June 01, 2026
For the Lindelöf Hypothesis concerning the Riemann zeta function \(\zeta(s)\), upper bounds as \(\mathop{\mathrm{Im}}(s)\to\infty\) have been extensively studied for many years. In particular, the Lindelöf Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise upper bounds and integral mean values of zeta-functions.
In this paper, we consider an analogue of the Lindelöf Hypothesis for the Barnes multiple zeta function \(\zeta_r (s,a,(w_1,\dots,w_r)) = \sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty (a+m_1 w_1+\cdots+m_r w_r)^{-s}\), and establish equivalent conditions in terms of integral mean values. In particular, the situation depends essentially on the \(\mathbb{Q}\)-rank of \(\langle w_1,\dots,w_r\rangle\), and it is especially interesting that phenomena peculiar to the Barnes multiple zeta function appear according to this rank.
Let \(r\) be a positive integer, \(s=\sigma+it \;(\sigma, t \in \mathbb{R})\) a complex variable, \(a>0\), and \(w_1,\ldots,w_r>0\). The Barnes multiple zeta function, denoted by \(\zeta_r(s,a,(w_1, \dots,w_r))\) and introduced in [1]–[3], is defined as \[\begin{align} \label{zeta95r} \zeta_r (s,a,(w_1,\dots,w_r)) =\sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty \frac{1}{(a+m_1 w_1+\cdots+m_r w_r)^s}. \end{align}\tag{1}\] The series on the right-hand side converges absolutely for \(\mathrm{Re}(s) > r\) and can be meromorphically continued to the entire complex \(s\)-plane. The function has simple poles located at \(s=1,\dots,r\).
The Barnes multiple zeta function was originally introduced by Barnes in the course of developing the theory of multiple gamma functions, which can be regarded as higher-dimensional analogues of the classical gamma function. This construction is based on Lerch’s formula for the Hurwitz zeta function, which expresses the logarithm of the classical gamma function as \[\log \Gamma(a) = \zeta'(0,a) + \frac{1}{2}\log 2\pi.\] In particular, the multiple gamma function can be defined in terms of the derivative of the Barnes multiple zeta function at \(s=0\) as \[\log \Gamma_r(a,\mathbf{w}) = \left. \frac{\partial}{\partial s} \zeta_r(s,a,\mathbf{w}) \right|_{s=0} + \rho_r(\mathbf{w}),\] where \(\rho_r(\mathbf{w})\) is a normalization constant depending only on \(\mathbf{w}\). Since then, Barnes-type zeta functions have been studied extensively in connection with special functions and analytic number theory.
Next, we will introduce some results of the most fundamental order evaluations for the Riemann zeta-function \[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\] These evaluations concern the order of \(\zeta(\sigma+it)\) as the imaginary part \(t\) tends to infinity. In particular, research on the order evaluation for \(\zeta(1/2 + it)\) as \(t \rightarrow \infty\) is particularly prominent. As a classical asymptotic formula, \[\zeta(\sigma + it) \ll |t|^{(1-\sigma)/2 +\varepsilon} \quad (0 \leq \sigma \leq 1, |t| \geq 2)\] is known, especially when \(\sigma=1/2\) is \(\zeta( 1/2 + it ) \ll t^{1/4+\varepsilon}\). This result is obtained using the functional equation of \(\zeta(s)\) and the Phragmén-Lindelöf convexity principle. Another classical asymptotic formula for \(\zeta(s)\) was proved by Hardy and Littlewood using the following formula; \[\label{order95of95zeta} \zeta(s) = \sum_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} + O(x^{-\sigma}) \qquad (x \rightarrow \infty),\tag{2}\] uniformly for \(\sigma \geq \sigma_0 > 0,\;|t| < 2\pi x/C\), when \(C > 1\) is a constant. This formula gives an indication in the discussion in the critical strip of \(\zeta(s)\). Hardy and Littlewood improved this to \[\label{Hardy-Littlewood} \zeta \left( \frac{1}{2} + it \right) \ll t^{1/6 + \varepsilon}\tag{3}\] by the van der Corput method, applying to 2 . Furthermore, the problem of improving upper bounds for \(\zeta(1/2+it)\) with respect to \(t\) has been extensively studied by many mathematicians. To mention a few results, in 1988, Bombieri and Iwaniec established the bound \(\zeta(1/2 + it) \ll t^{9/56 + \varepsilon}\). Since then, several refinements have been obtained, and Huxley proved \(\zeta(1/2 + it) \ll t^{32/205 + \varepsilon}\) in 2005. Furthermore, in 2017, \[\zeta\left( \frac{1}{2} + it \right) \ll t^{13/84 + \varepsilon}\] was proved by Bourgain (see [4]). At present, this is the best known bound.
