Abstract
In this paper we study the absolute convergence of general multiple Dirichlet series defined by
where \( a_j\,(1\le j\le r)\) are arithmetic functions. In particular we completely determine the region of absolute convergence under certain conditions on the arithmetic functions.
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1 Introduction
For any integer \( r\ge 1 \), the multiple zeta function (Euler–Riemann–Zagier type) of depth r is defined by
where \( s_i\,(1\le i\le r) \) are complex variables. Throughout the article, we denote \(\Re (s_i)=\sigma _i\). It is well known (see Theorem 3, [4]) that the series (1) is absolutely convergent in the region \( \{(s_1,s_2,\ldots ,s_r)\in {\mathbb {C}}^r:\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}> i+1\,\,\text {for}\,\,0\le i\le r-1\} \). For \( r=1 \), it is nothing but the Riemann zeta function. So multiple zeta function of depth r is multi-variable generalization of Riemann zeta function. Zhao [5] and Akiyama et al. [1] independently have shown that \( \zeta _r(s_1,s_2,\ldots ,s_r) \) can be extended meromorphically to the whole \( {\mathbb {C}}^r \).
One can consider the generalization of the series defined in (1) in the following manner
where \( a_j\,(1\le j\le r)\) are arithmetic functions. For each j, if \( a_j(m)=1,\,\forall \, m\in {\mathbb {N}} \), then \( \Phi _r((s_j);(a_j))= \zeta _r(s_1,s_2,\ldots ,s_r)\).
The first question that one can ask, is to find the region of absolute convergence of the multiple Dirichlet series defined in (2). In this context we have the following result.
Theorem 1
For each \( j\,(1\le j\le r)\), let \(\varphi _{j}(s)=\sum _{m=1}^{\infty }\frac{a_j(m)}{m^s}\) be absolutely convergent for \(\Re (s)>\alpha _j>0\). Then the multiple Dirichlet series defined in (2) is absolutely convergent in the region \( U_r:=\{(s_1,s_2,\ldots ,s_r)\in {\mathbb {C}}^r:\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}>\alpha _r+\alpha _{r-1}+\cdots +\alpha _{r-i}\,\,\text {for}\,\,0\le i\le r-1\}. \)
The series defined in (2) is already considered by Matsumoto and Tanigawa in [3], where they have mentioned that this series is absolutely convergent in the trivial region \( \{(s_1,s_2,\ldots ,s_r)\in {\mathbb {C}}^r:\sigma _i>\alpha _i\,\,\text {for}\,\,1\le i\le r\} \). Since their primary goal was to study the meromorphic continuation, so there is no need to start with exact region of absolute convergence.
Remark 1
In Proposition 3.1 of [2], Matsumoto et al. have given a region of absolute convergence for certain double Dirichlet series that region is equal to \( U_r\) for \( r=2 \) in the Theorem 1.
In the following theorem we give the necessary and sufficient conditions for the series (2) to converge absolutely under certain conditions on the arithmetic functions.
Theorem 2
For each \( j\,(1\le j\le r) \), let the arithmetic function \( a_j(m) \) and the positive real number \( \alpha _j \) satisfy the following conditions \( \bullet \) \( \sum _{m=1}^{\infty }\frac{a_j(m)}{m^s} \) has abscissa of absolute convergence \( \alpha _j, \) \( \bullet \) \( \sum _{m\le t}|a_j(m)|\gg t^{\alpha _j} \) for every \( t\ge 1. \) Then we have that the series defined in (2) is absolutely convergent at \((s_1,s_2,\ldots ,s_r)\in {\mathbb {C}}^r\) if and only if \(\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}>\alpha _r+\alpha _{r-1}+\cdots +\alpha _{r-i}\) for \(0\le i\le r-1\).
As an application of Theorem 2, we have the following result.
Corollary 1
The region of absolute convergence of the series defined in (1) is
Remark 2
In Proposition 2.1 of [6], Zhao et al. have derived the necessary and sufficient conditions for absolute convergence of certain generalized multiple zeta functions. Corollary 1 is also followed from this proposition.
