## Abstract

In this paper, we reconsider the large-*a* asymptotic expansion of the Hurwitz zeta function *ζ*(*s*,*a*). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes *G*-function and the *s*-derivative of the Hurwitz zeta function *ζ*(*s*,*a*) are provided. A detailed discussion on the sharpness of our error bounds is also given.

## 1. Introduction and main results

In this paper, we reconsider the large-*a* asymptotic expansion of the Hurwitz zeta function which is defined for complex *s* and *a* by the series expansion
*s*)>1 and *s*-plane with a simple pole of residue 1 at *s*=1. The Riemann zeta function is a special case since *ζ*(*s*,1)=*ζ*(*s*). As a function of *a*, with *s* (≠1) fixed, *ζ*(*s*,*a*) is analytic in the domain *a*.

It is well known that, as *s* (≠1) and *δ*>0 being fixed, the Hurwitz zeta function has the asymptotic expansion
*B*_{2n}’s denote the even-order Bernoulli numbers and (*w*)_{p}=*Γ*(*w*+*p*)/*Γ*(*w*) is the Pochhammer symbol [1, p. 25] or [2, eq. 25.11.43].

In the special case that *a*=*N* is a large positive integer, (1.1) can be regarded as an asymptotic expansion for the tail of the Dirichlet series expansion of the Riemann zeta function *ζ*(*s*), because
*ζ*(*s*) for *s*=2,3,…,15,16. Gram [4] combined (1.1) and (1.2) to evaluate numerically the first several zeros of the Riemann zeta function *ζ*(*s*) in the critical strip 0<ℜ(*s*)<1. A refinement of Gram’s analysis was provided by Backlund [5], who also gave precise bounds for the remainder term of the asymptotic expansion (1.1) in the case that *a* is a positive integer. Recently, Johannson [6] gave a numerical treatment of *ζ*(*s*,*a*) based on the asymptotic expansion (1.1).

The main aim of this paper is to derive new representations and bounds for the remainder of the asymptotic expansion (1.1). Thus, for *s*≠1 and any positive integer *N*, we define the *N*th remainder term *R*_{N}(*s*,*a*) of the asymptotic expansion (1.1) via the equality
*R*_{N}(*s*,*a*) are based on new representations of this remainder term. Before stating the main results in detail, we introduce some notation. We denote
*Γ*(1−*p*,*w*) is the incomplete gamma function. The function *Π*_{p}(*w*) was originally introduced by Dingle [7, pp. 407] and, following his convention, we refer to it as a basic terminant (but note that Dingle’s notation slightly differs from ours, e.g. *Π*_{p−1}(*w*) is used for our *Π*_{p}(*w*)). The basic terminant is a multivalued function of its argument *w* and, when the argument is fixed, is an entire function of its order *p*. We shall also use the concept of the polylogarithm function Li_{p}(*w*) which is defined by the power series expansion
*w*|<1 and by analytic continuation elsewhere [2, §25.12]. The order *p* of this function can take arbitrary complex values. Finally, we shall denote
*p* with ℜ(*p*)>0.

We are now in a position to formulate our main results. In theorem 1.1, we give new representations for the remainder term *R*_{N}(*s*,*a*).

### Theorem 1.1.

*Let N be a positive integer and let s be an arbitrary complex number such that s*≠1 *and* ℜ(*s*)>1−2*N. Then
**provided* *, and
**and
**provided* *, where the B*_{2N}*(t)’s denote the even-order Bernoulli polynomials.*

Our derivation of (1.5) is based on the integral representation (1.4). Alternatively, formula (1.5) can be deduced from the more general expansion (1.11) of Paris or from an analogous result about the Lerch zeta function by Katsurada [8, Theorem 1].

The subsequent theorem provides bounds for the remainder *R*_{N}(*s*,*a*). The error bound (1.7) may be further simplified by employing the various estimates for

### Theorem 1.2.

*Let N be a positive integer and let s be an arbitrary complex number such that s*≠1 *and* ℜ(*s*)>1−2*N. Then
**and
**provided that* *, and
**provided that*

In the special case that *a* is a positive integer, the bound (1.8) is equivalent to that obtained by Backlund.

For the case of positive *a* and real *s*, we shall show that the remainder term *R*_{N}(*s*,*a*) does not exceed the first neglected term in absolute value and has the same sign provided that *s*≠1 and *s*>1−2*N*. More precisely, we will prove that the following theorem holds.

### Theorem 1.3.

*Let N be a positive integer, a be a positive real number, and let s be an arbitrary real number such that s*≠1 *and s*>1−2*N. Then
**where* 0<*θ*_{N}(*s,a*)<1 *is a suitable number that depends on s, a and N.*

We remark that the special case of this theorem when *a* is a positive integer is known and it appears, for example, in [9, exer. 3.2, p. 292].