Improvements in order estimates for \(\zeta(1/2+it)\) are still ongoing, and the true order is conjectured to be \[\label{Lindelof95hypothesis} \zeta \left( \frac{1}{2} + it \right) \ll t^{\varepsilon}\tag{4}\] for any \(\varepsilon >0\). This conjecture is called the Lindelöf hypothesis. Furthermore, it is well known that the Lindelöf hypothesis (4 ) is equivalent to \[\label{Lindelof95hypothesis95int} \int_2^T \left| \zeta\left(\frac{1}{2}+it\right) \right|^{2k} dt = O(T^{1+\varepsilon})\tag{5}\] for any \(k\in\mathbb{N}\) and any \(\varepsilon>0\) (see, for example, [5] and [6]).
For convenience in this paper, we introduce the following boldface symbols to represent index tuples: \[\begin{align} &\mathbf{m}=(m_1,\dots,m_r), \quad \mathbf{n}=(n_{1},\dots,n_{r}), \quad\\ &\mathbf{w}=(w_1,\dots,w_r), \;\, \quad \mathbf{1}=(1,\dots,1). \end{align}\] We assume that \(w_1,\dots,w_r\in \mathbb{R}_{>0}\) throughout this paper. Using this notation, 1 can be rewritten as \[\begin{align} \zeta_r(s,a,\mathbf{w}) =\sum_{m_1,\dots,m_r\ge 0} \frac{1}{(a+\mathbf{m}\cdot\mathbf{w})^s}. \end{align}\] As an analogue of the classical formula (2 ), the following result was obtained in [7].
Proposition 1 (Theorem 2.1 in [8]). Let \(r-1<\sigma_1<\sigma_2\), \(x\ge 1\), and \(C>1\). If \(s=\sigma+it \in \mathbb{C}\) with \(\sigma_1<\sigma<\sigma_2\) and \(|t|\le 2\pi x/C\), then \[\begin{align} \zeta_r(s,a,\mathbf{w}) &=\sum_{0\le m_1\le x} \cdots \sum_{0\le m_r\le x} \frac{1}{(a+\mathbf{m}\cdot\mathbf{w})^{s}}\\ &\quad - \sum_{\substack{E\subseteq\{w_1,\dots,w_r\}\\E\ne\emptyset}} (-1)^{\#E} \frac{(a+x\sum_{ e\in E }e)^{r-s}}{(s-1)\cdots (s-r) w_1\cdots w_{r}} +O(x^{r-1-\sigma}) \end{align}\] as \(x\rightarrow \infty\).
Furthermore, by applying the finite sum approximation in Proposition 1 with \(x=t\), we immediately obtain the following upper bound for \(\zeta_r(\sigma+it,a,\mathbf{w})\) as \(t \to \infty\):
Proposition 2 (Theorem 1.5 in [9]). Let \(s=\sigma+it\) with \(t\ge 2\), \(a>0\), and \(\mathbf{w}\in\mathbb{R}_{>0}^r\). Then the following bounds hold: \[\zeta_r(\sigma+it,a,\mathbf{w}) \ll \begin{cases} 1 & \text{if } \quad \sigma> r,\\ \log t & \text{if } \quad \sigma=r,\\ t^{\,r-\sigma} & \text{if } \quad r-1< \sigma < r, \end{cases}\] as \(t \to \infty\), uniformly for \(\sigma\) in any fixed compact subinterval of each region. The implied constants depend on \(a\), \(r\), \(\mathbf{w}\), and the chosen strip.
As an analogue of the equivalence between 4 and 5 for the classical Lindelöf Hypothesis, we obtain the following theorem for the Barnes multiple zeta function.
