Where are the zeros of the zeta function?
The following papers discuss the location of the zeros of the zeta function:
The zeros of the Riemann zeta function, denoted by ζ(s), are the values of s for which ζ(s) = 0. The Riemann zeta function is defined for complex numbers s with real part greater than 1 by the following infinite sum:
ζ(s) = 1/(1^s) + 1/(2^s) + 1/(3^s) + ...
It has been analytically continued to the entire complex plane except for a simple pole at s = 1.
Trivial zeros: These are the negative even integers, i.e., s = -2, -4, -6, ...
Nontrivial zeros: These are the complex numbers s = σ + it, where σ and t are real numbers, that lie in the "critical strip" 0 < Re(s) < 1. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, asserts that all nontrivial zeros have a real part equal to 1/2, i.e., they lie on the critical line Re(s) = 1/2. While the Riemann Hypothesis remains unproven, extensive numerical computations have provided strong evidence for its validity. Voros explores the analytic features of zeta functions built from the Riemann zeros, explicitly detailing their polar structure and special values. Ghişa uses global mapping properties to study nontrivial zeros, and Lee attempts to prove the Riemann Hypothesis, arguing that the zeros should lie on the critical line. Raman investigates zeros of the Dirichlet Eta function.
The Riemann zeta function ζ(s) has two types of zeros:
• Trivial zeros: These occur at the negative even integers −2, −4, −6, … .
• Nontrivial zeros: These lie in the so-called critical strip 0 < Re(s) < 1.
It is conjectured (the famous Riemann Hypothesis) that all nontrivial zeros lie on the critical line Re(s) = 1/2, but this has not been proven.
The zeros of the Riemann zeta function, denoted as $\zeta(s)$, are the complex numbers $s$ for which $\zeta(s) = 0$.
The zeros can be classified into two types: \begin{enumerate} \item \textbf{Trivial Zeros:} These are the negative even integers: $s = -2, -4, -6, \dots$ \item \textbf{Non-trivial Zeros:} These are complex numbers $s = \sigma + it$ where $\sigma$ and $t$ are real numbers, such that $0 < \sigma < 1$. \end{enumerate}
The Riemann Hypothesis (one of the most famous unsolved problems in mathematics) states that all non-trivial zeros have a real part equal to $\frac{1}{2}$. In other words, if $s$ is a non-trivial zero, then $s = \frac{1}{2} + it$ for some real number $t$.
Here's a summary:
What is known about the non-trivial zeros:
In short:
The zeros of the Riemann zeta function, denoted as ζ(s), are of significant interest in number theory and complex analysis. Understanding their distribution is crucial for insights into the distribution of prime numbers. Here's an overview of where these zeros are located:
The Riemann zeta function has what are known as trivial zeros. These occur at the negative even integers: [ s = -2, -4, -6, -8, \ldots ] These zeros are well-understood and arise from the functional equation of the zeta function.
The more intriguing zeros are the non-trivial zeros, which lie in the "critical strip" where the real part of ( s ) satisfies: [ 0 < \Re(s) < 1 ] According to the Riemann Hypothesis, one of the most famous unsolved problems in mathematics, all non-trivial zeros are conjectured to lie on the critical line where: [ \Re(s) = \frac{1}{2} ]
Selected Zeros on the Critical Line: A vast number of non-trivial zeros have been computationally verified to lie on the critical line ( \Re(s) = \frac{1}{2} ). As of my knowledge cutoff in April 2023, billions of zeros have been checked and all conform to this pattern.
Symmetry: The non-trivial zeros exhibit symmetry with respect to the critical line and the real axis. If ( \rho = \frac{1}{2} + i\gamma ) is a zero, then so are ( \rho = \frac{1}{2} - i\gamma ), ( \rho = \frac{1}{2} + i(-\gamma) ), and ( \rho = \frac{1}{2} - i(-\gamma) ).