It is well known that if the Dirichlet series \( \sum _{m=1}^{\infty }\frac{1}{m^s}\) converges at \( s=s_{0} \), then this series converges for all s such that \( \Re (s)>\Re (s_0)\). In the following theorem we prove an analogues result for general multiple Dirichlet series in case of absolute convergence.
Theorem 3
For the arithmetic functions \( a_j(m)\,(1\le j\le r) \), let the series defined in (2) be absolutely convergent at \( (s_1^{\prime },s_2^{\prime },\ldots ,s_r^{\prime })\in {\mathbb {C}}^r\). Then the series converges absolutely at each \( (s_1,s_2,\ldots ,s_r)\in {\mathbb {C}}^r \) such that \(\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}\ge \sigma _r^{\prime }+\sigma _{r-1}^{\prime }+\cdots +\sigma _{r-i}^{\prime }\) for \(0\le i\le r-1\), where \( \Re (s_i^{\prime })=\sigma _i^{\prime } \) for \(1\le i\le r\).
2 Preliminaries
In this section we give some lemmas which are necessary ingredients to prove Theorem 2.
Lemma 1
For \(n\ge 1\) and \( \sigma \in {\mathbb {R}} \), we have
\( \sum _{m=1}^{\infty }\frac{|a(m)|}{m^{\sigma }}<\infty \) if and only if \( \sum _{m=1}^{\infty }\frac{|a(m)|}{(n+m)^{\sigma }}<\infty \) .
Proof
The lemma easily follows from the fact that
\(\square \)
Lemma 2
For the positive real number \( \alpha \), let the arithmeitc function a(m) satisfies \( \sum _{m\le t}|a(m)|\gg t^{\alpha } \) for every \( t\ge 1 \). Then \( \sum _{m=1}^{\infty }\frac{|a(m)|}{m^\alpha } \) diverges.
Proof
Using Abel summation formula, we have
This implies that \( \sum _{m\le t}\frac{|a(m)|}{m^\alpha }\gg \alpha \log t \) and hence \( \sum _{m=1}^{\infty }\frac{|a(m)|}{m^\alpha } \) diverges. \(\square \)
Lemma 3
Let the arithmetic function a(m) and positive real number \( \alpha \) satisfy the following conditions
-
\( \sum _{m=1}^{\infty }\frac{a(m)}{m^s} \) has abscissa of absolute convergence \( \alpha , \)
-
\( \sum _{m\le t}|a(m)|\gg t^{\alpha } \) for every \( t\ge 1. \) Then for \( \sigma >\alpha \) and \( n\ge 1 \), we have \( \sum _{m=1}^{\infty }\frac{|a(m)|}{(n+m)^{\sigma }}\gg _{\sigma }\frac{1}{n^{\sigma -\alpha }}\).
Proof
Using Abel summation formula, we have
Since \( \sum _{m=1}^{\infty }\frac{a(m)}{m^s} \) has abscissa of absolute convergence \( \alpha \), it is well known that \(\sum _{m\le x}|a(m)|=o(x^{\alpha +\epsilon })\) for any \( \epsilon >0 \). So the term \( \left( \sum _{m\le x}|a(m)|\right) \frac{1}{(n+x)^{\sigma }} \) tends to zero when x tends to \( \infty \). Therefore by taking x tends to \( \infty \) in Eq. (3), we get that
Now using the fact \( \sum _{m\le t}|a(m)|\gg t^{\alpha } \) for every \( t\ge 1\) in Eq. (4), we have
\(\square \)
3 Proof of Theorem 1
Proof
We will prove the theorem by induction on r.
The case \( r=1 \) easily follows from the hypothesis.
