The moments of the Riemann zeta function, ζ(s), are typically considered along the critical line, i.e., for s = 1/2 + it, where t is a real number. Specifically, the k-th moment is defined as:
I_k(T) = ∫0T |ζ(1/2 + it)|2k dt
These moments are extremely important in analytic number theory, particularly in connection with the distribution of prime numbers and the Riemann Hypothesis. Here's a summary of what's known, and what's conjectured:
Known Results (Asymptotic Formulas):
k = 0: This is a trivial case. I0(T) = T.
k = 1/2 (Hardy and Littlewood, Ingham, more recent works): It has been long known that I1/2(T) has the approximate main term C√T * log T, where C is an explicit constant. More recent work shows that I1/2(T) ~ C √T log T, and even provides lower-order terms and good error bounds.
k = 1 (Hardy and Littlewood, 1918):
I1(T) = ∫0T |ζ(1/2 + it)|2 dt ~ T log T
More precisely, Hardy and Littlewood showed that I1(T) = T log T + (2γ - 1 - log(2π))T + O(T1/2+ε), where γ is the Euler-Mascheroni constant. Ingham later improved the error term.
k = 2 (Ingham, 1926):
I2(T) = ∫0T |ζ(1/2 + it)|4 dt ~ (1 / (2π2)) T (log T)4
Ingham showed that I2(T) = (1/(2π2)) T (log T)4 + O(T (log T)3).
Conjectures (and related results):
General k (Conjectures): For k > 2, precise asymptotic formulas are conjectured, not proven. These conjectures are of the form:
Ik(T) ~ gk ak T (log T)k2
where: * gk is a constant that depends on k (and is difficult to determine). It's often described as a product over primes, capturing the arithmetic nature of the problem. * ak is an arithmetic factor given by ak = ∏p prime (1 - 1/p)k2 ∑m=0∞ (Γ(k + m) / (Γ(k)m!))2 p-m
k = 3 and k = 4 (Conjectures by Conrey, Farmer, Keating, Rubinstein, and Snaith, 2005): Based on random matrix theory (RMT) heuristics and number-theoretic arguments, specific values for g3 and g4 have been conjectured:
Connections to Random Matrix Theory (RMT): The conjectures for the moments are deeply connected to RMT. The idea is that the distribution of the values of |ζ(1/2 + it)| (for large t) should resemble the distribution of the characteristic polynomials of large random unitary matrices. This analogy has led to very precise predictions for the constants gk.
Lower Bounds: For general k, even proving good lower bounds of the form Ik(T) >> T(log T)k2 is a major open problem. Some progress has been made, particularly by Soundararajan, Radziwiłł, and others. The best such lower bounds are conditional on the Riemann Hypothesis.
Upper Bounds: Upper bounds for Ik(T) are also a significant area of research. The Lindelöf Hypothesis is equivalent to the statement that Ik(T) = O(T1+ε) for all k > 0 and any ε > 0. Unconditionally, we have weaker bounds, using techniques like Weyl differencing. The current best known upper bounds, due to work of Bourgain, show some improvement over older methods, but fall far short of the Lindelof Hypothesis.
Key Difficulties and Open Problems:
Proving the Asymptotic Formulas for k > 2: This is the most significant open problem. Even proving the correct order of magnitude (T(log T)k2) is extremely difficult.
Determining the Constants gk: Random matrix theory provides very strong conjectures, but rigorous proofs are lacking.
Improving Upper and Lower Bounds: Making progress towards the Lindelöf Hypothesis (for upper bounds) or towards the conjectured order of magnitude (for lower bounds) is a major challenge.
Understanding the Arithmetic Nature: The constants ak reflect the influence of prime numbers. Understanding how this arithmetic information is encoded in the moments is a deep question.
In summary, the moments of the Riemann zeta function are a central topic in analytic number theory. While the first two moments are well-understood, higher moments remain shrouded in mystery, with deep connections to random matrix theory and the distribution of prime numbers. The conjectures surrounding these moments are some of the most important unsolved problems in mathematics.
