The inverses of tails of the generalized Riemann zeta function within the range of integers

: In recent years, many mathematicians researched inﬁnite reciprocal sums of various sequences and evaluated their value by the asymptotic formulas. We study the asymptotic formulas of the inﬁnite reciprocal sums formed as (cid:16)(cid:80) ∞ k = n 1 k r ( k + t ) s (cid:17) − 1 for r , s , t ∈ N + , where the asymptotic formulas are polynomials.


Introduction
Throughout the years, many mathematicians have been working on partial infinite sums of reciprocal linear recurrence sequences.
In 2011, Takao Komatsu [8] researched the nearest integer of the sum of reciprocal Fibonacci numbers and derived (1.1) where • denoted the nearest integer; in other words, x = x − 1 2 .In 2020, Ho-Hyeong Lee and Jong-Do Park [9] gave the concept of asymptotic formulas, which were more accurate.The conclusions were as follows: where a n ∼ b n meant lim n→∞ (a n − b n ) = 0.For more results related to the infinite reciprocal sums of linear recurrence sequences, see [1,13,15] and references therein.
The zeta function ζ(z) is undoubtedly the most famous function in analytic number theory.Initially studied by Euler and achieved prominence with Riemann, it abstracted the attention of many mathematicians.Another well-known sequence harmonic number H n is the sum of the first n terms of ζ(z) when z = 1, and the generating function of harmonic numbers ∞ n=1 H n x n is an important tool to study the property of H n .Kim [4][5][6][7] derived many worthy and interesting results associated with the zeta function, harmonic number and its generating function, which inspired us deeply.
At the same time, many researchers began to study the tails of well-known functions such as the Riemann zeta function and the Hurwitz zeta function in [2,3,10,12,14].
For example, Kim Donggyun and Song Kyunghwan [3] studied the inverses of tails of the Riemann zeta function.Derived for s on the critical strip 0 < s < 1, Ho-Hyeong Lee and Jong-Do Park [10] dealt with the inverses of tails of Hurwitz zeta function when s ≥2, s ∈ N and 0 ≤ a < 1, and derived where x s j A s l+ j , x s j = s−2+ j j B j and B j are Bernoulli numbers.In this paper, we extend their asymptotic formulas for the methods and results by considering the tails of and further revealing the property of reciprocal sums of the various sequences.

Main results
Before our conclusion, we define i −1 = 0 for all i ∈ N + , which will take effect in expressing the asymptotic formulas in Theorem 3.
Theorem 1.For all m ∈ N, we have Theorem 2. For all m ∈ N, we have .
Remark.From Theorem 3, we derive that coefficients b j (0 ≤ j ≤ r + s − 1) are determined by r, s and t.At the same time, if we calculate b 0 by using the representation of b i−r−s+1 , there will appear In order to make b 0 the representation of b j , we give the definition i −1 = 0 for i ∈ N + in this paper.Undoubtedly, it satisfies n+1 Corollary 2. If s = 2 and t = 0, we have

Proof of Theorem 1
We need to solve several lemmas for the proof.Proof.See Lemma 2.1 [11].
Lemma 2. For all t ≥ 2 and t ∈ N, we have Proof.It is equivalent with The left side = n(n + t) − (n .
Hence, we have This completes the proof.
Proof.It is equivalent with The left side = .
We can restrict ε < 1 and fix t, then we have which proves (3.2) and completes the proof.
Proof of Theorem 1.
By Lemmas 1-3, we have for all ε > 0, there exists N 0 > 2, subject to Case 3. When t = 0, the proof is similar with Case 1, and we can easily deduce the result.

Proof of Theorem 2
Lemma 4. Let f (n, t, ε) = 3n 3 + an 2 + bn + c and a, b, c are defined in Theorem 2. Then for all ε > 0, there exists N 1 > 0, subject to Proof.It is equivalent with (3.4) The left side = 2 , where A(t) is a function with variable t, then we fix t for all ε > 0, and there exists N 1 > 0, subject to and this completes the proof.Lemma 5. Let g(n, t, ε) = 3n 3 + an 2 + bn + c − ε, then for all ε > 0 there exists N 2 > 0, subject to Proof.The proof is similar with Lemma 4, and we can easily deduce the result.
Proof of Theorem 2. By Lemmas 1, 4 and 5, we have for all ε > 0. There exists hence

Proof of Theorem 3
Proof of Theorem 3.According to the method of proving Theorem 2, it is enough to prove that there exists polynomial The left side = B(n hence we have the necessary condition (1): the notation α( f (x)) means the order of f (x), then we have We note the number of coefficients is r + s, then we have We get the necessary condition (2): If r + s − 1 ≤ i ≤ 2r + 2s − 2, the coefficients of C(n)D(n) and E(n)F(n) are equal, which is equivalent to the system of equations as follows: where i = 2r + 2s − 2, 2r + 2s − 3, • • • , r + s and r + s − 1.We rewrite the system of (3.9) as

Lemma 1 .
Let {a n } ∞ n=1 and {b n } ∞ n=1 be sequences of a positive real number with lim n →∞ a n = lim n →∞ b n = 0.If a n < b n + a n+1 hold for any n ∈ N + , then we have a n < ∞ k=n b n for n ∈ N + .