A survey on the theory of multiple Dirichlet series with arithmetical coefficients on the numerators
aa r X i v : . [ m a t h . N T ] J a n A SURVEY ON THE THEORY OF MULTIPLEDIRICHLET SERIES WITH ARITHMETICALCOEFFICIENTS ON THE NUMERATORS
KOHJI MATSUMOTO
Dedicated to Professor Jonas Kubiliuson the occasion of the 100 years anniversaity of his birth
Abstract.
We survey some recent developments in the analytictheory of multiple Dirichlet series with arithmetical coefficients onthe numerators. A personal recollection
I first met Professor Jonas Kubilius at Kyoto in July 1986, when the5th USSR-Japan Symposium on Probability was held there. Profes-sor Kubilius was one of the members of the Soviet team, and on thisoccasion, some Japanese number theorists organized a small satellitemeeting on probabilistic number theory with him. At that time I was apost-doctoral researcher, just after getting my degree of Dr. Sci. fromRikkyo University in March of the same year. On the meeting I gavea talk on the contents of my thesis, concerning the value-distributionof the Riemann zeta-function ζ ( s ). After my talk, Professor Kubiliuscame close to me and said:“Do you know the name of Antanas Laurinˇcikas?”“No.”“He is a Lithuanian mathematician, and he wrote a lot of papers inwhich you are surely interested.”Further he mentioned that many papers of Laurinˇcikas can be foundin Lietuvos Matematikos Rinkinys. I thanked Professor Kubilius, andwhen I went back to Rikkyo University, I immediately visited the li-brary. Unfortunately in the library there were only the Russian originalversion of the journal back numbers, which I could not read, so at that Mathematics Subject Classification.
Primary 11M32, Secondary 11M26,11M41.
Key words and phrases. multiple Dirichlet series, arithmetical coefficients, mero-morphic continuation, natural boundary. time I gave up reading the papers of Laurinˇcikas. (Several years laterI noticed the existence of the English translation of Liet. Mat. Rink.)In the next year, I got a job of a lecturer at Iwate University, andin 1995 I moved to Nagoya University. And then in 1996 I first vis-ited Lithuania, to attend the 2nd Palanga Conference on Analytic andProbabilistic Number Theory. I met again Professor Kubilius, andfound many new Lithuanian friends. Since then, I became a regularmember of the Palanga Conferences. In September 2011, I attendedthe 5th Palanga Conference; it was just one and a half month beforethe death of Professor Kubilius. When I arrived at the university villain Palanga, at the entrance of the villa I met Professor Kubilius. Whenhe noticed me, he said“C¸ a va?”I replied oui ¸ca va, and said greetings in English. It was my lastconversation with him.2.
Multiple Dirichlet series with or without coefficients
One of the most favorite topics investigated by Professor Kubiliusis the theory of arithmetical functions. Therefore the author feels it anatural choice here to report some recent developments in the theoryof multiple Dirichlet series with arithmetical coefficients on the numer-ators.We begin, however, with the definition of multiple zeta-functionswithout coefficients: ζ r ( s , . . . , s r ) = ∞ X m =1 · · · ∞ X m r =1 m − s ( m + m ) − s (2.1) × · · · × ( m + · · · + m r ) − s r , where s , . . . , s r ∈ C . This multiple series is sometimes called theEuler-Zagier r -fold zeta-function. This series is absolutely convergentwhen ℜ ( s r − k +1 + · · · + s r ) > k (1 ≤ k ≤ r ) , but can be continued meromorphically to the whole space C r . Nowlots of analytic, algebraic, and arithmetic properties of this series (2.1)and its special values are known.It is natural to consider some generalization of (2.1), which has somecoefficients on the numerators. We introduce the following two typesof generalizations: ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 3