We would like to emphasize that the requirement ℜ(*s*)>1−2*N* in the above theorems is not a serious restriction. Indeed, the index of the numerically least term of the asymptotic expansion (1.1) is *n*≈*π*|*a*|. Therefore, it is reasonable to choose the optimal *N*≈*π*|*a*|, whereas the condition *s*=*o*(|*a*|) has to be fulfilled in order to obtain proper approximations from (1.3).

Paris [10] studied in detail the Stokes phenomenon associated with the asymptotic expansion of *ζ*(*s*,*a*) by establishing an exponentially improved extension of (1.1). In particular, he proved that if *s* is any complex number such that *s*≠1 and ℜ(*s*)>1−2*N*_{k} for each *k*≥1, then the Hurwitz zeta function has the exact expansion

as long as *N*_{k}’s are chosen so that *k*th term behaving like (|*a*|*k*)^{ℜ(s)−1}e^{−2π|a|k}.

The remaining part of the paper is structured as follows. In §2, we prove the representations for the remainder term stated in theorem 1.1 and the convergent expansion (1.11). In §3, we prove the error bounds given in theorems 1.2 and 1.3. Section 4 provides applications of our results to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes *G*-function and the *s*-derivative of the Hurwitz zeta function *ζ*(*s*,*a*). The paper concludes with a discussion in §5.

## 2. Proof of the representations for the remainder term

In this section, we prove the representations for the remainder *R*_{N}(*s*,*a*) stated in theorem 1.1 and the convergent expansion (1.11).

We begin by deriving formula (1.4). It is known that the remainder term *R*_{N}(*s*,*a*) may be expressed in the form
*s*≠1 and ℜ(*s*)>1−2*N* (cf. [1, p. 24] or [2, eq. 25.11.28]). Temme [11, eq. (3.26), p. 64] showed that
*u*>0 and positive integer *N*. Suppose for a moment that *a* is real and positive. Substitution of the right-hand side of (2.2) into (2.1) yields
*u* to *t* by *t*=*au*/*k*. The changes in the orders of summation and integration are permitted because of absolute convergence. Now an analytic continuation argument shows that this result is valid for any complex *a* satisfying

The representation (1.5) can be proved as follows. By making the change of variable from *t* to *v* via *v*=*kt* in (2.3) and employing the integral representation (A 2) of the basic terminant, we find
*N*_{k}=*N* for each *k*≥1.

We now turn to the proof of formula (1.6). Our starting point is the integral representation
*a*>0, *s*≠1 and ℜ(*s*)>2−2*N* (cf. [2, eq. 25.11.7]). Integrating once by parts, we obtain
*s*≠1 and ℜ(*s*)>1−2*N*. This finishes the proof of formula (1.6).

We close this section by proving the convergent expansion (1.11). For this purpose, suppose that *s*≠1 and ℜ(*s*)>−1. Under these assumptions, formula (2.4) applies and gives
*s*≠1 and ℜ(*s*)>1−2*N*_{k} for each *k*≥1. By the definition (1.3) of the remainder term *R*_{1}(*s*,*a*), the expansion (2.9) is seen to be equivalent to (1.11).

## 3. Proof of the error bounds

In this section, we prove the bounds for the remainder term *R*_{N}(*s*,*a*) given in theorems 1.2 and 1.3.

We begin with the proof of the bound (1.7). Suppose that *N* is a positive integer and *s* is an arbitrary complex number such that *s*≠1 and ℜ(*s*)>1−2*N*. From (1.5), we infer that

We proceed by proving formula (1.10). Note that 0<*Π*_{p}(*w*)<1 whenever *w*>0 and *p*>0 (see proposition A.1). Therefore, employing the representation (1.5) and observing that the terms of the infinite sum in (1.5) are all positive, we can assert that
*θ*_{N}(*s*,*a*)<1 depending on *s*, *a* and *N*. The identity (3.1) then shows that this expression is equivalent to the required one in (1.10).

The bound (1.8) can be proved as follows. It is easy to see that, for any positive real *t* and complex *a* with *t*>0. Consequently, if *N*≥1 is fixed, the function *B*_{2N}−*B*_{2N}(*t*−⌊*t*⌋) does not change sign. From (1.6), using (3.2)–(3.4), we can conclude that
*R*_{N}(ℜ(*s*),|*a*|)| may be bounded as follows:

We close this section by proving the estimate (1.9). It follows from the first equality in (3.3) that for any positive real *t* and complex *a* with *t*+*a*|^{2}≥*t*^{2}+|*a*|^{2} holds true. We also have |*B*_{2N}(*t*−⌊*t*⌋)|≤|*B*_{2N}| for any *t*>0 [2, eq. 24.9.1]. Applying these inequalities and (3.2) in (2.6), we deduce that
*t* to *u* by *u*=*t*^{2}/|*a*|^{2} and used the known integral representation of the beta function [2, eq. 5.12.3] and the definition of *χ*(*p*).

## 4. Application to related functions

In this section, we show how the remainder terms of the known asymptotic expansions of the polygamma functions, the gamma function, the Barnes *G*-function and the *s*-derivative of the Hurwitz zeta function *ζ*(*s*,*a*) may be expressed in terms of the remainder *R*_{N}(*s*,*a*). Thus, the theorems in §1 can be applied to obtain representations and bounds for the error terms of these asymptotic expansions. Some of the resulting representations and bounds are well known in the literature but many of them, we believe, are new.

The asymptotic expansion of the digamma function *N* is any positive integer and *δ*>0 being fixed (cf. [1, p. 18] or [2, eq. 5.11.2]). (Throughout this section, we use subscripts in the

The polygamma function *ψ*^{(k)}(*z*)=(d^{k}/d*z*^{k})*ψ*(*z*), with *k* being any fixed positive integer, has the asymptotic expansion
*N*≥1, where *δ*>0 (see, for instance, [1, p. 18]). The functional relation [2, eq. 25.11.12]

The standard asymptotic expansion of the logarithm of the gamma function takes the form
*N* is any positive integer and *δ*>0 being fixed (see, e.g. [1, p. 12] or [2, eq. 5.11.1]). The asymptotic expansion (4.3) is usually attributed to Stirling, however, it was first discovered by De Moivre (for a detailed historical account, see [13, pp. 482–483]). If we substitute (1.3) into the right-hand side of the equality [2, eq. 25.11.18]

The *G*-function, introduced and studied originally by Barnes [17], has the asymptotic expansions
*N*≥1, where *δ*>0. The asymptotic expansion (4.5) was established by Ferreira & López [18], whereas (4.6) is due to Barnes [19]. The Barnes *G*-function is related to the Hurwitz zeta function through the functional equation [20, eq. (2.2)]

Elizalde [21] studied the large-*a* asymptotic behaviour of the *s*-derivative of the Hurwitz zeta function *ζ*(*s*,*a*) evaluated at negative integer values of *s*. In particular, he showed that
*N* is any positive integer and *δ*>0 being fixed. From (4.7) and (4.8), we infer that
*s* and comparing the result with (4.9), we obtain
*N*≥2. An immediate consequence of these relations is that Elizalde’s expansions are actually valid in the wider sector *δ*>0.

## 5. Discussion

In this paper, we have derived new representations and estimates for the remainder term of the asymptotic expansion of the Hurwitz zeta function. In this section, we shall discuss the sharpness of our error bounds.

First, we consider the bound (1.7) with *s* being real. In particular, the asymptotic expansions discussed in §4 belong to this case. Let *N* be any non-negative integer, *s* a real number and *a* a complex number. Suppose that *s*≠1, *s*>1−2*N* and

By the definition of an asymptotic expansion, *N*≥0. Therefore, when

Consider now the case when *π*/2. As *π*/2, the factor *N*. However, this is not the case, as the following argument shows. We assume that *s* is fixed, |*s*+2*N*−1−|2*πa*|| is bounded, *N* is large. The saddle point method applied to (A 11) shows that when *p*−|*w*|| is bounded, the asymptotics
*π*/2, the estimate (5.1) and thus the error bound (1.7) cannot be improved in general.

Finally, assume that *N*, remains bounded, provided that *R*_{N}(*s*,*a*). Otherwise, *N* is large, making the bound (5.1) completely unrealistic in most of the sectors

Let us now turn our attention to the case when *s* is allowed to be complex in (1.7). Let *N* be a positive integer and let *s* and *a* be complex numbers such that *s*≠1, ℜ(*s*)>1−2*N* and *s*)=*o*(*N*^{1/2}) as *N*. Consequently, if *N* is large, the estimates (5.5) and (1.7) are reasonably sharp in most of the sector *s*)=*o*(*N*^{1/2}) is replaced by the weaker condition *s*)+2*N*−1 is small and ℑ(*s*) is much larger than *N*^{1/2}, this quotient may grow exponentially fast in |ℑ(*s*)|, which can make the bound (5.5) completely unrealistic. Therefore, if the asymptotic expansion (1.1) is truncated just before its numerically least term, i.e. when *N*≈*π*|*a*|, the estimate (5.5) is reasonable in most of the sector *s*=*o*(|*a*|), the stronger assumption *a*.

When *π*/2, the estimate (5.5) may be replaced by
*s* satisfies the requirements posed in the previous paragraph and *N*. By an argument similar to that used in the case of real *s*, it follows that (5.6) is a realistic bound. Otherwise, if ℜ(*s*)+2*N*−1 is small and ℑ(*s*) is much larger than *N*^{1/2} or

Similarly to the case of real *s*, for values of *a* outside the closed sector

We continue by discussing the bound (1.9). Let *N* be a positive integer, *s* a real number and *a* a complex number. Suppose that *s*≠1, *s*>1−2*N* and *s* with ℑ(*s*)≠0, the bound (1.9) can be sharper than (5.5) and (5.6) since it does not involve the gamma function ratio which grows exponentially fast in |ℑ(*s*)|.

We conclude this section with a brief discussion of the error bound (1.8). With the same assumptions as those for the estimate (5.1), the bound (1.8) simplifies to
*s* is complex, the estimate (1.8) is reasonably sharp as long as

## Data accessibility

This work does not have any experimental data.

## Competing interests

The author has no competing interests.

## Funding

The author’s research was supported by the research grant no. GRANT11863412/70NANB15H221 from the National Institute of Standards and Technology.

## Acknowledgements

The author appreciates the help of Dorottya Sziráki in improving the presentation of some parts of the paper.

## Appendix A. Bounds for the basic terminant

In this appendix, we prove some estimates for the absolute value of the basic terminant *Π*_{p}(*w*) with ℜ(*p*)>0. These estimates depend only on *p* and the argument of *w* and therefore also provide bounds for the quantity

### Proposition A.1.

*For any complex p with* ℜ(*p*)>0 *it holds that*
*Moreover, when w and p are positive, we have* 0<*Π*_{p}(*w*)<1.

We remark that it was shown by the author [22] that *p* is real and positive, which improves on the bound (A 1) near

### Proof.

Our starting point is the integral representation [22]
*p*)>0. For *t*≥0, we have
*w* and *p*, notice that 0<1/(1+(*t*/*w*)^{2})<1 for any *t*>0. Therefore, the integral representation (A 2) combined with the mean value theorem of integration imply that 0<*Π*_{p}(*w*)<1. ▪

### Proposition A.2.

*For any complex p with* ℜ(*p*)>0, *we have*
*and*
*for*

### Proof.

It is enough to prove (A 3) when *p*)>0. This integral representation follows from the definition of the basic terminant *Π*_{p}(*w*) and a well-known representation of the incomplete gamma function [2, eq. 8.6.5]. To estimate the right-hand side of (A 5), we note that for any positive real *t* and complex *w* with *p*)>0. This integral representation may be deduced from the definition of the basic terminant *Π*_{p}(*w*) and two different representations of the incomplete gamma function [2, eqs. 8.6.4 and 8.6.5]. ▪

The following estimate was proved by the author in [22] and is valid for positive real values of the order *p*.

### Proposition A.3.

*For any p*>0 *and w with* *we have*
*where φ is the unique solution of the implicit equation*
*that satisfies* *if* *if* *if* *if* *if* *and* *if*

We remark that the value of *φ* in this proposition is chosen so as to minimize the right-hand side of the inequality (A 7).

### Proposition A.4.

*For any complex p with* ℜ(*p*)>0, *we have*
*and*
*for*

### Proof.

It is sufficient to prove (A 8) for *t* to *u* by *u*=*t*/|*w*|. The *u*-integral can be evaluated in terms of the hypergeometric function [22], giving the first estimate in (A 8). To obtain the second inequality in (.8), note that

### Proposition A.5.

*For any complex p with* ℜ(*p*)>0, *we have*
*for*

The dependence on |*w*| in these estimates may be eliminated by employing the bounds for |*Π*_{p}(*we*^{∓πi})| that were derived previously.

### Proof.

The proof is based on the functional relation [22]
*r*^{q}e^{−rα}, as a function of *r*>0, takes its maximum value at *r*=*q*/*α* when *α*>0 and *q*>0. We therefore find that
*q*>0 (see, for instance, [2, eq. 5.6.1]). We derive the second bound in (A 12) from the result that
*q*>0 (see, e.g. [2, eq. 5.6.4]) and the definition of *χ*(*q*). ▪

Finally, we mention the following two-sided inequality proved by Watson [23] for positive real values of *p*:
*χ*(*p*).

- Received May 24, 2017.
- Accepted June 5, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.