Keating-Snaith Conjecture on Moments of the Riemann Zeta Function

Keating-Snaith Conjecture on Moments of the Riemann Zeta Function

Definitions and mathematical statement

For fixed real $k>0$, define $I_k(T)=\int_0^T|\zeta(\tfrac12+it)|^{2k}\,\mathrm d t$. Let $G$ be the standard Barnes $G$-function, normalized by $G(1)=1$ and its usual canonical product, with $G(z+1)=\Gamma(z)G(z)$. Define \[ a_k=\prod_{p\ \mathrm{prime}}(1-p^{-1})^{k^2} \sum_{m=0}^{\infty}\left(\frac{\Gamma(m+k)}{\Gamma(k)m!}\right)^2p^{-m}. \] The assertion is $I_k(T)\sim a_kG(1+k)^2G(1+2k)^{-1}T(\log T)^{k^2}$ as $T\to\infty$. The parameter $k$ is fixed before this limit, and the target includes the leading constant, not just the logarithmic exponent. The functional equation alone is not being used as a unique definition of $G$.

Formulation and concept references: [S1].

Short English statement

Do all positive real moments of the zeta function on its central line have the precise leading asymptotic predicted by the arithmetic product and random-matrix factor?

Research attempt

Current frontier.

The exact leading asymptotic is classical for $k=1,2$, whereas the all-positive-real-$k$ assertion remains the conjecture stated in [S1]. Florea's 2026 account distinguishes the leading-constant problem from sharp order-of-magnitude bounds [E1], Sections 1--3. Harper proves the upper order $T(\log T)^{k^2}$ for each fixed $k\ge0$ assuming RH, without identifying the leading constant [E2]. The hybrid Euler--Hadamard approach separates prime and zero factors but its requisite moment splitting is conjectural in general [E3]. These distinctions are retained below; RH and a random-matrix model are not assumed to be theorems.

Attack route.

Direct mean-square evaluation of a polynomial approximating $\zeta^k$ loses control once its length becomes too large. A hybrid factorization could preserve the arithmetic Euler factor while transferring only the remaining component to a zero statistic. The route pursued here proves the arithmetic-side statements with a slowly growing cutoff, then writes the remaining requirement as a weighted moment. The important question is whether the prime weight can legitimately be decoupled from that residual; its variance will show why a naive Cauchy--Schwarz step is inadequate.

Attempt: checking the arithmetic constant.

Put $d_k(p^m)=(k)_m/m!$, for fixed $k>0$. On any compact interval of such $k$, its polynomial growth in $m$ implies \[ \sum_{m\ge0}d_k(p^m)^2p^{-m}=1+\frac{k^2}{p}+O_k(p^{-2}). \] Multiplication by $(1-p^{-1})^{k^2}$ cancels the term of order $p^{-1}$. The resulting local factors are positive and $1+O_k(p^{-2})$, so the Euler product $a_k$ converges to a positive number. This argument also justifies finite-prime product manipulations made below; it does not assert convergence of the Euler product for $\zeta$ on the critical line.

For $k=1$, $a_1=1$ and $G(2)^2/G(3)=1$. For $k=2$, \[ \sum_{m\ge0}(m+1)^2x^m=\frac{1+x}{(1-x)^3},\qquad a_2=\prod_p(1-p^{-2})=\zeta(2)^{-1}, \qquad\frac{G(3)^2}{G(5)}=\frac1{12}. \] Thus the proposed fourth-moment coefficient is $1/(2\pi^2)$, as required by the established case. The next integer Barnes factor is $G(4)^2/G(7)=1/8640$. These identities are independently checked by the script; their correctness supplies no estimate for the sixth moment.

Attempt: an explicit mean-square barrier.