The moments of the Riemann zeta function, denoted as (\zeta(s)), particularly along the critical line (\text{Re}(s) = \frac{1}{2}), are a central topic in analytic number theory. These moments are defined as integrals of the form:
[ I_k(T) = \int_0^T |\zeta(\frac{1}{2} + it)|^{2k} \, dt ]
where (k) is a positive integer or sometimes a real number. Although exact results for these moments are known only for some small values of (k), there are conjectures and asymptotic formulas for general (k). Here's a summary of known results and conjectures:
Trivially, (I_0(T) = T).
(k = 1) Moment:
This result stems from the approximate formula for the number of zeros of (\zeta(s)) up to height (T).
(k = 2) Moment:
This follows from work by Hardy and Littlewood. The coefficient can vary slightly depending on normalization conventions.
(k = 3) and Higher Moments:
These conjectures are grounded in connections between the distribution of values of the zeta function along the critical line and eigenvalues of random matrices from certain ensembles. Though rigorous proofs are still forthcoming for general (k), these moment conjectures have been verified numerically and contribute profoundly to our understanding of the zeta function's behavior.
A convenient way to state the “moments” of the Riemann zeta function along the critical line is via the integrals
M₍2k₎(T) = ∫₀^T |ζ(1/2 + i t)|^(2k) dt,
where k ≥ 1 is an integer (so 2k is the power on |ζ|). One is interested in the asymptotic behavior of M₍2k₎(T) as T → ∞. Here is a brief summary of what is known and what is conjectured:
1) The second moment (k = 1).
This is the integral
M₂(T) = ∫₀^T |ζ(1/2 + i t)|² dt.
It has been known classically (using techniques going back to Hardy–Littlewood, Ingham, and others) that
M₂(T) ∼ T log T
plus lower-order terms in T. In fact, there are more precise asymptotic expansions of the form
M₂(T) = T log T + (2γ − 1 − log(2π)) T + O(T^(1/2+ε)),
where γ is the Euler–Mascheroni constant. Thus the leading term T log T (and even the next constant times T) is rigorously established.
2) The fourth moment (k = 2).
This is
M₄(T) = ∫₀^T |ζ(1/2 + i t)|⁴ dt.
It is widely believed (and supported by large-scale computations, partial results, and heuristic arguments) that
M₄(T) ∼ a₄ ⋅ T (log T)⁴
for some positive constant a₄. Random-matrix heuristics (in particular, the so-called CFKRS “recipe” due to Conrey, Farmer, Keating, Rubinstein, and Snaith) give a conjectural value for a₄. However, unlike the case k = 1, there is currently no unconditional proof of the full main term. There are various partial results and upper/lower bounds consistent with this conjecture.
3) Higher moments (k ≥ 3).
For the sixth moment M₆(T) = ∫₀^T |ζ(1/2 + i t)|⁶ dt and beyond, there are no fully proven main-term asymptotics. The general conjecture (again supported by random-matrix theory) is that
M₍2k₎(T) ∼ a₂₍k₎ ⋅ T (log T)^(k²),
where a₂₍k₎ is a (conjecturally) computable constant. Thus the exponent of log T is k² for the 2k-th moment.
While these conjectural forms are backed up by some partial results, moment bounds, and special-case verifications, none is proven in full for k ≥ 2. In particular, the constants a₂₍k₎ for k ≥ 2 remain conjectural in the unconditional setting.
4) Connection to random-matrix theory.
One of the main reasons we believe the above asymptotic forms so precisely is the analogy between the Riemann zeta function on the critical line and characteristic polynomials of large random matrices (in the unitary group). The “CFKRS formula” (due to Conrey, Farmer, Keating, Rubinstein, and Snaith) uses these random-matrix ideas to predict very detailed asymptotics for M₍2k₎(T), including the constant factors.
Summary of what is “known.”
• The 2nd moment is rigorously known in full leading-order form (and even secondary terms).
• The 4th moment has strong conjectures about its leading term and constant but is not fully proven.
• For the 6th and higher even moments, only partial results and bounds exist.
• Random-matrix theory (and various “recipes”) give precise conjectural forms for all integer moments, but the proofs remain incomplete except in the 2nd-moment case.
Hence, aside from the fully resolved case of the second moment, the higher moments of |ζ(1/2 + i t)| remain one of the major open problems in analytic number theory.
I remember that moments are often studied in the context of the critical line, which is Re(s) = 1/2. So maybe the moments refer to integrals like ∫_{0}^{T} |ζ(1/2 + it)|^{2k} dt for some integer k. Yes, that sounds familiar. These are called the 2k-th moments of the zeta function.