Density of Zeros: The zeros become denser as the imaginary part ( t ) of ( s = \sigma + it ) increases. The Riemann-von Mangoldt formula provides an estimate for the number of zeros with imaginary parts between 0 and ( T ): [ N(T) \approx \frac{T}{2\pi} \log\left(\frac{T}{2\pi}\right) - \frac{T}{2\pi} ]
If the Riemann Hypothesis is true, it implies a precise distribution of prime numbers and has profound implications across number theory and related fields. Despite extensive numerical evidence supporting the hypothesis, a general proof remains elusive.
While the Riemann zeta function is the most studied, other zeta functions (such as the Hurwitz zeta function or zeta functions of algebraic varieties) also have zeros with their own distributions and conjectures, but these are distinct from the Riemann case.
In summary:
Understanding the zeros of the zeta function continues to be a central pursuit in analytic number theory, with ongoing research exploring their properties and implications.
The Riemann zeta function, ζ(s), has two types of zeros:
Trivial zeros:
These occur at the negative even integers. That is,
ζ(s) = 0 for s = –2, –4, –6, …
These zeros come from the sine factor in the completed (or “xi”) form of the zeta function when one writes down its functional equation.
Nontrivial zeros:
These lie in the “critical strip,” which is the region of the complex plane where the real part of s satisfies 0 < Re(s) < 1. More precisely, all of the known nontrivial zeros lie in this strip.
Furthermore, the famous Riemann Hypothesis asserts that every nontrivial zero actually lies on the “critical line,” where Re(s) = 1/2. Although billions of zeros have been computed and all of them lie on this critical line, a complete proof of the hypothesis remains one of the major unsolved problems in mathematics.
Additional symmetry properties:
• The nontrivial zeros appear in complex conjugate pairs.
• They are symmetric about the line Re(s) = 1/2 because of the functional equation satisfied by ζ(s).
In summary, the zeros of ζ(s) are:
• Trivial zeros at s = –2, –4, –6, …
• Nontrivial zeros within the strip 0 < Re(s) < 1 (and conjecturally all on Re(s) = 1/2).
The zeros of the Riemann zeta function are a fundamental topic in analytic number theory, and their locations have significant implications for the distribution of prime numbers. The zeta function, denoted by ζ(s), is defined for complex numbers s with real part greater than 1 by the series:
[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} ]
This function can be analytically continued to the entire complex plane, except for a simple pole at s = 1.
Trivial Zeros: These are the zeros that occur at the negative even integers, i.e., s = -2, -4, -6, etc. These zeros are relatively well-understood and are not particularly mysterious.
Nontrivial Zeros: These are the zeros that lie in the critical strip, which is the region of the complex plane where the real part of s is between 0 and 1. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, conjectures that all nontrivial zeros of the zeta function have real part equal to 1/2.
The nontrivial zeros of the zeta function are symmetric about the real axis and the critical line Re(s) = 1/2. This means that if ρ is a nontrivial zero, then so are its complex conjugate ρ* and 1 - ρ.
Extensive computations have been performed to locate the nontrivial zeros of the zeta function. As of now, it has been verified that the first tens of billions of nontrivial zeros lie on the critical line Re(s) = 1/2.
The Riemann Hypothesis remains unproven, despite much effort from mathematicians over the past century and a half.
There are also results that provide bounds on the proportion of zeros on the critical line. For example, it has been shown that at least 40% of the nontrivial zeros lie on the critical line.
The distribution of the zeros of the zeta function is closely linked to the distribution of prime numbers. The explicit formulae in analytic number theory relate sums over the zeros of the zeta function to sums over primes, making the study of these zeros essential for understanding the primes.
In summary, while the trivial zeros of the zeta function are known to be at the negative even integers, the location of the nontrivial zeros remains a major open question in mathematics, with the Riemann Hypothesis proposing that they all lie on the critical line Re(s) = 1/2.
The zeta function, ζ(s), is defined for complex numbers s with real part greater than 1 by the infinite series:
ζ(s) = Σₙ₌₁^∞ 1/nˢ
But this series doesn't converge for other values of s, right? So there's something called analytic continuation that extends the zeta function to the entire complex plane except for s=1, where it has a simple pole. Okay, so the zeta function is holomorphic everywhere except at s=1. Got it.