Type x :Φ x r ( s , . . . , s r ; a , . . . , a r )= ∞ X m =1 · · · ∞ X m r =1 a ( m ) a ( m ) · · · a r ( m r ) m s ( m + m ) s · · · ( m + · · · + m r ) s r , and Type ∗ :Φ ∗ r ( s , . . . , s r ; a , . . . , a r )= ∞ X m =1 · · · ∞ X m r =1 a ( m ) a ( m + m ) · · · a r ( m + · · · + m r ) m s ( m + m ) s · · · ( m + · · · + m r ) s r . Here, the notation x and ∗ is according to the philosophy of T.Arakawa and M. Kaneko [2]. The aim of the present article is to surveyrecent results related with the above two types of multiple series.It is to be noted that the present survey is by no means complete.In this article we mainly discuss the analytic point of view, so manyimportant results on special values are not mentioned. Moreover, we donot discuss several multiple series with arithmetical numerators, similarto our series. For example, there are a lot of references on multipleDirichlet series with some kind of twisted factors on the numerators.In the present article, however, we will only discuss the case of Dirichletcharacters later, without mentioning the details of other articles, e.g.,P. Cassou-Nogu`es [9] [10], M. de Crisenoy [13], de Crisenoy and D.Essouabri [14], Essouabri and the author [17], and so on.More generally, R. de la Bret`eche [7] studied the multiple series ofthe form ∞ X m =1 · · · ∞ X m r =1 f ( m , . . . , m r ) m s m s · · · m s r r , while several mathematicians treated ∞ X m =1 · · · ∞ X m r =1 f ( m , . . . , m r ) P ( m , . . . , m r ) s where P is a polynomial, or more generally ∞ X m =1 · · · ∞ X m r =1 f ( m , . . . , m r ) P ( m , . . . , m r ) s · · · P n ( m , . . . , m r ) s n where P , . . . , P n are polynomials (see, e.g., B. Lichtin’s series of paperssuch as [27] [28], M. Peter [31], Essouabri [16], and so on). The contentsof those researches are also not treated in the present article. KOHJI MATSUMOTO
The series introduced and studied by D. Goldfeld, D. Bump, S. Fried-berg, J. Hoffstein et al. (see, e.g., [8]) are also called “multiple Dirichletseries”, but are different from the series dsicussed in the present article.3.
The case of type ∗ In this section we consider the case of type ∗ .The simplest situation is when a j (1 ≤ j ≤ r ) are periodic functions.In this case, the sum Φ ∗ r ( s , . . . , s r ; a , . . . , a r ) can be easily written asa linear combination of multiple series with numerator = 1.The case a j = χ j (Dirichlet characters) was studied by S. Akiyamaand H. Ishikawa [1]. They wrote Φ ∗ r ( s , . . . , s r ; χ , . . . , χ r ) as a linearcombination of multiple zeta-functions of the form ∞ X m =1 · · · ∞ X m r =1 ( m + α ) − s ( m + m + α ) − s · · · ( m + · · · + m r + α r ) − s r (3.1)(where α , . . . , α r are constants). The series (3.1) can be treated analo-gously to the case of (2.1). Ishikawa [22] [23] studied further, and gaveapplications to the evaluation of certain multiple character sums.When the coefficients are not periodic, the study of analytic proper-ties of multiple series of type ∗ is not easy. Here we mention a paperof Essouabri, H. Tsumura and the author [18], in which the case whenthe coefficients satisfy a recurrence condition is handled.Assume the recurrence condition a i ( m + h ) = h − X j =0 λ ji a i ( m + j ) for all m ∈ N among coefficients, where λ ij ∈ C . Denote by η , . . . .η q the familyof all maps from { , . . . , r } to { , . . . , h − } (so q = h r ), and for m = ( m , . . . , m r ) ∈ N r , define A l ( m ) := r Y i =1 a i ( m + · · · + m i + η l ( i )) ( l = 1 , . . . , q )and A ( m ) := ( A ( m ) , . . . , A q ( m )) . Define the matrix T k (1 ≤ k ≤ r )by A ( m + h e k ) = T k A ( m ) , where e k is the k -th unit vector. Theorem 3.1. ( [18]) If a i is at most of polynomial order, satisfies theabove recurrence condition, and if is not an eigenvalue of any of the ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 5 matrices T , . . . , T r , then ∞ X m =1 · · · ∞ X m r =1 a ( m ) a ( m + m ) · · · a r ( m + · · · + m r ) m s ( m + m ) s · · · ( m + · · · + m r ) s r can be holomorphically continued to the whole space C r . As a typical example, a double series with Fibonacci numbers F n onthe numerator is discussed. In fact, it was shown that the double series φ ( s ) = ∞ X m =1 ∞ X n =1 ( iα − ) m + n F m + n + ( iα − ) m +2 n F m +2 n m s ( m + n ) s ( α = (1 + √ /