I think the study of these moments is important in understanding the distribution of the values of the zeta function on the critical line and has connections to the distribution of prime numbers. But I need to recall what exactly is known about these moments.
First, let's consider the case when k=1. The second moment, which is ∫_{0}^{T} |ζ(1/2 + it)|^2 dt. I believe this was first studied by Hardy and Littlewood, and they found that it's asymptotic to T log T as T tends to infinity. Then Ingham later improved this result, getting a more precise asymptotic formula with a main term T log T plus some lower-order terms. So for the second moment, we have a pretty good understanding.
What about the fourth moment, which is when k=2? This is ∫_{0}^{T} |ζ(1/2 + it)|^4 dt. I recall that this was investigated by Ingham as well, and he found an asymptotic formula involving T (log T)^4. But I think the exact asymptotic with the leading coefficient was established later. Maybe by Heath-Brown? Wait, actually, I think the fourth moment was handled by Motohashi, who related it to the cubic moment of the zeta function and used spectral theory. But I might be mixing up some results here.
Wait, let me check. The fourth moment conjecture states that ∫_{0}^{T} |ζ(1/2 + it)|^4 dt ~ C T (log T)^4 for some constant C. I think the value of C is known, and it's related to the product of zeta functions at certain points. Maybe involving zeta(2) or something. But I need to be precise here.
I recall that for the fourth moment, the leading coefficient is 1/(2π²) multiplied by some factors. Wait, actually, the exact asymptotic formula is known and involves an arithmetic factor and a geometric factor. The main term is (T/(2π)) (log T)^4 times a constant, and there are lower-order terms with powers of log T down to a constant term. But I might need to look up the exact expression.
But since I can't look things up right now, I have to rely on memory. Let me think. The general conjecture for the 2k-th moment is that ∫_{0}^{T} |ζ(1/2 + it)|^{2k} dt ~ C_k T (log T)^{k^2} as T → ∞, where C_k is a constant depending on k. This is known as the Keating-Snaith conjecture, which connects these moments to random matrix theory. They conjectured that the leading order coefficient C_k can be expressed in terms of products of Barnes G-functions or something similar, based on comparisons with the moments of characteristic polynomials of random unitary matrices.
But the known results are only for specific values of k. For k=1 and k=2, the moments are known. For k=1, as I mentioned, it's T log T. For k=2, it's T (log T)^4 with a specific coefficient. For higher k, like k=3, the moments are not known rigorously, but conjectures exist based on random matrix theory and other heuristics.
Wait, but I should confirm if the fourth moment is actually proven. Yes, I think it is. The fourth moment was proven by Ingham for the upper bound, and then later more precise results were obtained. Maybe by Heath-Brown? Let me recall. Heath-Brown did work on the fourth moment of the zeta function, and I think he established an asymptotic formula with a power-saving error term. So the result is that ∫_{0}^{T} |ζ(1/2 + it)|^4 dt ~ \frac{1}{2\pi^2} T (log T)^4. Wait, but I might be mixing up the constants.
Alternatively, the leading coefficient might involve values of the Riemann zeta function itself. For example, in the fourth moment, the coefficient could be related to ζ(2), but ζ(2) is π²/6, so maybe 1/(2π²) would be 3/(π^4) or something. Hmm, this is getting a bit fuzzy.
Alternatively, maybe the exact asymptotic is given by a main term of a constant times T (log T)^4 plus lower-order terms, and the constant is an explicit product over primes. For example, in the second moment, the coefficient is 1/(2π²) * something. Wait, no, the second moment asymptotic is T log T / (2π) + ...? Wait, actually, the mean square of |ζ(1/2 + it)|^2 is asymptotic to (1/(2π)) T log T. Wait, no, let me think again.
The mean value theorem of Ingham states that ∫{0}^{T} |ζ(1/2 + it)|^2 dt ~ T log T. Wait, but with a coefficient? Wait, actually, the result is that ∫{0}^{T} |ζ(1/2 + it)|^2 dt ~ T (log T - (1 + log 2π)). Wait, no, maybe not. Let me recall the exact formula.
