The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function

1.   Introduction

The Riemann zeta function $\zeta(s)$ and the Hurwitz zeta function $\zeta(s, a)$ are two important functions in the analytic number theory. The Riemann zeta function has an intimate connection with the distribution of primes, and the Hurwitz zeta function plays an important role in studying the properties of the Dirichlet L-function. The properties of $\zeta(s)$ and $\zeta(s, a)$ have attracted considerable attention from mathematicians.

In the past ten years, many mathematicians have been working on the tails of various zeta functions. For example, Xin ^[1] gives the asymptotic formulas of the Riemann zeta function with $s = 2$ and $s = 3$,

Kim and Song ^[2] studied the inverses of the tails of the Riemann zeta function and derived results for $s$ in the critical strip $0 < s < 1$,

Lee and Park ^[3] dealt with the inverses of the tails of the Hurwitz zeta function when $s\geq$2, $s\in \mathbb{N}$, and $0\leq a < 1$, and derived

where

and $B_j$ is the Bernoulli numbers.

Ohtsuka and Nakamura ^[4] derived the asymptotic formulas for the tails of the Fibonacci zeta function with s = 1 and s = 2,

More results associated with the asymptotic formulas can be found in ^{[5,6,7,8,9,10,11,12]}, where Hwang and Song ^[5] and Xu ^[6] study the tails related to the Riemann zeta function, Choi and Choo ^[7], Komatsu ^[8], Lee and Park ^[9,10] and Marques and Trojovsky ^[11] study the tails of the Fibonacci zeta function, and Xu and Wang ^[12] studied the tails of Pell zeta function.

The above results reveal the relations between zeta functions and polynomials. From other perspectives, these sums are the First Newton's identity of an infinite polynomial with roots $k^{-s}$ or $F_k^{-s}$ for $k \geq n$; therefore, they also reflect the importance of Newton's identities, which are elaborately introduced in ^[13].

In general, the consideration is limited to cases where $s\in\mathbb{N^+}$ or $s\in (0, 1)$. Due to the application of the estimate of the tails of the Riemann zeta function in various other problems, such as the upper of the Dirthlet L-function, the average of many well-known arithmetical functions, and so on, we hope to generalize the results to the complex plane. For this proposal, in this paper, the generalized Riemann zeta function $R(r, s, t)$ and its n-th tail $R_n(r, s, t)$ are defined for $Re(r+s) > 1$, $t\in \mathbb{C}$ by series

and then we study the asymptotic formulas of $R_n(r, s, t)$ for $r, s$ in the half-plane $Re(r+s) > 1$ and $t\in \mathbb{C}$. In addition, inspired by the expressions of $\zeta(s)$ with $0 < s < 1$ in ^[2], we can similarly estimate the inverses of the tails of $\zeta(s)$ for $Re(s) < 0$. Similarly, we can derive the asymptotic formulas for the inverses of the tails of the Hurwitz zeta function and the Dirichlet L-function $L(1, \chi)$ for $Re(s) > 1$ and $Re(s) < 0$. It is worth noting that the inverses of the Hurwitz zeta function and the Dirichlet L-function possess new characteristics, and we obtain their dominant terms.

2.   Preliminaries

Theorem 1. Let the function $R(r, s, t)$ and the n-th tail $R_n(r, s, t)$ be as above, for any $r, s, t \in \mathbb{C}$. If

we have

where

and the coefficients

are constants determined by $r, s, t$ as follows:

where $[x]$ denotes the integer part of the real part of $x$, and $\{x\} = x-[x].$

Theorem 2. Let

be the n-th tail of the Riemann zeta function, then for all $Re(s) < 0$, we have

where

and the coefficients

are constants determined by $s$ as follows:

and $[x]$ denotes the integer part of the real part of $x$, and $\{x\} = x-[x].$

Corollary 1. Let

be the n-th tail of the Riemann zeta function, then for $Re(s) > 1$, we have

where

and

Corollary 2. If $r = 0$, $0 < t = a \leq 1$, and $|s| > 1$, we have

where

and

Now we start considering the asymptotic formulas for inverses of the tails of the Hurwitz zeta function $\zeta(s, a)$ for $Re(s) < 0$ and $a = \frac{p}{q}\in \mathbb{Q}, p, q \in \mathbb{N}$, $(p, q) = 1$.