Theorem 1. Suppose that \(a>0\) and that \(\mathbf{w}=\lambda (p_1,\ldots,p_r)\), where \(\lambda>0\), \(p_j\in \mathbb{N}\), and \(\gcd(p_1,\ldots,p_r)=1\). For any \(\varepsilon>0\), \[\zeta_r(\sigma+it,a,\mathbf{w}) =O(t^\varepsilon) \qquad \left(r-\frac{1}{2}\leq \sigma\leq r\right)\] holds if and only if \[\int_2^T \left| \zeta_r(\sigma+it,a,\mathbf{w}) \right|^{2k}dt = O(T^{1+\varepsilon}) \qquad \left(r-\frac{1}{2}\leq \sigma\leq r\right)\] holds for any \(k\in \mathbb{N}\) and any \(\varepsilon>0\). Here the implied constants may depend on \(\varepsilon\), \(k\), \(\sigma\), \(a\), and \(\mathbf{w}\).
Theorem 2. Suppose that \(a>0\), \(w_1,\dots,w_r>0\), and \(\dim_{\mathbb{Q}}\langle w_1,\dots,w_r\rangle=d \;(1\le d \le r)\). Assume that \(\zeta_r'(s,a,\mathbf{w})\) with respect to \(s\) satisfy polynomial growth conditions in vertical strips in the region \(r-d/2\leq \sigma\leq r\). Then, for any \(\varepsilon>0\), \[\zeta_r(\sigma+it,a,\mathbf{w}) = O(t^\varepsilon) \qquad \left(r-\frac{d}{2}\leq \sigma\leq r\right)\] holds if and only if \[\int_2^T \left| \zeta_r(\sigma+it,a,\mathbf{w}) \right|^{2k}dt = O(T^{1+\varepsilon}) \qquad \left(r-\frac{d}{2}\leq \sigma\leq r\right)\] holds for any \(k\in\mathbb{N}\) and any \(\varepsilon>0\). Here the implied constants may depend on \(\varepsilon\), \(k\), \(\sigma\), \(a\), and \(\mathbf{w}\).
Remark 3. The assumption on the polynomial growth condition in Theorem 2 is indispensable when the \(\mathbb{Q}\)-rank \(d\) is greater than or equal to \(2\). Without this assumption, the sufficiency part of the theorem may fail to hold. To see this breakdown, consider the rank two case (\(r=2\)) with \(w_1=1\) and \(w_2=\theta\), where \(\theta\) is a Liouville number. By definition, there exist infinitely many positive integers \(q\) such that the distance to the nearest integer, denoted by \(\|q\theta\|\), becomes extremely small.
In the functional equation in [10] for \(\zeta_2(s, a, (1, \theta))\), we encounter the following series containing the small denominators:\[\label{eq:small95denom95series}\sum_{n=1}^\infty \frac{e^{2\pi in(y_1+\theta y_2)}}{(e^{2\pi in\theta}-1)n^{1-s}}.\tag{6}\] Let \(s = \sigma + it\) with \(1 \leq \sigma \leq 2\). For every positive integer \(N\), there exist infinitely many pairs of integers \(p\) and \(q\) \((q\ge 2)\) such that \(0<\left|\theta-p/q\right|<q^{-N}\). Choose one such pair and denote it by \((p_k,q_k)\). Then \(\|q_k \theta\| = |q_k \theta - p_k| < q_k^{-(N-1)}\), and hence the target factor for \(n=q_k\) satisfies \[\left| e^{2\pi i q_k \theta} - 1 \right|^{-1} \asymp \|q_k \theta\|^{-1} > q_k^{N-1}.\] Hence the coefficients occurring in 6 can become arbitrarily large along suitable subsequences. This suggests that the functional equation by itself does not automatically yield polynomial growth estimates for \(\zeta_r(s,a,\mathbf{w})\). Therefore, the polynomial growth assumption in Theorem 2 should not be regarded merely as a technical condition.
In this section, we collect several auxiliary lemmas that will be used in the proofs of the main theorems.
Lemma 1. Suppose that \(\zeta_r'(\sigma+it,a,\mathbf{w})\ll t^B\) holds in a fixed vertical strip. If \(|\zeta_r(\sigma+it_0,a,\mathbf{w})|\geq t_0^\eta\) for some \(\eta>0\), then \[|\zeta_r(\sigma+it,a,\mathbf{w})| \geq \frac{1}{2} t_0^\eta\] holds whenever \(|t-t_0|\leq c t_0^{\eta-B}\) with a sufficiently small constant \(c>0\).
