Related papers: Explicit formulas of a generalized Ramanujan sum
We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Toth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose…
We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…
Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial…
In this paper, we consider a general form of the analogue of Ramanujan's sum in the ring of polynomials over a finite field. We first prove some multiplicative properties of such functions before considering their finite Fourier series and…
In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.
In this paper, we state some $q$-analogues of the famous Ramanujan's Master Theorem. As applications, some values of Jackson's $q$-integrals involving $q$-special functions are computed.
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…
The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as…
In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}\Omega(k)$, where $\Omega(k)$ is the total number of prime factors of $k$.
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…
Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…
A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are…
In this paper we obtain some new transformation formula for Ramanujan summation formula and also establish some eta-function identities. we also deduce a q-Gamma function identity, n q-integral and some interesting series representation.
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…
Several methods are used to evaluate finite trigonometric sums. In each case, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation…
A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0<a\leq 1,\ x>0$, $N\in\mathbb{N}$ and $h\in\mathbb{Z}$, is undertaken. Two of the general…
In this short note we use the umbral formalism to derive the Ramanujan Master Theorem and discuss its extension to more general cases.
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…