In combination with the following functional equation related to the Hurwitz zeta function ^[14]:

For any positive integer $n$, write

for some $q \geq r\geq 1$ and $l, r \in \mathbb{N_+}.$ The $n$-th tails of the Hurwitz zeta function for $Re(s) < 0$ are defined by

where

for $m = 1, 2, \cdots, q.$

When

rewrite above $n$-th tails of the Hurwitz zeta function for $Re(s) < 0$ as

Theorem 3. Let function $\zeta(s, \frac{p}{q})$ and the $n$-th tail $\zeta_n(s, \frac{p}{q})$ be as above for $Re(s) < 0$ and $p, q \in \mathbb{N_+}, \; (p, q) = 1$.

(i) Let $n = q(l-1)+1$ for some $l \in \mathbb{N_+}$, if

holds, we have

where

for $m = 1, 2, \cdots, q.$

(ii) Let

for some $q \geq r\geq 1$ and $l, r \in \mathbb{N_+},$ if

hold, we have

where

for $m = 1, 2, \cdots, q.$

Remark 1. Since $\sum^{q}_{m = 1}a_m = 0$ (see the proof of Theorem 3 in this paper), we observe that the result (i) of Theorem 3 follows (ii) when $r = 1.$

Similarly, according to the following functional equation related to the Dirichlet L-function ^[14]:

where $\chi$ is a character module $k$.

The asymptotic formula of the inverse of the Dirichlet L-function $L(s, \chi)$ is trivial when $\chi = \chi_1$ is the principal character module $k$. Therefore, we only consider the case for $\chi \neq \chi_1.$ Similar to the Hurwitz zeta function, the $n$-th tail of $L(s, \chi)$ could be defined for $Re(s) > 1$ by

where

for some $k \geq r\geq 1$ and $l, r \in \mathbb{N_+}$.

When $r = 1$, we have

In combination with (2.7) and (2.8), for $Re(s) < 0$, we have

where

for $j = 1, 2, \cdots, k$ and $m = 1, 2, \cdots, k$.

The $n$-th tail of $L(s, \chi)$ could be defined for $Re(s) < 0$ by

where

for some $k \geq u\geq 1$, $k \geq r\geq 1$, and $l, u, r \in \mathbb{N_+}.$

When $r = 1$, we have

Theorem 4. Let the function $L(s, \chi)$ and the $n$-th tail $L_n(s, \chi)$ be as above for $\chi \neq \chi_1$ is the nonprincipal character module $k$.

(i) When $Re(s) > 1$, let $n = k(l-1)+1$ for some $l \in \mathbb{N_+}$, if

holds, we have

(ii) When $Re(s) > 1$, let $n = k(l-1)+r$ for some $k \geq r\geq 1$, and $l, r \in \mathbb{N_+},$ if

hold, we have

(iii) When $Re(s) < 0$, let $n = k^2(l-1)+1$ for some $l \in \mathbb{N_+}$, if

holds, we have

where

for $j = 1, 2, \cdots, k$ and $m = 1, 2, \cdots, k$.

(iv) When $Re(s) < 0$, let

for some $k \geq u \geq1$, $k \geq r\geq 1$ and $l, u, r \in \mathbb{N_+}$, if

hold, we have

where

for $j = 1, 2, \cdots, k$ and $m = 1, 2, \cdots, k$.

3.   The proof of Theorem 1

Before our proof, we first propose several lemmas.

Lemma 1. For all $s, t\in \mathbb{C}$ and $n\in \mathbb{N}$, if $n > |t|$, we have

Proof. Let

and the third equation holds for $| \frac{t}{n}| < 1.$ □

Lemma 2. For all $r, s, t \in \mathbb{C}$ and $Re(r+s) > 1$, there exists a unique generalized complex coefficient polynomial

where

is determined by $r, s$, and $t$, subject to the real part of its order being greater than $0$ and

Proof. First, we assume

satisfies

Since the absolute value of the order of $[B^*_{r, s, t}(n+1)-B^*_{r, s, t}(n)\, ]\, n^r(n+t)^s$ is greater than $|r+s+\{r+s\}-2|$, then we have the necessary condition (1): The order of $[B^*_{r, s, t}(n+1)-B^*_{r, s, t}(n)\, ]\, n^r(n+t)^s$ is equal to the order of $B^*_{r, s, t}(n+1)B^*_{r, s, t}(n)$.

In other words, we have $l-1+r+s = 2l,$ which implies

Then we rewrite

with $b^*_{r+s-1} \neq0,$ by Lemma 1, we have

Let $n > 1$, we have

and

where $[x]$ denotes the integer part of the real part of $x$.

Now we derive the necessary condition (2): The following system of equations has at least one solution,

where $k = 0, 1, 2, \cdots, [r+s-2]$ and $[r+s-1]$.

