The Inequality For The Riemann Zeta Function

Introduction

The Riemann zeta function, denoted by $\zeta(s)$, is a fundamental object of study in analytic number theory. It is defined as the infinite series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$. The zeta function has numerous applications in number theory, and its properties have been extensively studied. In this article, we will focus on the inequality for the Riemann zeta function, specifically the function $x \mapsto x(\zeta(x+1) - 1)$ and the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ for all $x > 0$.

The Function $x \mapsto x(\zeta(x+1) - 1)$

To prove that the function $x \mapsto x(\zeta(x+1) - 1)$ is decreasing, we need to show that its derivative is negative for all $x > 0$. Let's start by computing the derivative of this function.

Computing the Derivative

We have that

$$\frac{d}{dx} x(\zeta(x+1) - 1) = \zeta(x+1) + x \frac{d}{dx} \zeta(x+1) - 1.$$

Using the definition of the zeta function, we can rewrite this as

$$\frac{d}{dx} x(\zeta(x+1) - 1) = \sum_{n=1}^{\infty} \frac{1}{(n+x)^2} + x \sum_{n=1}^{\infty} \frac{-2}{(n+x)^3} - 1.$$

Simplifying this expression, we get

$$\frac{d}{dx} x(\zeta(x+1) - 1) = \sum_{n=1}^{\infty} \frac{1}{(n+x)^2} - \frac{2x}{(n+x)^3} - 1.$$

Showing the Derivative is Negative

To show that the derivative is negative, we need to show that the sum

$$\sum_{n=1}^{\infty} \frac{1}{(n+x)^2} - \frac{2x}{(n+x)^3} - 1$$

is negative for all $x > 0$. We can do this by showing that each term in the sum is negative.

For the first term, we have

$$\frac{1}{(n+x)^2} < 0$$

for all $n \geq 1$ and $x > 0$. For the second term, we have

$$\frac{2x}{(n+x)^3} > 0$$

for all $n \geq 1$ and $x > 0$. Finally, the third term is always negative.

Therefore, we have shown that the derivative of the function $x \mapsto x(\zeta(x+1) - 1)$ is negative for all $x > 0$, which implies that the function is decreasing.

The Inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$

To prove the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ for all $x > 0$, we can use the fact that the function $x \mapsto x(\zeta(x+1) - 1)$ is decreasing.

Using the Decreasing Function

Since the function $x \mapsto x(\zeta(x+1) - 1)$ is decreasing, we have that

$$x(\zeta(x+1) - 1) \leq x(\zeta(1) - 1)$$

for all $x > 0$. Simplifying this expression, we get

$$\zeta(x+1) \leq \zeta(1) + \frac{1}{x}.$$

Using the Definition of the Zeta Function

Using the definition of the zeta function, we have that

$$\zeta(1) = \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots.$$

This is a well-known result in number theory, and it can be shown that the sum converges to a finite value.

Simplifying the Inequality

Substituting the value of $\zeta(1)$ into the inequality, we get

$$\zeta(x+1) \leq 1 + \frac{1}{x} + \frac{1}{x}.$$

Simplifying this expression, we get

$$\zeta(x+1) \leq \frac{1}{x} + 1 + \frac{1}{x}.$$

Using the Triangle Inequality

Using the triangle inequality, we have that

$$\zeta(x+1) \leq \frac{1}{x} + \frac{1}{x} + 1.$$

Simplifying this expression, we get

$$\zeta(x+1) \leq \frac{2}{x} + 1.$$

Using the Fact that $x > 0$

Since $x > 0$, we have that

$$\frac{2}{x} + 1 \leq \frac{1}{x^2} + 1.$$

Therefore, we have shown that

$$\zeta(x+1) \leq \frac{1}{x^2} + 1$$

for all $x > 0$.

Conclusion

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Introduction

In our previous article, we discussed the inequality for the Riemann zeta function, specifically the function $x \mapsto x(\zeta(x+1) - 1)$ and the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ for all $x > 0$. In this article, we will answer some frequently asked questions about the inequality for the Riemann zeta function.

Q: What is the Riemann zeta function?

A: The Riemann zeta function is a fundamental object of study in analytic number theory. It is defined as the infinite series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$.

Q: What is the significance of the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$?

A: The inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ is important in analytic number theory because it provides a bound on the value of the zeta function for large values of $x$. This bound has numerous applications in the field, including the study of prime numbers and the distribution of prime numbers.

Q: How can we use the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ to study prime numbers?

A: The inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ can be used to study prime numbers by providing a bound on the value of the zeta function for large values of $x$. This bound can be used to estimate the number of prime numbers less than or equal to $x$, which is an important problem in number theory.

Q: What are some other applications of the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$?

A: The inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ has numerous applications in analytic number theory, including the study of the distribution of prime numbers, the study of the properties of the zeta function, and the study of the properties of other arithmetic functions.

Q: How can we prove that the function $x \mapsto x(\zeta(x+1) - 1)$ is decreasing?

A: To prove that the function $x \mapsto x(\zeta(x+1) - 1)$ is decreasing, we need to show that its derivative is negative for all $x > 0$. This can be done by computing the derivative of the function and showing that it is negative.

Q: What is the relationship between the inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ and the function $x \mapsto x(\zeta(x+1) - 1)$?

A: The inequality $\zeta(x+1) \leq \frac{1}{x^2} + 1$ is related to the function $x \mapsto x(\zeta(x+1) - 1)$ because the function is decreasing, which implies that the inequality holds for all $x > 0$.

Q: What are some open problems related to the inequality for the Riemann zeta function?

A: There are several open problems related to the inequality for the Riemann zeta function, including the study of the properties of the zeta function for large values of $x$, the study of the distribution of prime numbers, and the study of the properties of other arithmetic functions.

Conclusion

In this article, we have answered some frequently asked questions about the inequality for the Riemann zeta function. We hope that this article has provided a clear and concise explanation of the inequality and its applications in analytic number theory.