On the Riesz Sums in Number Theory
Hailong Li, Qianli Yang
Departmant of Mathematics and Information Science, Weinan Normal University, Shannxi, P. R. China
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  and . 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 . 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  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
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  and . 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
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  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
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  in both of which the functional equation is not assumed. Riesz sums with the functional equation can be found e.g. in , where by differencing, the asymptotic formula for the original sum is deduced. The principle goes back to Landau  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
If has the derivative , then
The Riesz kernel which produces the Riesz sum is defined by
Remark 1. Notes on (1.5). Let denote the order of the Riesz mean and set . Then (1.5) reads ()
This can be found in Hardy-Riesz  and Chandrasekharan and Minakshisundaram  and used in the context of Perron's formula
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  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
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
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
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
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.
where is a constant large enough. Suppose that satisfies the following growth conditions: there exists a constant such that
where are positive, integrable and as , and is some constant. Then with
with constant and if is given by
with constants such that
Then for anyand with (2.7) we have the asymptotic formula
for some , where is the sum of the residues of in ,
Theorem 2.2. Under the same condition as in Theorem 2.1, if is given by
Then with a constant and we have the asymptotic formula
where is the sum of the residues of
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
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
with a constant. Then
where and all ’s amount to the reducing factor with possibly different constants etc. in (2.14).
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
Then we have
Theorem 3.1. (A version of the Siegle-Walfisz prime ideal theorem) If
with an arbitrary constant however large it may be, we have
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 ,  with sharp estimate on the error term.
Example 3.2. For set
Since where indicates the Dedekind zeta-function of , we have
Where are the Laurent coefficients of the Dedekind zeta:
The Riesz sums may be thought of as integration or Abelian process () 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  and  for them for which the generating function satisfies the functional equation.
For general modular relations, we refer to ,  and the most comprehensive . In the last ref., the Riesz sums are treated in Chapter 6. Some extracts and generalization have been made in .
We would like to thank Professor Shigeru Kanemitsu for allowing to use the material in  freely.
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