Some problems on mean values and theuniversality of zeta and multiple zeta-functions
Kohji Matsumoto
Abstract
Here I propose and discuss several problems on analytic properties ofsome zeta-functions and multiple zeta-functions, which are related to mytalk or to talks of other participants at the conference.

1

Mean values of multiple zeta-functions
Recall the definition of generalized multiple zeta-functions:ζr ((s1 , . . . , sr ); (α1 , . . . , αr ), (w1 , . . . , wr ))=∞X···

m1 =0∞
X(α1 + m1 w1 )−s1 (α2 + m1 w1 + m2 w2 )−s2

mr =0

× · · · × (αr + m1 w1 + · · · + mr wr )−sr ,(1.1)

where s1 , . . . , sr are complex variables, and α1 , . . . , αr , w1 , . . . , wr are complex
parameters. The series (1.1) was first introduced in [30] [31], and then in [32][33], under certain assumptions on α1 , . . . , αr , w1 , . . . , wr , it has been shown that
(1.1) can be continued meromorphically to the whole Cr space. Moreover, if
α1 , . . . , αr , w1 , . . . , wr are positive real, then (1.1) is polynomial order with respect
to =sr (Theorem 4 of [32]). The order of the magnitude of (1.1) has been further
studied by [35] [8], but how is the real order? Does the analogue of the Lindelöfhypothesis hold?
A standard way of studying those problems is to consider the mean value of(1.1). Is it possible to obtain an asymptotic formula, or at least a non-trivial
upper bound, of the mean value
Z T10···Z Tr0|ζr ((σ1 + it1 , . . . , σr + itr );(α1 , . . . , αr ), (w1 , . . . , wr ))|2 dt1 · · · dtr(1.2)

in a certain “critical” region? Let us consider the simplest case r = 2. I proveda functional equation forζ2 (s1 , s2 ) = ζ2 ((s1 , s2 ); (1, 2), (1, 1)) =∞X∞
X

m1 =1 m2 =1

1−s2
1m−s
1 (m1 + m2 )by using a certain contour integral ([34]). By modifying the argument in [34]suitably, it is probably possible to deduce a certain approximate functional equation for ζ2 (s1 , s2 ), which would produce various mean value results (analogousto the classical case of the Riemann zeta-function ζ(s) or Dirichlet L-functions
L(s, χ)) for ζ2 (s1 , s2 ). It is desirable to carry out this program. For r ≥ 3, eventhe functional equation is not known; to search for such an equation is clearly a
big problem.
In the case of ζ(s), there are two major approaches to the mean square problem; the approximate functional equation, and the Atkinson method (see Ivić [9][10]). How to generalize Atkinson’s method to the multiple case? An essential
idea of Atkinson is to consider the product ζ(s1 )ζ(s2 ), and divide it asζ(s1 )ζ(s2 ) = ζ(s1 + s2 ) + ζ2 (s1 , s2 ) + ζ2 (s2 , s1 ).
Then the problem is reduced to the study of the double zeta-function ζ2 (s1 , s2 ).
Similarly, we can write the product ζ2 (s1 , s2 )ζ2 (s3 , s4 ) as a sum of several multiple
zeta-functions ζr with r ≤ 4. But we have already developed some basic theory ofζr ([32] [33] etc.). Hence we may expect that Atkinson’s treatment for ζ2 (s1 , s2 )could be extended to ζr (r ≤ 4) in the frame of our theory, and an explicit formula(of Atkinson type) for the mean square of ζ2 could be deduced. Moreover themean square problem for ζr (r ≥ 3) can be reduced similarly to the study ofanalytic properties of ζk (k ≤ 2r).
Of course various other mean values different from (1.2) may also be considered. Concerning the Hurwitz zeta-function
ζ(s, α) =∞X(n + α)−s(0 < α ≤ 1),(1.3)

n=0mean square values with respect to the parameter α have been studied in detailby Katsurada and Matsumoto [14] [15] [16] [17] and Katsurada [13]. Dr. J.
Furuya pointed out that, similarly, the mean square of the multiple zeta-function(1.1) with respect to the parameters α1 , . . . , αr may be considered, and it will be
more accessible than (1.2).

2

Universality of multiple zeta-functions

Another interesting problem on multiple zeta-functions is to consider the universality property. Detailed accounts on the universality of ζ(s) can be found in
Karatsuba-Voronin [12] or Laurinčikas [18]. Probably an accessible case is theBarnes multiple zeta-function, that is the special case s1 = s2 = · · · = sr−1 = 0in (1.1):
ζr (sr ; αr , (w1 , . . . , wr )) =∞X

m1 =0···∞
X

mr =02(αr + m1 w1 + · · · + mr wr )−sr .