Now, the zeros of the zeta function are the points s where ζ(s)=0. From what I remember, there are two types of zeros: the "trivial" zeros and the "non-trivial" ones. The trivial zeros are supposed to be at the negative even integers: -2, -4, -6, and so on. But wait, why are those zeros? Maybe it's related to the functional equation of the zeta function. Let me recall that the functional equation relates ζ(s) to ζ(1-s). The exact form is:
ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s)
Where Γ is the gamma function, the generalization of factorial. So if s is a negative even integer, say s = -2n for some positive integer n, then sin(πs/2) becomes sin(-nπ) which is zero. So sin(πs/2) would be zero whenever s is an even negative integer, leading ζ(s) to be zero. Hence the trivial zeros at s = -2, -4, -6, etc. That makes sense.
But then there are the non-trivial zeros, which are the ones everyone cares about. The Riemann Hypothesis states that all non-trivial zeros lie on the critical line Re(s) = 1/2. But even without assuming the hypothesis, I think it's known that non-trivial zeros lie in the critical strip 0 < Re(s) < 1. Wait, how do we know that?
So, first, the region where the zeros can be. The critical strip is between 0 and 1 in the real part. The critical line is the line where Re(s) = 1/2. But how do we know that all non-trivial zeros are in the critical strip?
Well, from the functional equation, ζ(s) relates to ζ(1-s). So if s is a zero in the region Re(s) > 1, then 1-s would be in the region Re(s) < 0. But except for the trivial zeros, ζ(s) doesn't have zeros in Re(s) < 0 because the trivial zeros are already accounted for by the sin term. Wait, is that right?
Wait, maybe another approach. Euler's product formula for ζ(s) is:
ζ(s) = Πₚ (1 - 1/p^s)^(-1)
where the product is over all primes p. This converges for Re(s) > 1. Because the product is over primes, and each term is non-zero, ζ(s) can't have zeros in the region Re(s) > 1. Therefore, any zero in the critical strip must be in 0 ≤ Re(s) ≤ 1. But actually, since ζ(s) has a pole at s=1, we know there are no zeros for Re(s) ≥ 1 except possibly on the line Re(s)=1. But it was proven by Hadamard and de la Vallée Poussin that ζ(s) has no zeros on Re(s)=1, which was crucial in proving the Prime Number Theorem. So all non-trivial zeros must lie in the critical strip 0 < Re(s) < 1.
Therefore, the non-trivial zeros are in 0 < Re(s) < 1. But the Riemann Hypothesis says they are actually on Re(s) = 1/2. But until now, this hasn't been proven. However, a lot of numerical evidence supports this conjecture. For example, many zeros have been computed and they all lie on the critical line.
So, putting it all together, the zeros of the Riemann zeta function are:
The trivial zeros at the negative even integers: s = -2, -4, -6, ..., which come from the poles of the gamma function in the functional equation, causing ζ(s) to be zero there.
The non-trivial zeros in the critical strip 0 < Re(s) < 1. The Riemann Hypothesis predicts they all lie on the critical line Re(s) = 1/2, but this remains unproven.
But wait, let me check again. So the functional equation is ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s). So if s is a zero, then either sin(πs/2) is zero (which gives the trivial zeros) or ζ(1-s) is zero. So if s is a non-trivial zero in the critical strip, then 1-s is also a zero. Hence the zeros are symmetric around the critical line Re(s) = 1/2. Therefore, if we have a zero at s = σ + it, then there's another zero at 1 - σ + it. So if σ = 1/2, then 1 - σ = 1/2, so they are the same point. That explains why the critical line is the axis of symmetry.
Therefore, if a zero is not on the critical line, there's another one mirrored across the line Re(s) = 1/2. So proving that all zeros lie on Re(s) = 1/2 would mean that all the zeros are self-conjugate in a sense. But so far, mathematicians have not been able to prove this, though it's widely believed.
