Several generalizations on Wolstenholme Theorem

This paper generalizes Wolstenholme theorem on two aspects. The first generalization is a parameterized form: let p > k + 2, k ≥ 1, ∀t ∈ ℤ, then ${{(pt + p - 1)!} \over {(pt)!}}\mathop \sum \limits_{m = 0}^{k - 1} {( - 1)^m}\mathop \sum \limits_{1 \le {i_l} < \cdots < {i_{k - m}} \le p - 1} {{{p^{k - (m + 1)}}} \over {\mathop \prod \limits_{l = 1}^{k - m} (pt + {i_l})}} \equiv 0{\left( {\bmod {p^{k + 1}}} \right)^.}$ The second generalization is on composite number module: Let 1overa be the x in congruent equation ax ≡ 1(mod m)(1 ≤ x < m), if m ≥ 5, then $$\matrix{ {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^2}} } \hfill & \equiv \hfill & {{m \over 6}[2m\varphi (m) + \prod\limits_{p|m} {(1 - p)]{{(\bmod m)}^{\;;}}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^3}} } \hfill & \equiv \hfill & {{{{m^2}} \over 4}[m\varphi (m) + \prod\limits_{p|m} {(1 - p)](\bmod m){\;^;}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^4}} } \hfill & \equiv \hfill & {{m \over {30}}[6{m^3}\varphi (m) + 10{m^2}\prod\limits_{p|m} {(1 - p) - \prod\limits_{p|m} {(1 - {p^3})](\bmod m){\;^;}} } } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^r}} } \hfill & \equiv \hfill & {{m^r}\sum\limits_{d|m} {\mu (d){{\left( {{m \over d}} \right)}^{ - r}}\sum\limits_{k = 1}^{{m \over d}} {{k^r}(\bmod m){\;^.}} } } \hfill \cr } $$ Where φ(x) is Euler function , μ(x) is Möbius function.

Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

By using the properties of Euler function, an upper bound of solutions of Euler function equation is given, where is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation are given, where is the number of distinct prime factors of n.

Another generalization of Euler’s arithmetic function and Menon’s identity

We define the k-dimensional generalized Euler function $$\varphi _k(n)$$ φ k ( n ) as the number of ordered k-tuples $$(a_1,\ldots ,a_k)\in {\mathbb {N}}^k$$ ( a 1 , … , a k ) ∈ N k such that $$1\le a_1,\ldots ,a_k\le n$$ 1 ≤ a 1 , … , a k ≤ n and both the product $$a_1\cdots a_k$$ a 1 ⋯ a k and the sum $$a_1+\cdots +a_k$$ a 1 + ⋯ + a k are prime to n. We investigate some of the properties of the function $$\varphi _k(n)$$ φ k ( n ) , and obtain a corresponding Menon-type identity.

Reciprocity of poly-Dedekind-type DC sums involving poly-Euler functions

The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind-type DC sums, which are obtained from the Dedekind-type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).