Hardy and Littlewood proved that ∫_{0}^{T} |ζ(1/2 + it)|^2 dt ~ T log T. Then later, Ingham found the next term, which is something like T (log T - (1 + log 2π - γ)) where γ is Euler's constant. But I might be misremembering.
Alternatively, the formula is ∫_{0}^{T} |ζ(1/2 + it)|^2 dt = T log T + (2γ - 1 - log 2π)T + E(T), where E(T) is an error term. So the leading term is T log T, and the next term is a constant times T.
Similarly, for the fourth moment, the leading term is conjectured to be C T (log T)^4, where C is a specific constant. But I think for the fourth moment, the asymptotic formula is known with the leading term and some lower-order terms.
In any case, the key point is that for the second and fourth moments, we have asymptotic formulae, but for higher moments (sixth, eighth, etc.), only conjectures exist based on random matrix theory and other heuristic methods.
Additionally, there are results on what are called "twisted moments" where you integrate the zeta function times some Dirichlet polynomial or other functions, which can sometimes be used to approximate higher moments.
Another thing to consider is the work of Conrey, Ghosh, Gonek, and others who have worked on moments of the zeta function. For example, they have conjectured formulas for the moments involving contour integrals and ratios of zeta functions, but these are still conjectural.
Also, there's the work of Soundararajan and Radziwill on upper bounds for moments. They showed that the 2k-th moment is bounded by T (\log T)^{k^2 + ε} for any ε > 0, which is in line with the conjectured asymptotic.
But in terms of exact asymptotics, only the second and fourth moments are known. For the sixth moment, even the leading term is not known rigorously.
Wait, but I should check if the fourth moment is actually known. Let me think. Yes, I believe the fourth moment was established by Ingham with an error term, and then improved by Heath-Brown who got a sharper error term. The result is that ∫_{0}^{T} |ζ(1/2 + it)|^4 dt ~ \frac{1}{2\pi^2} T (log T)^4. Wait, but I need to confirm the constant.
Alternatively, the exact formula might involve a residue from some integral. For example, using the approximate functional equation or the moments conjecture.
Alternatively, the leading coefficient for the fourth moment is known to be 6/(π^2) times some factor. Wait, this is getting confusing. Let me try to recall the precise statement.
In the work of Heath-Brown on the fourth moment of the Riemann zeta function, he proved that
∫_{0}^{T} |ζ(1/2 + it)|^4 dt = \frac{T}{2\pi^2} (log T)^4 + O(T (log T)^3).
But I need to check the constants. Wait, actually, the coefficient might involve ζ(2). Since ζ(2) = π²/6, so 1/ζ(2) = 6/π². So if the coefficient is 1/(2π²), that would be 6/(2π²) * ζ(2) = 3/π² * π²/6 = 1/2. Wait, that seems off.
Alternatively, maybe the coefficient is 1/(2π²) multiplied by something else. Alternatively, the main term could be (1/(2π²)) T (log T)^4. But I need to verify.
Alternatively, let's think about the conjecture. The Keating-Snaith conjecture suggests that the leading coefficient for the 2k-th moment is a product over primes involving the value of the zeta function and some arithmetic factors. For k=2, this would give a specific constant.
But perhaps instead of getting bogged down in the exact constants, which I might not remember precisely, I should state that the second and fourth moments are known asymptotically with their leading terms and some lower-order terms, while higher moments are conjectured but not proven.
Additionally, there are results on moments averaged over other ranges or with weights, but the main moments on the critical line are the ones usually referred to.
Another aspect is the use of the Riemann Hypothesis (RH) in these results. Some results on moments are conditional on RH, but the second and fourth moments are known unconditionally.
Also, there's the work of Selberg on the distribution of log ζ(1/2 + it) and its connection to the moments, showing that it's distributed like a Gaussian variable, which is related to the moments growing in a specific way.
But focusing back on the original question: "What are the known moments of the ζ function?" So the answer would be that for the Riemann zeta function on the critical line Re(s) = 1/2, the second moment (k=1) and the fourth moment (k=2) are known asymptotically, while higher moments are not rigorously established but are conjectured based on random matrix theory and other heuristics.
To be more precise, the second moment ∫{0}^{T} |ζ(1/2 + it)|² dt is asymptotically T log T, and the fourth moment ∫{0}^{T} |ζ(1/2 + it)|⁴ dt is asymptotically (1/(2π²)) T (log T)^4, up to a constant factor. However, I need to check if the constant is correctly recalled.
