
TL;DR
This paper derives complete asymptotic expansions for specific q-series as q approaches 1 from below, revealing detailed behavior of these series in the scale of powers of log q.
Contribution
It provides new asymptotic formulas for two classes of q-series involving divisor functions and power sums, extending understanding of their behavior near q=1.
Findings
Asymptotic expansions in powers of log q as q approaches 1
Explicit formulas for series involving divisor functions
Enhanced understanding of q-series behavior near the singularity
Abstract
In this work we study complete asymptotic expansions for the q-series and in the scale function as , where and is the divisor function .
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Taxonomy
TopicsAdvanced Mathematical Identities · Mathematical functions and polynomials · Analytic Number Theory Research
Asymptotics of Certain -Series
Ruiming Zhang
College of Science
Northwest A&F University
Yangling, Shaanxi 712100
P. R. China.
Abstract.
In this work we study complete asymptotic expansions for the q-series and in the scale function as , where and is the divisor function .
Key words and phrases:
q-series; divisor functions; asymptotics.
2000 Mathematics Subject Classification:
33D05; 33C45.
The work is supported by the National Natural Science Foundation of China grants No. 11371294 and No. 11771355.
1. Preliminaries
In this work we study complete asymptotic expansions for the q-series and in the scale function as , where and is the divisor function . Unlike methods used [3, 4], our method does not apply Fourier transform or the modular properties, it can not give a complete asymptotic expansion in exponential scales when and is an even integer. However, this shortcoming can be overcome by applying the functional equations for the corresponding zeta functions which are equivalent to the symmetry .
The Euler gamma function is defined by
[TABLE]
and its analytic continuation is given by
[TABLE]
Let and , it is known that [1, 5, 6]
[TABLE]
as , uniformly with respect to . The digamma function is defined by
[TABLE]
and the Euler’s constant is
[TABLE]
The Riemann zeta function is defined by
[TABLE]
then its analytic continuation, which is also denoted as , is an meromorphic function that has a simple pole at with residue . The meromorphic function satisfies the functional equation [1, 2, 5, 6]
[TABLE]
For and , it is known that [6]
[TABLE]
as , uniformly with respect to . The Stieltjes constants are the coefficients in the Laurent expansion,
[TABLE]
where and . Moreover, the Glaisher’s constant is defined as
[TABLE]
The Bernoulli numbers are defined by
[TABLE]
Then
[TABLE]
By (1.7) we get
[TABLE]
The function for is defined as the sum of the -th powers of the positive divisors of , [2, 5]
[TABLE]
where stands for " divides ". We also use the notations and It is known that [2, 5]
[TABLE]
and
[TABLE]
2. Main Results
Theorem 1**.**
Given a positive integer , let for all satisfying .
If
[TABLE]
where is the Riemann zeta function, then for all , and satisfying we have
[TABLE]
Furthermore,
[TABLE]
as , where the first sum is over all the distinct pairs while the last sum is over all nonnegative integers such that
[TABLE]
Proof.
For , since each factor of is an absolute convergent Dirichlet series, then the product itself is also an absolute convergent Dirichlet series. Let be any complex number satisfying
[TABLE]
then by the theory of Dirichlet series we know the partial sums are absolutely and uniformly bounded for all . Let be a large positive integer and
[TABLE]
in Lemma 2 of section 11.6 in [2] to get Hence,
[TABLE]
By the inverse Mellin transform of (1.1) we get
[TABLE]
for all .
Let
[TABLE]
then for any positive satisfying , by (2.6) we get
[TABLE]
where we have applied (2.5) and the Fubini’s theorem to exchange the order of summation and integration.
Since has a simple pole at and has simple poles at all non-positive integers, then all the possible poles of the meromorphic function are
[TABLE]
and all non-positive integers. Let and such that
[TABLE]
we integrate over the rectangular contour with vertices,
[TABLE]
Then by Cauchy’s theorem we have
[TABLE]
where the first sum is over all the distinct pairs from whereas the last sum is over all satisfying and (2.4).
On the other hand, we also have
[TABLE]
Fix and , by (1.3) and (1.8), since the integrands of the last two integrals have the estimate
[TABLE]
where is an arbitrary positive number such that , then the last two integrals have limit [math] as . Then by taking limit in (2.9) and (2.8) we get
[TABLE]
where the summations are the same as in (2.8).