Also, I remember that there's a so-called "Z-function" or the Riemann-Siegel Z function which is a real-valued function on the critical line that is used to study the zeros of ζ(s). The zeros of ζ(s) on the critical line correspond to real zeros of the Z-function.
Additionally, the number of non-trivial zeros in the critical strip up to some height T is known asymptotically. The Riemann-von Mangoldt formula gives that the number of zeros with 0 < Im(s) < T is about (T/(2π)) log(T/(2π)) - T/(2π) + O(log T). So there are infinitely many zeros in the critical strip.
Furthermore, it's known that at least a positive proportion of the zeros lie on the critical line, thanks to the work of Hardy, Littlewood, Selberg, and others. But this proportion is not 100%, so the full conjecture is still open.
So, summarizing:
The Riemann zeta function has zeros at the negative even integers (trivial zeros) and infinitely many non-trivial zeros in the critical strip 0 < Re(s) < 1.
The non-trivial zeros are symmetric with respect to the critical line Re(s) = 1/2 and the real axis.
The Riemann Hypothesis asserts that all non-trivial zeros lie on Re(s) = 1/2, but this remains unproven.
Therefore, the answer to "Where are the zeros of the zeta function?" is that they are at the negative even integers (trivial zeros) and in the critical strip 0 < Re(s) < 1 (non-trivial zeros), with the latter conjectured to all lie on the critical line Re(s) = 1/2.
Wait, but let me check a source or two to make sure I haven't made a mistake here. For example, the trivial zeros are at s = -2, -4, -6, etc., because of the sine term in the functional equation. Since sin(πs/2) is zero when s is an even negative integer. So that's correct. And the non-trivial zeros are in the critical strip. The critical line is the line Re(s)=1/2, which is the vertical line in the complex plane crossing 1/2 on the real axis.
Yes, so all non-trivial zeros known so far are on the critical line, and they come in pairs symmetrical around the critical line and the real axis. So if 1/2 + it is a zero, then so is 1/2 - it, and also, due to the functional equation, if σ + it is a zero, then 1 - σ + it is a zero. So combining these symmetries, if a zero is not on the critical line, then there would be four zeros: σ + it, 1 - σ + it, σ - it, and 1 - σ - it. But if it's on the critical line, then σ = 1/2, so they collapse to two points: 1/2 + it and 1/2 - it.
Hence, the non-trivial zeros are either on the critical line or come in quadruples symmetrical with respect to the critical line and the real axis. But since we haven't found any off the critical line, it's believed they are all on it.
Therefore, my conclusion is that the zeros of the zeta function are at the negative even integers and in the critical strip (0 < Re(s) < 1), with the non-trivial ones conjectured to lie precisely on the critical line Re(s) = 1/2.
Final Answer The zeros of the Riemann zeta function are the trivial zeros at negative even integers and the non-trivial zeros in the critical strip. All non-trivial zeros are conjectured to lie on the critical line. Thus, the zeros are at ( s = -2, -4, -6, \ldots ) and those in the critical strip ( 0 < \text{Re}(s) < 1 ), with the latter hypothesized to satisfy ( \text{Re}(s) = \frac{1}{2} ). The final answer is:
The nontrivial zeros lie in the critical strip ( 0 < \text{Re}(s) < 1 ) and are conjectured to all lie on the critical line ( \text{Re}(s) = \frac{1}{2} ). The trivial zeros occur at the negative even integers. So, the zeros of the zeta function are (\boxed{s = -2, -4, -6, \ldots}) and the nontrivial zeros in the critical strip ( \boxed{0 < \text{Re}(s) < 1} ), with the latter conjectured to be on ( \boxed{\text{Re}(s) = \frac{1}{2}} ).
But wait, according to the problem statement, it asks "Where are the zeros of the zeta function?" So maybe they expect the answer to mention both the trivial and non-trivial zeros. The standard answer is that the Riemann zeta function has trivial zeros at the negative even integers and non-trivial zeros in the critical strip 0 < Re(s) < 1. The Riemann Hypothesis states that all non-trivial zeros lie on the line Re(s) = 1/2.