Next we will prove the case \( r=2 \). Let \( (s_1,s_2)\in U_2 \), then \( \sigma _2>\alpha _2 \) and \( \sigma _2+\sigma _1>\alpha _2+\alpha _1 \). Then we can choose \( \epsilon _2>0 \) such that
Now using the hypothesis and (5), we get that
Suppose the theorem is true for \( r-1,\,r\ge 3 \). Now we will prove for r. Let \( (s_1,s_2,\ldots ,s_r)\in U_r \), then
So there exists an \( \epsilon _r >0 \) such that
Now consider
where the above inequality follows from induction hypothesis and (6). \(\square \)
4 Proof of Theorem 2
Proof
Let \(\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}>\alpha _r+\alpha _{r-1}+\cdots +\alpha _{r-i}\) for \(0\le i\le r-1\), then it follows from hypothesis and Theorem 1 that the series defined in (2) is absolutely convergent at \( (s_1,s_2,\ldots ,s_r) \).
We will prove the converse part by induction on r.
The case \( r=1 \) easily follows from Lemma 2.
Suppose the theorem is true for \( r-1 \), \( r\ge 2 \). Now we will prove for r. By hypothesis, we have
The above inequality implies that
and then by applying Lemmas 1 and 2, we have \( \sigma _r>\alpha _r \). Now \( \text {for any}\,\,(m_1,m_2,\ldots ,m_{r-1})\in {\mathbb {N}}^{r-1}\), we get from Lemma 3 that
Therefore from expressions (7) and (8), we have
Then from induction hypothesis, it follows that
\(\square \)
5 Proof of Theorem 3
Proof
It is enough to show that for \(\sigma _r+\sigma _{r-1}+\cdots +\sigma _{r-i}\ge \sigma _r^{\prime }+\sigma _{r-1}^{\prime }+\cdots +\sigma _{r-i}^{\prime }\), \(0\le i\le r-1\), we have
Now consider
Here \(\left( \frac{m_1+\cdots +m_{r-1}}{m_1+\cdots +m_r}\right) ^{\sigma _r-\sigma _r^{\prime }}\le 1 \) as \( \sigma _r\ge \sigma _r^{\prime } \), therefore we have
where the last inequality follows from the fact \( \left( \frac{m_1+\cdots +m_{r-2}}{m_1+\cdots +m_{r-1}}\right) ^{\sigma _r+\sigma _{r-1}-\sigma _r^{\prime }-\sigma _{r-1}^{\prime }}\le 1 \) as \( \sigma _r+\sigma _{r-1}\ge \sigma _r^{\prime }+\sigma _{r-1}^{\prime } \).
Continuing in the same manner, we obtain (9). \(\square \)
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References
Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta functions and their values at non-positive integers. Acta Arith. 98(2), 107–116 (2001)
Matsumoto, K., Nawashiro, A., Tsumura, H.: Double Dirichlet series associated with arithmetic functions. Kodai Math. J. 44(3), 437–456 (2021)
Matsumoto, K., Tanigawa, Y.: The analytic continuation and the order estimate of multiple Dirichlet series. J. Théor. Nombres Bordeaux 15(1), 267–274 (2003)
Matsumoto, K.: On the Analytic Continuation of Various Multiple Zeta Functions, Number Theory for the Millennium, II (Urbana, IL, 2000), 417–440. A K Peters, Natick (2002)
Zhao, J.: Analytic continuation of multiple zeta functions. Proc. Am. Math. Soc. 128(5), 1275–1283 (2000)
Zhao, J., Zhou, X.: Written multiple zeta values attached to \(\mathfrak{sl} (4)\). Tokyo J. Math. 34(1), 135–152 (2011)
Acknowledgements
The author would like to thank Professor Kohji Matsumoto for his useful suggestions that improved the presentation of the article. He also so indebted to Dr. G. K. Viswanadham for his valuable discussions. The research of the author is supported by the UGC research fellowship.
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Sahoo, D.K. Absolute convergence of general multiple dirichlet series. Res. number theory 9, 37 (2023). https://doi.org/10.1007/s40993-023-00444-y
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DOI: https://doi.org/10.1007/s40993-023-00444-y