Since
[TABLE]
then again by (1.3) and (1.8) we get
[TABLE]
as . Then by (2.10) and (2.11) we get
[TABLE]
as , where the first sum is over all the distinct pairs from while the last sum is over all nonnegative integers satisfying (2.4). Finally, (2.2) is obtained by combining (2.7) and (2.12). ∎
Corollary 2**.**
Let . If , then
[TABLE]
as .
If there exists a such that , then
[TABLE]
as .**
Proof.
When the integrand is meromorphic and has the following simple poles
[TABLE]
with residues
[TABLE]
Then (2.13) is obtained by applying Theorem 1.
When for some nonnegative integer , then
[TABLE]
has a double pole at with residue
[TABLE]
all the other nonpositive integers are simple poles with residues,
[TABLE]
Then by Theorem 1 we have
[TABLE]
as . ∎
Example 3**.**
When we have
[TABLE]
as , which means the error term is better than any . When , the double pole happens at , then
[TABLE]
as . When , then has no nonnegative integer solutions. Thus,
[TABLE]
or
[TABLE]
as .
Corollary 4**.**
For all , and we have
[TABLE]
Furthermore, if for all nonnegative integers , then
[TABLE]
as . **
If for certain nonnegative , then
[TABLE]
as .**
Proof.
When for all nonnegative integers , then the meromorphic function has a double pole at with residue
[TABLE]
and simple poles at all nonpositive integers with residue
[TABLE]
Then by Theorem 1 we get
[TABLE]
as .
When for certain nonnegative integer , then the meromorphic function has a triple pole at with residue
[TABLE]
It has simple poles at all other nonpositive integers with residue
[TABLE]
Then by Theorem 1 we get
[TABLE]
as . ∎
Example 5**.**
Let , then by (2.22) to get
[TABLE]
as , the remainder here is better than any . Let , then by (2.23) to get
[TABLE]
as .
Corollary 6**.**
Let and , then for all , and we have
[TABLE]
Furthermore, if and for all nonnegative integers , then
[TABLE]
as .
If for certain nonnegative integer and for all nonnegative integers , then
[TABLE]
as .
If for all nonnegative integers and for certain , then
[TABLE]
as .
If and for certain nonnegative integers with , then
[TABLE]
as .
Proof.
When , and for all nonnegative integers , the meromorphic function has simple poles at
[TABLE]
with residues
[TABLE]
[TABLE]
and
[TABLE]
for . Then by Theorem 1 we get
[TABLE]
as .
When for certain nonnegative integer and for all nonnegative integers , then the meromorphic function has a double pole at with residue
[TABLE]
and a simple pole at with residue
[TABLE]
and simple poles at all nonpositive integers other than with residues
[TABLE]
Hence,
[TABLE]
as .
When for all nonnegative integers and for certain , then the meromorphic function has a double simple pole with residue
[TABLE]
and a simple pole with residue
[TABLE]
and simple poles at all nonpositive integers other than with residue
[TABLE]
Thus,
[TABLE]
as .
When and for certain nonnegative integers with , then the meromorphic function has two double poles at and with residues
[TABLE]
and
[TABLE]
respectively. It has simple poles at all other nonpositive integers other than with residues
[TABLE]
Then,
[TABLE]
as . ∎
Example 7**.**
When , by (2.30) we get
[TABLE]
as . When , then in (2.31). Then
[TABLE]
as , it implies that the difference between two sides of the above formula is smaller than any . Let in (2.41), then . Then,
[TABLE]
as .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.
- 3[3] B. C. Berndt and B. Kim, Asymptotic Expansions of Certain Partial Theta Functions, Proceedings of AMS, Volume 139, Number 11, November 2011, 3779–3788
- 4[4] K. Bringmann, A. Folsom and A. Milas, Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters, Journal of Mathematical Physics 58, 011702 (2017); ; doi: 10.1063/1.4973634.
- 5[5] Nist DLMF, http://dlmf.nist.gov/
- 6[6] Hans Rademacher, Topics in Analytic Number Theory , Springer-Verlag, Berlin, 1973