Wait, actually, in Ingham's result for the fourth moment, the asymptotic formula is ∫_{0}^{T} |ζ(1/2 + it)|^4 dt ~ (1/(2π²)) T (log T)^4. But I think the exact coefficient might be different. Let me think.
In the paper by Heath-Brown (1981), "The fourth power moment of the Riemann zeta function", he proved that
∫_{0}^{T} |ζ(1/2 + it)|^4 dt = T \frac{1}{2\pi^2} (log(T/2\pi))^4 + O(T (log T)^3).
So the main term is (1/(2π²)) T (log(T/(2π)))^4. If we expand that, it's (1/(2π²)) T (log T - log 2π)^4, which would give a quartic in log T, but the leading term is (1/(2π²)) T (log T)^4. So the leading coefficient is 1/(2π²).
Therefore, the fourth moment is known with the leading term (1/(2π²)) T (log T)^4.
So putting it all together, the known moments are the second and fourth moments on the critical line, with asymptotic formulas as T tends to infinity.
Additionally, there are results for moments at other points in the critical strip, but the most studied are those on the critical line.
In summary, the answer is that the second and fourth moments of the Riemann zeta function on the critical line are known asymptotically. The second moment grows like T log T, and the fourth moment grows like T (log T)^4 with specific constants. Higher moments are conjectured but remain unproven, with their asymptotics predicted by random matrix theory.
Final Answer The known moments of the Riemann zeta function on the critical line are the second and fourth moments. Specifically, as ( T \to \infty ),
[ \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^2 \, dt \sim T \log T ]
and
[ \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^4 \, dt \sim \frac{1}{2\pi^2} T (\log T)^4. ]
Higher moments remain conjectural. Thus, the boxed answer for the known moments is:
\boxed{\int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{2k} \, dt \sim C_k T (\log T)^{k^2} \text{ for } k = 1, 2}
The known moments of the Riemann zeta function on the critical line are the second and fourth moments. These moments are given by integrals of powers of the zeta function over certain intervals. Specifically, as ( T \to \infty ):
Second Moment (k=1): [ \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^2 \, dt \sim T \log T ] This result was established by Hardy and Littlewood, with further refinements by Ingham.
Fourth Moment (k=2): [ \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^4 \, dt \sim \frac{1}{2\pi^2} T (\log T)^4 ] This result was proven by Heath-Brown, building on earlier work by Ingham.
Higher moments (for ( k \geq 3 )) are not rigorously established but are conjectured to follow asymptotic forms predicted by random matrix theory. The conjectured form for the ( 2k )-th moment is ( \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{2k} \, dt \sim C_k T (\log T)^{k^2} ), where ( C_k ) is a constant depending on ( k ).
\boxed{\int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{2k} \, dt \sim C_k T (\log T)^{k^2} \text{ for } k = 1, 2} ```
The moments of the Riemann zeta function are a fascinating and challenging area of number theory. They refer to integrals of the form:
Integer Moments: $I_k(T) = \frac{1}{T} \int_0^T |\zeta(1/2 + it)|^{2k} dt$
Half-Integer Moments: Less commonly studied, but integrals of the form $\int_0^T |\zeta(1/2 + it)|^{k} dt$ where $k$ is a half-integer (e.g., 3/2, 5/2).
Understanding these moments provides insights into the distribution of values of the zeta function on the critical line and is related to important conjectures like the Riemann Hypothesis. However, obtaining exact formulas or even accurate asymptotic estimates for these moments is extremely difficult.
Here's a summary of what is known:
1. Known Integer Moments (Asymptotic Results):
k = 0: This is trivial: $\frac{1}{T} \int_0^T 1 \, dt = 1$.
k = 1: Hardy and Littlewood (1918) showed: $I_1(T) = \frac{1}{T} \int_0^T |\zeta(\frac{1}{2} + it)|^2 dt = \log T + (2\gamma - 1) + O(\frac{1}{\log T})$, where $\gamma$ is the Euler-Mascheroni constant.
k = 2: Ingham (1926) showed: $I_2(T) = \frac{1}{T} \int_0^T |\zeta(\frac{1}{2} + it)|^4 dt = \frac{1}{2\pi^2} (\log T)^4 + \text{lower order terms}$.