By calculation, the system of Eq (3.6) is equivalent with

and

for $k = 1, 2, \cdots, [r+s-1].$ Combining with $Re(r+s-1) > 0$, we can then derive the unique solution of the system of Eq (3.6) as follows:

and

where $k = 1, 2, \cdots, [r+s-1].$

So we derive the necessary condition of (3.2):

with

where $[x]$ denotes the integer part of the real part of $x$, $\{x\} = x-[x].$

Now we show that (3.9) and (3.10) are also sufficient conditions. If Eq (3.10) holds, the functions $\left[B^*(n+1, r, s, t)-B^*(n, r, s, t) \right] n^r(n+t)^s$ and $B^*(n+1, r, s, t)B^*(n, r, s, t)$ are analytic functions of $n$, which means the series of $k$ in formulas (3.4) and (3.5) are their Taylor expansion.

Then consider the following function

the coefficients of the first [r+s] terms are equal to zero, and the tail term is convergent, which proves the Eq (3.2). □

Lemma 3. If for all $r, s, t \in \mathbb{C}$ and $Re(r+s) > 1$, there exists a generalized complex coefficient polynomial

where

subjects to the real part of its order is greater than $0$, and

Then, we have

Proof. Using the equation

Then, we have

Hence, we have

and then

which is equivalent with

This completes the proof. □

Proof of Theorem 1. Since $Re(r+s) > 1$ and series

absolutely converges and is analytic in the half-plane $Re(r+s) > 1$. Then we have

Let $n > |t|$, by Lemmas 1 and 2, these are unique generalized complex coefficient polynomials

where

is determined by $r, s, t$ and

subject to the real part of its order being greater than $0$, and

Then, in combination with Lemma 3, we have

This completes thhe proof. □

4.   Proof of Corollaries 1 and 2 and Theorem 2

Proof of Corollary 1. If $r = 0$ and $t = 0$, we have

By Theorem 1, we have

where

and

□

Proof of Corollary 2. The proof of Corollary 2 follows from Corollary 1. □

Before the proof of Theorem 2, we introduce some basic preliminary properties that could be found in ^[15].

Proposition 5. The Riemann zeta function $\zeta(s)$ satisfies

Proposition 6. (1) The function $\frac{1}{\Gamma(s)}$ is entire and only possesses zeros at $s = 0, -1, -2, \cdots$.

(2) The function $\Gamma(s)$ is holomorphic for $s \in \mathbb{C}$ except poles at $s = 0, -1, -2, \cdots$ and $\Gamma(s)$ possesses no zero.

Proof of Theorem 2. By Proposition 5, we have

By Proposition 6, we have $\zeta(s)$ is entire in half-plane $Re(s) < 0$. Hence, for $Re(s) < 0,$ we have

By Theorem 1, we derive

which completes the proof.□

5.   Proof of Theorems 3 and 4

Proof of Theorem 3. (i) We first prove

where

for $m = 1, 2, \cdots, q.$ For any complex variable $s = \sigma +it$, we have

and then

Since

Lemma 1 and Theorem 1, let $n = q(l-1) +1,$ then for any sufficiently large positive integer $l \geq 1$ and $Re(s) < 0$, if

holds, we have

(ii) Let $n = q(l-1)+r$ for some $q \geq r\geq 1$ and $l, r \in \mathbb{N_+},$ if

and

hold, we have

□

Proof of Theorem 4. (i) First of all, since $\chi$ is a nonprincipal character mod k, we have

Let

if

holds, and the asymptotic formula of the inverse of the $n$-th tail of the L-function $L(s, \chi)$ for $Re(s) > 1$ is as follows:

(ii) Let

for some $k \geq r\geq 1$ and $l, r \in \mathbb{N_+},$ if

and

hold, we have

(iii) Let

for some $l \in \mathbb{N_+}$, if

holds, in combination with

we obtain

Then, we have

(iv) Let

for some $k \geq u \geq1$, $k \geq r\geq 1$, and $l, u, r \in \mathbb{N_+}$, if

and

hold, we have

□

6.   Conclusions

In this paper, we derive the asymptotic formulas for the reciprocal sums of the Riemann zeta function, the Hurwitz zeta function, and the Dirchlet L-function for $Re(s) > 1$ and $Re(s) < 0$.

Author contributions

Zhenjiang Pan: writing-review and editing, writing-original draft, validation, resources, methodology, formal analysis, conceptualization.

Zhengang Wu: writing-review and editing, resources, methodology, supervision, validation, formal analysis, funding acquisition.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors express their gratitude to the referee for very helpful and detailed comments. Supported by the National Natural Science Foundation of China (Grant No. 11701448).

Conflict of interest

The authors declare no conflicts of interest.