Therefore, the answer should mention both types, so arranging that:
Trivial zeros at s = -2, -4, -6, ... and non-trivial zeros in the critical strip, with the conjecture for the critical line. So written as separate boxed statements? Or perhaps combined. The problem is in Chinese, translated as "Where are the zeros of the zeta function?" so they might need to mention both.
But perhaps the answer is best given as two parts. For example:
The zeros of the Riemann zeta function are at the negative even integers (the trivial zeros) and in the critical strip ( 0 < \text{Re}(s) < 1 ) (the non-trivial zeros), which are conjectured to lie on the critical line ( \text{Re}(s) = \frac{1}{2} ).
But in terms of boxes, maybe they need separate boxes. However, the system might expect multiple boxes. Alternatively, stating it in one sentence with multiple boxed parts. Let me see.
Alternatively, perhaps:
Trivial zeros: \boxed{s = -2, -4, -6, \ldots}
Non-trivial zeros: \boxed{0 < \text{Re}(s) < 1}
Riemann Hypothesis: \boxed{\text{Re}(s) = \frac{1}{2}}
But the original question is "Where are the zeros of the zeta function?" So maybe including both the known and the conjectured. However, in mathematical literature, the zeros are divided into the trivial ones and the non-trivial ones. The non-trivial ones are known to be in the critical strip, and conjectured to be on the critical line.
Therefore, the answer should state the known locations and the conjecture. So perhaps:
The zeros of the Riemann zeta function are the trivial zeros at ( \boxed{s = -2, -4, -6, \ldots} ) and the non-trivial zeros, which all lie in the critical strip ( \boxed{0 < \text{Re}(s) < 1} ). According to the Riemann Hypothesis, these non-trivial zeros all lie on the critical line ( \boxed{\text{Re}(s) = \frac{1}{2}} ).
But how does the user want the answer? Since the user mentions "the zeros", which include both trivial and non-trivial. So the answer should mention both.
Alternatively, if they want the answer in one box, maybe:
The zeros of the zeta function are at ( \boxed{-2, -4, -6, \ldots} ) and in the critical strip ( \boxed{0 < \text{Re}(s) < 1} ), conjectured to lie on ( \boxed{\text{Re}(s) = \frac{1}{2}} ).
But perhaps combining the critical strip and critical line into one box with the conjecture. Alternatively, separate them.
Alternatively, as per the standard answer, it's:
The Riemann zeta function has trivial zeros at the negative even integers and nontrivial zeros in the critical strip 0 < Re(s) < 1. The Riemann Hypothesis states that all nontrivial zeros lie on the critical line Re(s) = 1/2.
Expressed with boxes:
Trivial zeros: \boxed{s = -2, -4, -6, \ldots}
Nontrivial zeros: \boxed{0 < \text{Re}(s) < 1} (conjectured \boxed{\text{Re}(s) = \frac{1}{2}})
But perhaps the user expects a single box containing all parts, but since boxes are separate, maybe multiple boxes. Alternatively, it might be:
The zeros of the zeta function are located at \boxed{s = -2, -4, -6, \ldots} and in the region \boxed{0 < \text{Re}(s) < 1}, with the latter conjectured to lie on the line \boxed{\text{Re}(s) = \frac{1}{2}}.
That might be acceptable. Since the user's question is in Chinese and translated, perhaps they expect all parts to be boxed separately. So, I will present it as three boxed statements.
Final Answer The zeros of the Riemann zeta function are located at (\boxed{s = -2, -4, -6, \ldots}) (trivial zeros) and in the critical strip (\boxed{0 < \text{Re}(s) < 1}) (nontrivial zeros), which are conjectured to all lie on the critical line (\boxed{\text{Re}(s) = \frac{1}{2}}).
The Riemann zeta function, ζ(s), has two types of zeros: trivial and non-trivial.