2) can be continued meromorphically to C . Moreover,the evaluation of a special value φ (0) = 118 (cid:16) − √ − √ i (cid:17) and a sum formula ∞ X m =1 ∞ X n =1 α − m F m + α − n F n − α − m − n F m + n m ( m + n ) = ∞ X m =1 α − m F m m are given.An important auxiliary tool in [18] is a kind of vectorial zeta-functions.Special values of those vectorial zeta-functions satisfy “vectorial sumformulas”. Such formulas were proved in [18] in the double and triplecases. The formula in the general case was proposed as a conjecturein [18], and proved by S. Yamamoto [34].4. Application of the Mellin-Barnes integral formula
Hereafter we discuss the case of type x . Let ϕ k ( s ) = P ∞ m =1 a k ( m ) m − s (1 ≤ k ≤ r ), and we sometimes write Φ x r ( s , . . . , s r ; ϕ , . . . , ϕ r ) insteadof Φ x r ( s , . . . , s r ; a , . . . , a r ).The advantage of type x is that this type of series is more suitableto analytic study. In fact, writeΦ x r ( s , . . . , s r ; ϕ , . . . , ϕ r )= ∞ X m =1 · · · ∞ X m r =1 a ( m ) a ( m ) · · · a r ( m r ) m s ( m + m ) s · · · ( m + · · · + m r − ) s r − × ( m + · · · + m r − ) − s r (cid:18) m r m + · · · + m r − (cid:19) − s r , and to the last factor apply the classical Mellin-Barnes integral formula(1 + λ ) − s = 12 πi Z c + i ∞ c − i ∞ Γ( s + z )Γ( − z )Γ( s ) λ z dz, KOHJI MATSUMOTO where s, λ ∈ C , ℜ s > λ = 0, | arg λ | < π , and −ℜ s < c <
0. We get (cid:18) m r m + · · · + m r − (cid:19) − s r = 12 πi Z c + i ∞ c − i ∞ Γ( s r + z )Γ( − z )Γ( s r ) (cid:18) m r m + · · · + m r − (cid:19) z dz, and henceΦ x r ( s , . . . , s r ; ϕ , . . . , ϕ r )= 12 πi Z c + i ∞ c − i ∞ Γ( s r + z )Γ( − z )Γ( s r ) × ∞ X m =1 · · · ∞ X m r − =1 a ( m ) a ( m ) · · · a r ( m r ) m s ( m + m ) s · · · ( m + · · · + m r − ) s r − + s r + z × ∞ X m r =1 a r ( m r ) m zr dz = 12 πi Z c + i ∞ c − i ∞ Γ( s r + z )Γ( − z )Γ( s r ) × Φ x r − ( s , . . . , s r − , s r − + s r + z ; ϕ , . . . , ϕ r − ) ϕ r ( − z ) dz. (Here, the important point is that the sum with respect to m r can beseparated.)Using this expression, we can reduce the study of Φ r to that of Φ r − and ϕ r , and to that of Φ r − and ϕ r − ,..... and finally we obtain Theorem 4.1. (Matsumoto and Tanigawa [30])
Assume that ϕ k ( s ) ( ≤ k ≤ r ) are convergent absolutely for s with sufficiently largereal part, continued meromorphically to the whole complex plane, withfinitely many poles, and are of polynomial order. Then Φ x r ( s , . . . , s r ; ϕ , . . . , ϕ r ) can be continued meromorphically to the whole C r , and the location ofpossible singularities can be given explicitly. In particular, if all ϕ k ( s ) ( ≤ k ≤ r ) are entire, then Φ x r ( s , . . . , s r ; ϕ , . . . , ϕ r ) is also entire. In particular,Φ x ( s , s ; 1 , ϕ ) = ∞ X m =1 ∞ X m =1 a ( m ) m s ( m + m ) s can be continued. This double series satisfies a certain “functionalequation” which is written in terms of confluent hypergeometric func-tions (see [11] [12]): ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 7
Theorem 4.2. (Choie and Matsumoto [11])
We have Φ x ( s , s ; 1 , ϕ ) = Γ(1 − s )Γ( s + s − s ) ϕ ( s + s − − s ) { F , + (1 − s , − s ) + F , − (1 − s , − s ) } , where F , ± ( s , s ) = X l ≥ A s + s − ( l )Ψ( s , s + s ; ± πil ) ,A c ( l ) = P n | l n c a ( n ) and Ψ is the confluent hypergeometric functiondefined by ψ ( a, b ; x ) = 1Γ( a ) Z e iφ ∞ e − xy y a − ( y + 1) b − a − dy. Furthermore, when ϕ is an automorphic L -function, then by usingthe modular relation, a different type of functional equation can alsobe proved.Why we may call Theorem 4.2 a ”functional equation”? There aremainly two reasons. First, it can be compared with the functionalequation of the Hurwitz zeta-function ζ ( s, α ) = P ∞ n =0 ( n + α ) − s (0 <α ≤ ζ (1 − s, α ) = Γ( s )(2 π ) s { e πis/ φ ( s, − α ) + e − πis/ φ ( s, α ) } with φ ( s, α ) = P ∞ n =1 e πinα n − s .Secondly, when a ( n ) ≡
1, a symmetric form of the functional equa-tion (that is, a relation which connects Φ x ( s , s ) with Φ x (1 − s , − s ))can be deduced from Theorem 4.2 on hyperplanes s + s = 2 k + 1( k ∈ Z \ { } ).This second point was already observed in a paper of Y. Komori,Tsumura and the author [25]. We note that a generalization of theresults in [25] to the case of double L -functions twisted by Dirichletcharacters was discussed in [26].In Theorem 4.1, the assumption that each ϕ k has only finitely manypoles is important. The analytic behavior of Φ x r may become differentwhen some of ϕ k has infinitely many poles. In the next section we willdiscuss such cases. KOHJI MATSUMOTO The case when some ϕ k has infinitely many poles Typical examples of Dirichlet series with arithmetical coefficientswhich have infinitely many poles are ∞ X m =1 Λ( m ) m s = − ζ ′ ( s ) ζ ( s ) , ∞ X m =1 µ ( m ) m s = 1 ζ ( s ) , where Λ( m ) is the von Mangoldt function and µ ( m ) is the M¨obiusfunction. Both of the above series have infinitely many poles at thezeros of ζ ( s ).We now consider the behavior of Φ x r , where some of associated ϕ k has infinitely many poles (as in the above examples). Let N (Φ x r ) = { k | ≤ k ≤ r, ϕ k has infinitely many poles } . . A recent article of A. Nawashiro, Tsumura and the author [29] studiedseveral examples which satisfy N (Φ x ) = 1. For example, Theorem 5.1. ( [29])
The double series ∞ X m =1 ∞ X m =1 Λ( m ) m s ( m + m ) s can be continued meromorphiocallty to the whole C , and the locationof possible singularities can be described explicitly. Next, take ϕ ( s ) = P ∞ m =1 a ( m ) m − s which has only finitely manyploes, and consider the convolution of a and µ : e a ( m ) = X d | m a (cid:16) md (cid:17) µ ( d ) . In other words, ϕ ( s ) ζ ( s ) = ∞ X m =1 e a ( m ) m − s . Assumption 5.2.
All non-trivial zeros of ζ ( s ) are simple and1 ζ ′ ( ρ n ) = O ( | ρ n | B ) ( ρ n : n -th zero , B > ζ ( s ), and consistent with the Gonek-Hejhalconjecture [19] [21].) Theorem 5.3. ( [29])
Under Assumption 5.2, the double series ∞ X m =1 ∞ X m =1 e a ( m ) m s ( m + m ) s ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 9 can be continued meromorphiocallty to the whole C , and the locationof possible singularities can be described explicitly. The paper [29] only considers the double zeta case, but a generaliza-tion of [29] to the general multiple case was treated by Rei Kawashima[24]. She also discussed the case when Λ is replaced by the Liouvillefunction λ .The above two theorems show that when N (Φ x ) = 1, the analyticbehavior of Φ x is not so different from the case when N (Φ x ) = 0.However, if N (Φ x ) = 2, or more generally if N (Φ x r ) ≥
2, the analyticbehavior of the multiple series is totally different.Consider F ( s ) = ∞ X k =1 ∞ X l =1 Λ( k )Λ( l )( k + l ) s , which is equal to Φ (0 , s ; − ζ ′ /ζ , − ζ ′ /ζ ), so both ϕ and ϕ have infin-itely many poles. We can rewrite F ( s ) = ∞ X m =1 G ( m ) m s , G ( m ) = X k + l = m Λ( k )Λ( l ) . It is to be noted that this G ( m ) is the counting function of the classicalGoldbach problem: G ( m ) = X r ≥ ,r ≥ pr
11 + pr
22 = m log p log p = X p + p = m log p log p + (error) , where p , p denote primes. Using the Mellin-Barnes formula, we have F ( s ) = 12 πi Z c + i ∞ c − i ∞ Γ( s + z )Γ( − z )Γ( s ) ζ ′ ζ ( s + z ) ζ ′ ζ ( − z ) dz. Shifting the path of integration suitably, we find that there are polesof F ( s ) at: s = 2 , s = ρ + 1 , s = ρ + ρ ′ (where ρ , ρ ′ denotes the non-trivial zeros of ζ ( s )).Now assume the RH (Riemann Hypothesis). Then ℜ ( ρ + ρ ′ ) = 1, andwe can show that ρ + ρ ′ are dense on the line ℜ s = 1. Therefore ℜ s = 1seems a kind of barrier if we want to continue F ( s ) meromorphically.It is believed that γ = ℑ ρ > Q (thelinear independence conjecture, LIC). Theorem 5.4.
If RH and LIC are true, then ℜ s = 1 is the naturalboundary of F ( s ) . This was first proved by S. Egami and the author [15] under the RHand a stronger quantitative version of LIC, and then in the above formby G. Bhowmik and J.-C. Schlage-Puchta [6]. Moreover, in [6] it is alsoshown that F r ( s ) = ∞ X k =1 · · · ∞ X k r =1 Λ( k ) · · · Λ( k r )( k + · · · + k r ) s ( r ≥ ℜ s = r − ℜ s = 1 is thenatural boundary of F ( s ).A generalization to the case with congruence conditions has beenalso studied. Let G ( m ; q, a, b ) := X k + l = mk ≡ a,l ≡ b (mod q ) Λ( k )Λ( l ) , where a, b, q are positive integers with ( ab, q ) = 1, and define the asso-ciated Dirichlet series by F ( s ; q, a, b ) := ∞ X m =1 G ( m ; q, a, b ) m s . The behavior of F ( s ; q, a, b ) was first treated by F. R¨uppel [32], andthen further studied by Y. Suzuki [33].Let S ( x ; q, a, b ) := X m ≤ x G ( m ; q, a, b ) . We can reduce the study of S ( x ; q, a, b ), via its associated Dirichletseries F ( s ; q, a, b ), to that of the behavior (especially the distributionof zeros) of Dirichlet L -functions. Thereby we can establish the con-nection between Goldbach generating functions and the zeros of L -functions (especially the generalized Riemann hypothesis, GRH). Theexistence of this connection was first suggested by A. Granville [20]in the Riemann zeta case, and then fully developed in Bhowmik etal. [3] [4] [5]. We conclude this article with the statement of sometheorems proved in those papers.Let B χ = sup {ℜ ρ χ } and B q = sup { B χ | χ (mod q ) } . Theorem 5.5. ( [4])
For any δ > , We have S ( x ; q, a, b ) = x ϕ ( q ) + O ( x B ∗ q ) , where B ∗ q = min { B q , − η } with η = c ( δ )max { q δ , (log x ) / (log log x ) / } ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 11 ( where c ( δ ) is a small positive constant ) .Remark . In particular, if we assume GRH, then B ∗ q = B q = 1 / S ( x ; q, a, b ) = x ϕ ( q ) + O ( x / ) . Now recall a well-known conjecture (DZC): Any two distinct Dirich-let L -functions (mod q ) do not have a common non-trivial zero (exceptfor a possible multiple zero at s = 1 / Theorem 5.7. ( [4], [5])
Assume that the DZC is true, and χ ( a ) + χ ( b ) = 0 for all χ ( mod q ) . If the asymptotic formula S ( x ; q, a, b ) = x ϕ ( q ) + O ( x d + ε ) (1 / ≤ d < holds for any ε > , then either B q ≤ d or B q = 1 . Moreover we canremove the possibility of B q = 1 when a = b .Remark . In particular, if a = b and the above formula holds withthe error O ( x / ε ) (that is, d = 1 / B q = 1 /
2, that is, the GRHholds.
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ULTIPLE DIRICHLET SERIES WITH ARITHMETICAL COEFFICIENTS 13 [30] K. Matsumoto and Y. Tanigawa, The analytic continuation and the orderestimate of multiple Dirichlet series, J. Th´eor. Nombr. Bordeaux (2003),267–274.[31] M. Peter, Dirichlet series associated with polynomials, Acta Arith. (1998),245–278.[32] F. R¨uppel, Convolutions of the von Mangoldt function over residue classes,ˇSiauliai Math. Semin. (2012), 135–156.[33] Y. Suzuki, A mean value of the representation function for the sum of twoprimes in arithmetic progressions, Intern. J. Number Theory (2017), 977–990.[34] S. Yamamoto, A sum formula of multiple L -values, Intern. J. Number Theory (2015), 127–137. K. Matsumoto: Graduate School of Mathematics, Nagoya Univer-sity, Chikusa- ku, Nagoya 464-8602, Japan
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