For $A(t)=\sum_{n\le N}a_n n^{-it}$ and $S=\sum_{n\le N}|a_n|^2$, direct integration gives \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-6d-65-61-6e-2d-73-71-75-61-72-65} \left|\frac1T\int_T^{2T}|A(t)|^2\,\mathrm d t-S\right| \le\frac{4N(1+\log N)}{T}S. \end{equation} To prove it, the off-diagonal integral has absolute value at most $2/(T|\log(n/m)|)$. For $n>m$, $\log(n/m)\ge(n-m)/n\ge(n-m)/N$. Hence the total error is at most \[ \frac{4N}{T}\sum_{d=1}^{N-1}\frac1d \sum_{m\le N-d}|a_ma_{m+d}| \le \frac{4N(1+\log N)}{T}S, \] where the inner sum is bounded by $S$ using Cauchy--Schwarz. All sums are finite. Thus $N\log N=o(T)$ is a sufficient diagonal regime for this particular estimate. It is not a necessary condition for all possible mean-value methods. In particular, the inequality does not dismiss the more delicate off-diagonal methods that establish the fourth moment. It merely shows exactly why extending this direct argument to long polynomials would be unjustified.

Attempt: a rigorous finite-prime factor.

For $y\ge2$ define the finite, never-vanishing product \[ P_y(t)=\prod_{p\le y}(1-p^{-1/2-it})^{-1},\qquad \langle F\rangle_T=\frac1T\int_T^{2T}F(t)\,\mathrm d t. \] For each fixed $y$ and $k>0$, \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-66-69-78-65-64-2d-79} \lim_{T\to\infty}\langle|P_y|^{2k}\rangle_T =A_y(k):=\prod_{p\le y}\sum_{m\ge0}d_k(p^m)^2p^{-m}. \end{equation} Here is a justification not invoking a random-prime assumption. A nonzero integer combination of the numbers $\log p$, $p\le y$, cannot be zero, by unique factorization. The average of each nonconstant character of the resulting torus flow therefore tends to zero by direct integration. Approximation by trigonometric polynomials extends this to continuous functions on the finite torus. The integrand in question is continuous because $p^{-1/2}<1$. On each circle the absolutely convergent binomial series for $(1-z)^{-k}$ and orthogonality give the local sum in (30006978:fixed-y). The finite torus integral factors.

The convergent Euler product and Mertens' product theorem give \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-41-79} A_y(k)\sim a_k(e^\gamma\log y)^{k^2} \qquad(y\to\infty). \end{equation} These familiar prime-factor asymptotics are also the arithmetic input in [E3]; the product $P_y$ used here is an elementary finite product, not an assertion of the full smoothed hybrid formula from that paper.

A diagonal choice now provides $y=y(T)\to\infty$ with $\log y=o(\log T)$ for which (30006978:fixed-y) holds with relative error $o(1)$ simultaneously for $k$ and $2k$. Explicitly, choose a sequence $y_j\to\infty$, then a strictly increasing sequence $T_j$ so that both fixed-$y_j$ relative errors are at most $1/j$ for every $T\ge T_j$, and also $\log y_j\le\sqrt{\log T_j}$. Put $y(T)=y_j$ between successive thresholds. This is an existence construction, not a quantitative cutoff such as $y=T^\theta$.

Attempt: the remaining weighted statement.

