{n}$ need not be regular.

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Remark.

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Problem 115 of Part I.

Problem 2.161

The function $f(x)$ is defined for $x > 0$ , is positive and decreasing and satisfies the inequalities

f(x) < x^{\alpha -\lambda } \quad \text{in the neighbourhood of } x=0

f(x) < x^{-\alpha -\lambda } \quad \text{in the neighbourhood of } x=\infty , \qquad \alpha > 0 .

The sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$ is defined as in Problem 160. Then

\liminf _{r \to \infty } \frac{1}{N(r)} \sum _{n=1}^{\infty } f\left(\frac{r_{n}}{r}\right) \leq \int _{0}^{\infty } f\left(x^{\frac{1}{\lambda }}\right) \, dx .

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Remark.

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Problem 116 of Part I.