(2.1)

This is a function of single variable sr , and is a generalization of the Hurwitz zetafunction (1.3). It is known that (1.3) satisfies the universality property, when αis transcendental or rational (Bagchi [1], Gonek [7]). By generalizing the proof ofsuch universality results, it is likely that we could obtain a universality theorem
for the Barnes multiple zeta-function (2.1).
More challenging is to consider the universality for general multi-variable series (1.1). In this case, the statement of the result should be formulated on theCr space. Again it is natural to begin with the case r = 2. In [34], I introduced
the double Hurwitz-Lerch zeta-function of the formζ2 ((s1 , s2 ); α, β, w) =∞X(α + m1 )−s1

m1 =0

∞
X

e2πim2 β (α + m1 + m2 w)−s2 .

(2.2)

m2 =1

Various limit theorems and universality theorems are known for the classicalLerch zeta-function∞
ζ(s, α, β) =X

e2πinβ (n + α)−sn=0(see [2] [3] [4] [5] [6] [19] [20] [21] [22] [25] [26] [28], due to Garunkštis, Laurinčikas
and myself). Is it possible to prove analogous results for (2.2), by generalizing
the methods developed in those papers?

3

Further problems on the universality

The theory of universality for usual (single variable, single sum) zeta andL-functions is now a very active research field. At the conference several talks
were devoted to the study of various discrete limit theorems. Hence it is natural
to go further and consider discrete universality theorems. In particular, it isan interesting problem to prove discrete universality theorem for L-functions
attached to cusp forms. This should be achieved by modifying the idea given in[29].
Discrete limit theorems in further general situations have been proved byKačinskaitė [11] and her subsequent papers (partly with Laurinčikas). It is desirable to obtain discrete universality theorems for this general class of zetafunctions, under suitable assumptions. A possible way is to modify the papers[23] [24] of Laurinčikas, but by combining with the idea in [29] it is likely that wecould weaken the assumptions considerably (cf. [27]).
Another direction is to study discrete limit theorems or discrete universality
for multiple zeta-functions. For instance, is it possible to prove such results for theBarnes multiple zeta-function (2.1), or the double Hurwitz-Lerch zeta-function(2.2)? Can one generalize the recent work of J. Ignatavičiūtė?
As a generalization of the Euler-Zagier multiple sum, multiple Dirichlet seriesof the formΦr ((s1 , . . . , sr ); (ϕ1 , . . . , ϕr ))
3

=∞
X

m1 =1···∞Xar (mr )a1 (m1 ) a2 (m2 )···s1s(m1 + m2 ) 2(m1 + · · · + mr )sr
mr =1 m1(3.1)

has been introduced in [35], where
ϕj (s) =∞Xaj (n)n−s(1 ≤ j ≤ r)n=1

are usual Dirichlet series. Assume ϕj (s) is (for all j) convergent absolutely inthe region <s > αj (> 0), can be continued meromorphically to the whole plane,
holomorphic except for the pole (of order at most 1) at s = αj , and of polynomial
order with respect to =s. Then in [35] it is shown that the multiple series (3.1)
can be continued to the whole Cr space. Consider the special case that ϕj (s)(1 ≤ j ≤ r) are L-functions attached to cusp forms. Then, is it possible to apply
the idea in [29] to obtain some universality (or discrete universality) result forthis multiple series? We may also consider various other interesting special casesof (3.1). In which case can one prove a kind of universality?

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4

[8] H. Ishikawa, K. Matsumoto and Y. Tanigawa, in preparation.
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37 (1997), 146-155).

5

[21] A. Laurinčikas, The universality of the Lerch zeta-function, Liet. Mat. Rink.
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[27] A. Laurinčikas and K. Matsumoto, On estimation of the number of zerosof linear combinations of certain zeta-functions, Liet. Mat. Draug. Mokslo
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independence for Lerch zeta-functions, Nagoya Math. J. 157 (2000), 211227.
[29] A. Laurinčikas and K. Matsumoto, The universality of zeta-functions attached to certain cusp forms, Acta Arith. 98 (2001), 345-359.
[30] K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes,of Shintani, and Eisenstein series, preprint.
[31] K. Matsumoto, On analytic continuation of various multiple zeta-functions,in “Number Theory for the Millenniam, the Millennial Conference on Number Theory”, B. Berndt et al. (eds.), A K Peters, to appear.
[32] K. Matsumoto, The analytic continuation and the asymptotic behaviour ofmultiple zeta-functions I, preprint.
[33] K. Matsumoto, The analytic continuation and the asymptotic behaviour ofmultiple zeta-functions II, preprint.
[34] K. Matsumoto, Functional equations for double zeta-functions, preprint.

6[35] K. Matsumoto and Y. Tanigawa, The analytic continuation and the order
estimate of multiple Dirichlet series, in preparation.
Graduate Schoolof Mathematics
Nagoya University
Chikusa-ku, Nagoya 464-8602
Japan7