Trivial Zeros: These occur at the negative even integers due to the functional equation of the zeta function, which involves the sine function. The trivial zeros are at: [ \boxed{s = -2, -4, -6, \ldots} ]
Non-trivial Zeros: These are located in the critical strip defined by (0 < \text{Re}(s) < 1). The Riemann Hypothesis conjectures that all non-trivial zeros lie on the critical line where the real part of (s) is (1/2). The critical strip and the conjectured critical line are: [ \boxed{0 < \text{Re}(s) < 1} ] [ \boxed{\text{Re}(s) = \frac{1}{2}} ]
Thus, the zeros of the zeta function are the trivial zeros at the negative even integers and the non-trivial zeros in the critical strip, with the latter conjectured to lie on the critical line.
The zeros of the Riemann zeta function, (\zeta(s)), are of significant interest, particularly in number theory. There are two types of zeros:
Trivial Zeros: These occur at the negative even integers, (s = -2, -4, -6, \ldots). These zeros are well-understood and uncontroversial.
Non-Trivial Zeros: These occur in the critical strip, where the real part of (s) is between 0 and 1. According to the Riemann Hypothesis, a conjecture that remains unproven, all non-trivial zeros lie on the critical line, which is defined by (\text{Re}(s) = \frac{1}{2}). This means their form is (s = \frac{1}{2} + it), where (t) is a real number.
The distribution of these non-trivial zeros, particularly whether all of them lie on the critical line, has deep implications for the distribution of prime numbers and is one of the most famous unsolved problems in mathematics.
The zeros of the Riemann zeta function, ζ(s), are a central topic in number theory, with profound implications for the distribution of prime numbers. They can be categorized into two types:
Trivial Zeros: These are the negative even integers: -2, -4, -6, -8, and so on. That is, ζ(-2n) = 0 for all positive integers n. These are relatively easy to prove and are called "trivial" because they are not considered as interesting or deep as the others. The proof stems directly from the functional equation of the zeta function and the presence of a sine term that vanishes at these values.
Non-trivial Zeros: These are all complex numbers that lie in what's called the "critical strip," which is the region of the complex plane where the real part of s is between 0 and 1 (i.e., 0 < Re(s) < 1). All non-trivial zeros found to date have a real part of 1/2. The statement that all non-trivial zeros have a real part of 1/2 is the famous Riemann Hypothesis, one of the most important unsolved problems in mathematics.
Here's a breakdown of the key ideas:
Critical Strip: The non-trivial zeros are confined to the vertical strip in the complex plane defined by 0 < Re(s) < 1. This was proven by Hadamard and de la Vallée Poussin. The Prime Number Theorem is equivalent to showing that there are no zeros on the lines Re(s) = 1 and Re(s) = 0.
Critical Line: The Riemann Hypothesis states that all non-trivial zeros lie on the "critical line," which is the vertical line Re(s) = 1/2 within the critical strip. This can be written as s = 1/2 + it, where t is a real number and i is the imaginary unit.
Infinite Non-trivial Zeros: There are infinitely many non-trivial zeros. This was proven by Hardy. Furthermore, Hardy and Littlewood showed that infinitely many of these zeros lie on the critical line.
Density of Zeros: The number of non-trivial zeros with imaginary part between 0 and T is approximately (T / (2π)) * log(T / (2π)) - T / (2π) + O(log T). This is known as the Riemann-von Mangoldt formula.
Computational Verification: The Riemann Hypothesis has been verified computationally for trillions of zeros. All zeros found so far lie on the critical line. However, this is not a proof, as there are infinitely many zeros, and a counterexample could, in principle, exist very high up.
Importance: The location of the non-trivial zeros is intimately connected to the distribution of prime numbers. The Riemann Hypothesis, if true, would provide very precise information about the error term in the Prime Number Theorem, giving a much tighter bound on how primes are distributed. Many other important results in number theory are conditional on the Riemann Hypothesis – they have been proven assuming the Riemann Hypothesis is true.
In summary: