Pure and Applied Mathematics Journal
Volume 4, Issue 5-1, October 2015, Pages: 15-19

On the Riesz Sums in Number Theory

Hailong Li, Qianli Yang

Departmant of Mathematics and Information Science, Weinan Normal University, Shannxi, P. R. China

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(Hailong Li)

To cite this article:

Hailong Li, Qianli Yang. On the Riesz Sums in Number Theory. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects of Engineering Disciplines. Vol. 4, No. 5-1, 2015, pp. 15-19. doi: 10.11648/j.pamj.s.2015040501.13


Abstract: The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.

Keywords: Riesz Sum, Riesz Mean, Dirichlet Series, Asymptotic Formula


1. Introduction

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with sum ability of Fourier series and of Dirichlet series [8] and [11]. Given an increasing sequence  of real numbers and a sequence  of complex numbers, the Riesz sum of order  is defined as in [8, p.2] and [11, p.21] by

(1.1)

where

(1.2)

where the prime on the summation sign means that when , the corresponding term is to be halved.

(1.1) or rather normalized  which appears in (G-8-2) is called the Riesz sum of order  If  approaches a limit  as , the sequence  is called Riesz summable or  summble to , which is called the Riesz mean of the sequence. Sometimes the negative order Riesz sum is considered, in which case the sum is taken over all  which are not equal to .

In number theory it is often the case that the main study of research is the behavior of an arithmetic function whose generating function is given explicitly, say in the form of a Dirichlet series or an Euler product. Then the problem amounts to extracting the essential main term from the data on generating functions.

In this number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the Cf. Remark 1. There is some mention on the divisor problem in [7] in the light of the Riesz sum and there are enormous amount of literature on the Riesz sums and we shall not dwell on well-known cases very in detail. We are concerned with the case where the generating function does not necessarily satisfy the functional equation and concentrate on asymptotic formulas rather than exact identities.

An example is given.

Recall the definition of the periodic Bernoulli polynomial etc. ([14, p.170]). Then

say, or

Integration of both sides amounts to (1.1):

where we used .

The application of the Riesz sum comes into play through Perron's formula (1.6) below, sometimes in truncated form. The application of the truncated first order Riesz sum appears on [10, p.105 ] and a truncated general order Riesz sum is treated in [13] in both of which the functional equation is not assumed. Riesz sums with the functional equation can be found e.g. in [9], where by differencing, the asymptotic formula for the original sum is deduced. The principle goes back to Landau [15] in which one can find the integral order Riesz sum and its reduction to the original partial sum by differencing.

The general formula for the difference operator of order  with difference  is given by

(1.3)

If  has the  derivative , then

(1.4)

The Riesz kernel which produces the Riesz sum is defined by

(1.5)

Remark 1. Notes on (1.5). Let denote the order of the Riesz mean and set . Then (1.5) reads ()

(1.6)

This can be found in Hardy-Riesz [11] and Chandrasekharan and Minakshisundaram [8] and used in the context of Perron's formula

(1.7)

where the left-hand side sum is called the Riesz sum of order  and denoted  as mentioned above and

.

The special case of (1.6) with  is known as the discontinuous integral whose truncated form can be found e.g. in Davenport [10, pp.109-110]. This and the general case (1.6) can be proved by the method of residues, distinguishing the cases  and .

Here as above, the prime on the summation sign means that when , the corresponding term is to be halved, and this halving comes from the peculiarity of the discontinuous integral.

If the order , then the right-hand side member of (1.6) is

and the Riesz sum amounts to the  times integration of the original sum . Thus Landau's differencing is an analogue of the integration and differentiation.

In view of this integration-differentiation aspect there are a number of cases in which the Riesz sum appears in disguised form. Especially, when there is a gamma factor  or  involved.

The very special case  of (1.5) (and of the corresponding logarithmic case () presents excessive complexities in notation, so that we follow Hardy and Riesz [11] to use (1.7) by suppressing the prime on the summation. We are to bear this special case in mind although not explicitly stated.

Remark 2. The Riesz summability is useful in summability of (Fourier) series. We recall e.g. the well-known result that the Riesz summability implies Abel summability [11, Theorem 24]. There is an explicit formula known for the transition.

Lemma 1.1. The sum of the Abel mean at all points of the sector  other than the origin is

(1.8)

Where  is the  Riesz sum defined in (1.1).

2. Riesz Sums

Definition 1. Let  be a real number (mostly we assume that it is nonnegative) and let , be arbitrary sequences of real numbers strictly increasing to infinity such that . Let  ne any sequence of complex numbers. Then we write

(2.1)

and refer to  (resp. ) as the Riesz sum of order  of the second (resp. first) kind associated to the series (resp. ), where absolute convergence of the series is assumed in some half-plane . For the special choice of (resp. ),   or , denoting the norm of the integral ideal  (resp.  or ), we dennote the corresponding Riesz sum  (resp. ) by  (resp. ) and refer to it as the arithmetic (resp/ logarithmic) Riesz sum of order  associated to the series or.

Theorem 2.1. (Kanemitsu) Let  denote the abscissa of absolute convergence of the Dirichlet series

(2.2)

Which we may assume  without loss of generality, for  and for  let  denote a Majorant of . Suppose that can be continued analytically to a meromorphic function in some region R0 extending vertically from top to bottom fo the complex plane and bounded on the left by a piecewise smooth Jordan curve

(2.3)

and that all the poles of lying in  are contained in a finite part of and are not on . Take a subregion  whose boundary consists of the line segments , overline DA and that part of BC of  with  with  large enough for all the poles of  are contained in  and is to be taken as (2.7). I.e.

and

where  is a constant large enough. Suppose that  satisfies the following growth conditions: there exists a constant  such that

(2.4)

(2.5)

(2.6)

where are positive, integrable and  as , and  is some constant. Then with

(2.7)

with  constant  and if  is given by

()

(2.8)

(2.9)

with constants  such that

 Then for anyand with (2.7) we have the asymptotic formula

(2.10)

provided that

(2.11)

for some , where  is the sum of the residues of  in ,

(2.12)

(2.13)

where

(2.14)

and .