For this cutoff write $b_T=e^\gamma\log y(T)$ and set \[ Q_T(t)=\frac{|P_{y(T)}(t)|^{2k}} {\langle|P_{y(T)}|^{2k}\rangle_T},\qquad D_T(t)=\left(\frac{b_T}{\log T}\right)^{k^2} \left|\frac{\zeta(1/2+it)}{P_{y(T)}(t)}\right|^{2k}. \] These are nonnegative measurable functions and $\langle Q_T\rangle_T=1$. There is an exact identity \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-77-65-69-67-68-74-65-64-2d-69-64-65-6e-74-69-74-79} \frac{\langle|\zeta(1/2+it)|^{2k}\rangle_T} {a_k(\log T)^{k^2}} =\frac{\langle|P_{y(T)}|^{2k}\rangle_T}{a_k b_T^{k^2}} \langle Q_TD_T\rangle_T. \end{equation} The prefactor tends to one. Thus the local moment asymptotic reduces to \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-6d-69-73-73-69-6e-67-2d-77-65-69-67-68-74-65-64} \langle Q_TD_T\rangle_T\longrightarrow g_k:=\frac{G(1+k)^2}{G(1+2k)}. \end{equation} For example, the two genuinely additional statements $\langle D_T\rangle_T\to g_k$ and $\langle(Q_T-1)D_T\rangle_T\to0$ would suffice. Neither has been proved in this argument. Summing local $[T,2T]$ asymptotics over dyadic intervals would then give the requested integral from $0$ to $T$: finitely many initial intervals contribute $o(T(\log T)^{k^2})$, and the geometric tail of the interval lengths controls the remaining error. Conversely the printed global asymptotic implies the local one by subtraction.

Stress test: the weight is not harmless.

The same fixed-prime computation yields \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-38-3a-76-61-72-69-61-6e-63-65} \langle Q_T^2\rangle_T \sim\frac{a_{2k}}{a_k^2}b_T^{2k^2}\longrightarrow\infty. \end{equation} Consequently $\langle D_T\rangle_T\to g_k$ alone says nothing about the covariance in (30006978:missing-weighted). A Cauchy--Schwarz argument would need, for example, $\|D_T-g_k\|_2=o(b_T^{-k^2})$, an extremely strong concentration statement not supplied by a moment model. Nonnegative random variables with the right separate means can still have substantial positive correlation; the exact identity has not removed this arithmetic obstacle.

Weak distributional convergence is likewise insufficient. To see this without making a claim about $\zeta$, let $L\to\infty$ and start with the model $Z_L=\exp(\sqrt{L/2}\,N)$ for a standard Gaussian $N$. Its $2k$-th moment is $e^{k^2L}$. With probability $e^{-(2kb-k^2)L}$, for any $b>k/2$, replace it by $e^{bL}$. This probability tends to zero, so the normalized logarithm has the same weak Gaussian limit, but the added contribution to its $2k$-th moment is exactly $e^{k^2L}$. The leading coefficient doubles asymptotically. This construction falsifies the proposed inference from a central-limit law to the desired moment constant.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The finite-product moment evaluation, diagonal cutoff construction, weighted identity, and variance calculation are justified. They isolate a weighted prime--residual correlation requirement and rule out two insufficient transfers: a bare Gaussian limit and an unweighted residual mean. The missing statement (30006978:missing-weighted) still contains the difficult zero statistics and their correlation with primes. No leading constant beyond the established cases has been proved here.

Sources

• S1 J. P. Keating and N. C. Snaith, "Random Matrix Theory and zeta(1/2+it)," Communications in Mathematical Physics 214 $2000$, 57-89. https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/keating-snaith.pdf. Catalogue formulation source.
• E1 Alexandra Florea, "A survey of moment bounds for $\zeta(s)$: From Heath-Brown's work to the present," J. London Math. Soc. 113 $2026$, no. 1, e70376. Sections 1--3. https://doi.org/10.1112/jlms.70376. Current distinction between exact moment asymptotics and moment-size bounds.
• E2 Adam J. Harper, "Sharp conditional bounds for moments of the Riemann zeta function," arXiv:1305.4618 $2013$. Main theorem, under the Riemann Hypothesis. https://arxiv.org/abs/1305.4618. Conditional upper bounds, with no asserted leading constant.
• E3 S. M. Gonek, C. P. Hughes and J. P. Keating, "A hybrid Euler--Hadamard product for the Riemann zeta function," Duke Math. J. 136 $2007$, 507--549; arXiv:math/0511182. Introduction and prime/zero moment-splitting discussion. https://www.sas.rochester.edu/mth/people/faculty/gonek-steve/assets/pdf/hybridformula.pdf. Motivation for separating arithmetic and zero factors; no conjectural splitting is used as a theorem.