Proof. Put \[F(t)=\zeta_r(\sigma+it,a,\mathbf{w}).\] Then \[F'(t)=i\zeta_r'(\sigma+it,a,\mathbf{w}),\] and hence, by the assumption, \[|F'(t)|\ll t^B\] in the strip under consideration. Therefore, for \(t\) sufficiently close to \(t_0\), the mean value theorem gives \[|F(t)-F(t_0)| \leq \sup_{\xi\in[t,t_0]} |F'(\xi)|\, |t-t_0| \ll t_0^B |t-t_0|.\] If \[|t-t_0|\leq c t_0^{\eta-B}\] with \(c>0\) sufficiently small, then \[|F(t)-F(t_0)|\leq \frac{1}{2} t_0^\eta.\] Thus \[|F(t)| \geq |F(t_0)|-|F(t)-F(t_0)| \geq t_0^\eta-\frac{1}{2} t_0^\eta = \frac{1}{2} t_0^\eta.\] This proves the lemma. ◻
Remark 4. \(\zeta_r(s,a,\mathbf{1})\) is referred to as the Hurwitz multiple zeta function, and it can be expressed as a linear combination of single Hurwitz zeta functions (see [11] p. 86): \[\begin{align} \zeta_r(s,a,\mathbf{1})=\sum_{j=0}^{r-1}p_{r,j}(a)\zeta_H(s-j, a). \label{HMZ-HZ} \end{align}\tag{7}\] In this formula, \(p_{r,j}(a)\) is given by \[\begin{align} p_{r,j}(a)=\frac{1}{(r-1)!}\sum_{l=j}^{r-1}(-1)^{r+1-j}\binom{l}{j}S(r,l+1) a^{l-j}, \end{align}\] where \(S(r,l+1)\) denotes the Stirling number of the first kind.
Proof of Theorem 1. Since \(w_j=\lambda p_j\), we have \[\zeta_r(s,a,\mathbf{w}) = \lambda^{-s} \sum_{m_1,\ldots,m_r\geq 0} \frac{1}{(a/\lambda+p_1m_1+\cdots+p_rm_r)^s}.\] Put \(b=a/\lambda\). For each integer \(n\geq 0\), let \[A(n)=\#\{(m_1,\ldots,m_r)\in \mathbb{Z}_{\geq 0}^r \mid p_1m_1+\cdots+p_rm_r=n\}.\] Then \[\zeta_r(s,a,\mathbf{w}) = \lambda^{-s}\sum_{n=0}^{\infty}\frac{A(n)}{(n+b)^s}.\] It is well known that \(A(n)\) is a quasi-polynomial of degree \(r-1\). Hence the above Dirichlet series can be expressed as a finite linear combination of Hurwitz zeta-functions: \[\zeta_r(s,a,\mathbf{w}) = \lambda^{-s} \sum_{\nu=0}^{q-1} \sum_{\ell=0}^{r-1} c_{\nu,\ell}\, \zeta_H(s-\ell,\beta_\nu),\] where \(q\) may be taken to be a common multiple of \(p_1,\ldots,p_r\), the constants \(c_{\nu,\ell}\) depend only on \(p_1,\ldots,p_r\) and \(a\), and \(\beta_\nu>0\).