k = 3: Conrey and Gonek (2001) showed: $I_3(T) = \frac{1}{T} \int_0^T |\zeta(\frac{1}{2} + it)|^6 dt = P_9(\log T) + O(T^{-1/2 + \epsilon})$ where $P_9$ is a polynomial of degree 9 in $\log T$ whose leading coefficient is $\frac{42}{9!}$.
k = 4: Conrey and Keating (1992) conjectured (based on Random Matrix Theory): $I_4(T) = \frac{1}{T} \int_0^T |\zeta(\frac{1}{2} + it)|^8 dt \sim c_4 (\log T)^{16}$, where $c_4 = \frac{24024}{16!}$. The conjecture has not been proven, but is strongly supported by numerical evidence and theoretical arguments. This is an active area of research. The best-known bounds are of the form $(\log T)^A \leq I_4(T) \leq (\log T)^B$ where $A<16$ and $B>16$.
k > 4: For $k > 4$, there are no proven asymptotic formulas. Keating and Snaith (2000) formulated general conjectures for all integer $k$ based on Random Matrix Theory. The conjectures are of the form: $I_k(T) \sim c_k (\log T)^{k^2}$, where $c_k$ is a constant that can be expressed in terms of certain arithmetic factors. These constants are closely related to the number of ways to write integers as sums of $k$ squares. 2. The Random Matrix Theory Connection:
Random Matrix Theory (RMT) has played a pivotal role in conjecturing the asymptotic behavior of the zeta function moments. The eigenvalues of large random matrices from certain ensembles (e.g., the Unitary Group) exhibit statistical properties that are remarkably similar to those conjectured for the zeros of the Riemann zeta function. Keating and Snaith, in particular, used RMT to make precise predictions about the moments.
3. The Recipe Method (and its limitations):
Conrey, Farmer, Keating, Rubinstein, and Snaith developed a method called the "recipe" for computing the conjectured moments. This involves:
The recipe method works well for $k \le 3$ and leads to the RMT conjectures for larger $k$. However, it relies on certain unproven assumptions about the analytic continuation and behavior of the multiple Dirichlet series.
4. Difficulties and Open Problems:
In summary: While significant progress has been made in understanding the moments of the Riemann zeta function, especially through the use of Random Matrix Theory, many challenging open problems remain. Proving the conjectures for $k > 4$ is a major goal in analytic number theory. The moments provide a crucial link between the Riemann Hypothesis, the distribution of prime numbers, and the theory of random matrices.
The Riemann zeta function, denoted as ( \zeta(s) ), is a significant function in mathematics, particularly in number theory. The moments of the zeta function refer to integrals or averages of powers of the zeta function over certain contours or intervals. These moments are of great interest because they provide insights into the distribution of the zeros of the zeta function and have implications for the distribution of prime numbers.
Several moments of the zeta function are known or have been studied:
The first moment of the zeta function on the critical line ( \Re(s) = \frac{1}{2} ) is given by integrals of the form: [ \int_{-\infty}^{\infty} \zeta\left(\frac{1}{2} + it\right) W(t) \, dt, ] where ( W(t) ) is a weighting function. The evaluation of such integrals has been studied extensively.
Second Moment:
The second moment, or the mean square, is particularly important and is given by: [ \int_{-T}^{T} \left| \zeta\left(\frac{1}{2} + it\right) \right|^2 \, dt. ] The asymptotic behavior of this integral is known and is related to the distribution of the zeros of the zeta function.
Fourth Moment:
The fourth moment: [ \int_{-T}^{T} \left| \zeta\left(\frac{1}{2} + it\right) \right|^4 \, dt, ] has been studied, and its asymptotic behavior has been determined.
Higher Moments:
Additionally, there are conjectures and heuristic results concerning the moments of the zeta function, such as the conjectures by Keating and Snaith which relate the moments to random matrix theory. These conjectures provide predictions for the asymptotic behavior of the moments and have led to significant advances in our understanding of the zeta function.
It's important to note that while some moments are known or conjectured, many aspects of the behavior of the zeta function's moments remain areas of active research in analytic number theory.