Theorem 2.2. Under the same condition as in Theorem 2.1, if  is given by

(2.15)

Then with a constant  and we have the asymptotic formula

(2.16)

where  is the sum of the residues of

 in ,

(2.17)

Corollary 2.1. Suppose that the conditions of Theorem 2.1 are satisfied and let be the maximum of the real parts of poles of  in , and be the maximum order of poles with real parts , and define  to be 1or 0 according as  has a pole in or not  to be 1 or 0 according as  or not . then

(2.18)

(2.19)

where and or  according to the choice of .

Similar results hold for the logarithmic Riesz sums with the following replacement to be made: Instead of  we have  the sum of the residues of  in , instead of  we have , . We state a very convenient corollary.

Corollary 2.2. Suppose that the conditions of Theorem 2.1 are satisfied and suppose that we have the asymptotic formula

(2.20)

with  a constant. Then

(2.21)

where and all ’s amount to the reducing factor  with possibly different constants  etc. in (2.14).

3. Examples

Example 3.1. Let  be an algebraic number field of degree  with discriminant . Let  be the ring of algebraic integers in  and let  be an artitrary, fixed non-zero ideal of . Let  be the group of all fractional ideals with numerators and denominators relatively prime to , and  denote the ray class group of ,  the quotient of  modulo the group  of principal ideals () with totally positive  such that .We define the Möbius function  on ideals in the same manner as in the rational case and for  we put

(3.1)

Then we have

Theorem 3.1. (A version of the Siegle-Walfisz prime ideal theorem) If

(3.2)

with an arbitrary constant  however large it may be, we have

(3.3)

with a constant  depending at most on  and  and so is the constant.

With Theorem 3.1 at hand, we may obtain generalizations of asymptotic formulas in [5], [16] with sharp estimate on the error term.

Example 3.2. For  set

(3.4)

Since  where  indicates the Dedekind zeta-function of , we have

(3.5)

Where  are the Laurent coefficients of the Dedekind zeta:

(3.6)

(3.7)

4. Quwllenangaben

The Riesz sums may be thought of as integration or Abelian process ([4]) while differencing the Riesz sum to deduce a formula for the Riesz sum of order 0 corresponds to differentiation or Tauberian process.

The logarithmic Riesz sums also appeared in various context and we refer to [1] and [2] for them for which the generating function satisfies the functional equation.

For general modular relations, we refer to [9], [3] and the most comprehensive [14]. In the last ref., the Riesz sums are treated in Chapter 6. Some extracts and generalization have been made in [17].

Acknowledgements

We would like to thank Professor Shigeru Kanemitsu for allowing to use the material in [13] freely.

International Cooperation Projects of Science and Technology Agency of Shaanxi Province (No.2015KW-022)


References

  1. Bruce C. Berndt and S. Kim, Logarithmic means and double series of Bessel functions, preprint 2014.
  2. Bruce C. Berndt and S. Kim, Identities for logarithmic means: A survey, preprint 2014.
  3. B. C. Berndt and M. I. Knopp, Hecke's theory of modular forms and Dirichlet series, World Sci., Singapore etc., 2008.
  4. W. E. Briggs, Some Abelian results for Dirichlet series, Mathematika 9 (1962), 49-53.
  5. R. G. Buschman, Asymptotic expressions for Pnaf(n) logr n, Pacic J. Math. 9 (1959), 9-12.
  6. K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functionsII, World Scienti_c, London Singapore New Jersey, 2009.
  7. K. Chandrasekharan, Arithmetical functions, Springer Verl., Berlin-Heidelberg-New York 1970.
  8. K. Chandrasekharan and S. Minakshisundaram, Typical means, Oxford UP, Oxford 1952.
  9. K. Chandrasekharan and Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93-136.
  10. H. Davenport, Multiplicative number theory, 1st ed. Markham, Chicago 1967, 2nd ed. Springer, New York etc. 1980.
  11. G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, CUP. Cambridge 1915; reprint, Hafner, New York 1972 .
  12. A. E. Ingham, The distribution of prime numbers. Cambridge Tracts Math. Math. Phys., No. 30 Stechert-Hafner, Inc., New York 1964.
  13. S. Kanemitsu, On the Riesz sums of some arithmetical functions, in p-adic L-functions and algebraic number thery Surikaiseki Kenkyusho Kokyuroku 411 (1981), 109-120.
  14. S. Kanemitsu and H. Tsukada, Contributions to the theory of zetafunctions modular relation supremacy, World Sci. London etc. 2014, 303 pp.
  15. E. Landau, Uber die Anzahl der Gitterpunkte in gewissen Bereichen (Zweite Mit.), Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. (1915), 209-243=Collected Works Vol. 6, Thales Verl., Essen 1985, 308-342.
  16. S. Swetharanyam, Asymptotic expansions for certain type of sums involving the arithmetic functions in the theory of numbers, Math. Student (2) 28 (1960), 9-26.
  17. X.-H. Wang and N. -L. Wang, Modular-relation theoretic interpretation of M. Katsurada's results, preprint 2014.

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