The implication from the Lindelöf-type bound to the moment estimates is immediate. Indeed, replacing \(\varepsilon\) by \(\varepsilon/(2k)\) in the pointwise estimate gives \[\left|\zeta_r(\sigma+it,a,\mathbf{w})\right|^{2k} \ll t^\varepsilon,\] and therefore \[\int_2^T \left|\zeta_r(\sigma+it,a,\mathbf{w})\right|^{2k}dt \ll T^{1+\varepsilon}.\] Conversely, assume that the moment estimate holds for every \(k\in\mathbb{N}\) and every \(\varepsilon>0\). Using the above finite linear representation in terms of Hurwitz zeta-functions, together with the standard polynomial growth estimates for the Hurwitz zeta-function in vertical strips, we obtain a polynomial growth estimate for \(\zeta_r(s,a,\mathbf{w})\) and also for its derivative with respect to \(t\) in the strip \(r-1/2\leq \sigma\leq r\). Then the standard local mean-value argument applies: if \(\left|\zeta_r(\sigma+it_0,a,\mathbf{w})\right|\) were larger than \(t_0^\eta\) for some \(\eta>0\), then the derivative bound would imply that this large value persists on a short interval around \(t_0\). Integrating over this short interval and using the assumed \(2k\)-th moment estimate gives a contradiction when \(k\) is chosen sufficiently large. Hence \[\zeta_r(\sigma+it,a,\mathbf{w})=O(t^\varepsilon)\] for every \(\varepsilon>0\). This proves the equivalence. ◻
Proof of Theorem 2. Assume that \[\zeta_r(\sigma+it,a,\mathbf{w}) = O\!\left(|t|^{\varepsilon/2k}\right) \qquad \left( r-\frac{d}{2}\le \sigma\le r \right)\] for any \(\varepsilon>0\) and \(k\in\mathbb{N}\). Then \[\int_1^T |\zeta_r(\sigma+it,a,\mathbf{w})|^{2k}dt \ll \int_1^T t^\varepsilon dt \ll T^{1+\varepsilon}.\] Conversely, assume that the moment estimate holds for every \(k\in\mathbb{N}\) and every \(\varepsilon>0\). Suppose, to the contrary, that the Lindelöf-type estimate fails. Then there exist \(\eta>0\) and a sequence \(t_n\to\infty\) such that \[\left|\zeta_r(\sigma+it_n,a,\mathbf{w})\right| \geq t_n^\eta .\] By the assumed polynomial growth of \(\zeta_r'(s,a,\mathbf{w})\) in vertical strips, there exists a constant \(B>0\) such that \[\zeta_r'(\sigma+it,a,\mathbf{w})\ll t^B\] in the strip under consideration. Hence, for \(|t-t_n|\leq c\,t_n^{\eta-B}\) with a sufficiently small constant \(c>0\), the mean value theorem gives \[\left|\zeta_r(\sigma+it,a,\mathbf{w})\right| \geq \frac{1}{2} t_n^\eta .\] Therefore, \[\int_{t_n-c t_n^{\eta-B}}^{t_n+c t_n^{\eta-B}} \left|\zeta_r(\sigma+it,a,\mathbf{w})\right|^{2k}dt \gg t_n^{2k\eta+\eta-B}.\] Choosing \(k\) sufficiently large, this contradicts the assumed bound \[\int_2^T \left|\zeta_r(\sigma+it,a,\mathbf{w})\right|^{2k}dt \ll T^{1+\varepsilon}.\] Thus the Lindelöf-type estimate follows. ◻
Remark 5. By Proposition 1, we have \[\zeta_r(s,a,\mathbf{w}) = \sum_{0\le m_1,\dots,m_r\le x} (a+\mathbf{m}\cdot\mathbf{w})^{-s} + O(x^{r-1-\sigma})\] for \(x\asymp t\) and \(r-1<\sigma<r\). The trivial estimate for the finite sum yields \[\zeta_r(\sigma+it,a,\mathbf{w}) \ll t^{r-\sigma}.\] However, the oscillatory factor \[(a+\mathbf{m}\cdot\mathbf{w})^{-it} = e^{-it\log(a+\mathbf{m}\cdot\mathbf{w})}\] suggests that additional cancellation may occur. Indeed, after fixing \(m_2,\dots,m_r\), the inner sum with respect to \(m_1\) can be regarded as a one-dimensional exponential sum of the form \[\sum_{0\le m_1\le x} e^{-it\log(a+B+m_1w_1)} (a+B+m_1w_1)^{-\sigma},\] where \[B=m_2w_2+\cdots+m_rw_r.\] This suggests the possible bound \[\zeta_r(\sigma+it,a,\mathbf{w}) \ll t^{1/2} \qquad \left(r-1<\sigma\le r-\frac{1}{2}\right),\] which would improve the trivial estimate \[\zeta_r(\sigma+it,a,\mathbf{w}) \ll t^{r-\sigma}\] in this region. A rigorous proof would require a more delicate analysis of the associated exponential sums, such as dyadic decomposition and van der Corput type estimates. This pointwise estimate also gives the mean-square bound \[\int_1^T |\zeta_r(\sigma+it,a,\mathbf{w})|^2dt \ll T^2 \qquad \left(r-1<\sigma<r-\frac{1}{2}\right),\] which improves the bound obtained directly from \(\zeta_r(\sigma+it,a,\mathbf{w})\ll t^{r-\sigma}\). These considerations suggest that stronger pointwise and mean-square bounds may hold in the region \(r-1<\sigma\le r-1/2\). However, a rigorous proof appears to require a substantially more delicate analysis of the associated exponential sums, and we leave this problem for future work.