The moments of the Riemann zeta function, particularly on the critical line (where the real part of the complex variable is ( \frac{1}{2} )), are fundamental objects of study in analytic number theory. These moments are typically defined as integrals of the form:
[ I_k(T) = \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{2k} \, dt ]
where ( \zeta(s) ) is the Riemann zeta function, ( t ) is a real variable, and ( k ) is a positive real number representing the moment order.
The first moment, involving the square of the absolute value of the zeta function, is well-understood. The asymptotic formula as ( T \to \infty ) is given by:
[ I_1(T) = \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{2} \, dt = T \log T + (2\gamma - 1)T + o(T) ]
Here, ( \gamma ) denotes the Euler–Mascheroni constant, and ( o(T) ) represents lower-order terms that become negligible as ( T ) grows.
The second moment, which involves the fourth power of the zeta function's absolute value, also has a well-established asymptotic formula:
[ I_2(T) = \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^{4} \, dt \sim \frac{1}{2\pi^2} T (\log T)^4 + \text{lower-order terms} ]
The leading term here involves ( (\log T)^4 ), and more precise coefficients can be determined through deep analytical methods.
For moments of order three and higher, the precise asymptotic behavior remains conjectural. Predictions for these higher moments are often guided by analogies with random matrix theory and involve intricate conjectures about the distribution of zeros of the zeta function. Notable conjectures in this area include:
Keating–Snaith Conjectures: These conjectures relate the moments of the zeta function to moments of characteristic polynomials of random matrices from the unitary group, providing precise predictions for the leading-order asymptotics of ( I_k(T) ) for all positive integers ( k ).
Conrey–Ghosh–Gonek Conjectures: These extend the understanding of moments by incorporating lower-order terms and finer structure beyond the leading asymptotics.
Despite significant progress and numerical evidence supporting these conjectures, rigorous mathematical proofs for moments with ( k \geq 3 ) remain elusive as of the latest knowledge up to October 2023.
Understanding these moments is not only a central problem in analytic number theory but also has deep connections to other areas such as random matrix theory, quantum chaos, and the statistical distribution of prime numbers.
There is a well‐developed body of work on the moments of the Riemann ζ–function on the critical line. For k a positive real number (typically, one studies even moments, so k is taken so that one looks at the 2k–th moment), one sets
Mₖ(T) = ∫₀ᵀ |ζ(½ + it)|^(2k) dt.
It is widely expected (and supported by extensive numerical work and random matrix theory heuristics) that
Mₖ(T) ~ Cₖ · T · (log T)^(k²) as T → ∞.
Here Cₖ is an explicit constant; for example, the Keating–Snaith conjectures (based on analogies with characteristic polynomials of random matrices) predict a precise value for Cₖ in terms of the Barnes G–function and other standard quantities.
However, from the rigorous side of analytic number theory, only a few of these asymptotic formulas have been established completely:
The second moment (k = 1):
It is known that
∫₀ᵀ |ζ(½ + it)|² dt = T log T + (2γ – 1 – log 2π) T + O(T^(½+ε)),
where γ is Euler’s constant. This result goes back to Hardy–Littlewood and has been refined by many authors.
The fourth moment (k = 2):
Ingham proved that
∫₀ᵀ |ζ(½ + it)|⁴ dt = (1/(2π²)) T (log T)⁴ + O(T (log T)³).
Later work by Heath-Brown and others has given further improvements in the error term.
For moments of order higher than 4 (that is, for k > 2), we do not yet have a full asymptotic evaluation. There are upper and lower bounds and many partial results, but no complete rigorous proof of an asymptotic formula of the same quality as for k = 1, 2. (The corresponding conjecture is that the 2k–th moment should be asymptotically Cₖ T (log T)^(k²) with the constant given by the Keating–Snaith prediction.)
To summarize: • For k = 1 (the second moment) and k = 2 (the fourth moment) we have precise asymptotic formulas: – M₁(T) = T log T + (2γ – 1 – log 2π)T + O(T^(½+ε)), – M₂(T) = (1/(2π²)) T (log T)⁴ + O(T (log T)³). • For k > 2, one has the conjectured behavior Mₖ(T) ~ Cₖ T (log T)^(k²), though proving such a result remains one of the challenging open problems in the field.
Thus, the “known moments” of ζ usually refer to these established results for k = 1, 2 together with the general conjectural picture for higher moments.