Related papers: Exact Summatory Functions for Prime $k$-tuples
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions as extensions of the class of multiplicative functions. We apply these concepts to Ramanujan's sum and its analogue with respect to…
We give a simple method for estimating the average of singular series' associated with prime k-tuples.
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the…
We give a full description of all sets of functions on the group $(\mathbb{ Z}_p, +)$ of prime order which are closed under the composition with the clone generated by $+$ from both sides. Thereby, we also get a description of all iterative…
Inspired from a joint work by A. Beckmann, S. Buss and S. Friedman, we propose a class of set-theoretic functions, predicatively computable functions. Each function in this class is polynomial time computable when we restrict to finite…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
This paper offers a solution method that allows one to find exact values for a large class of convergent series of rational terms. Sums of this form arise often in problems dealing with Quantum Field Theory.
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
By applying hybrid functions of general block-pulse functions and the second Chebyshev polynomials,integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of integrodifferential systems are…
In this paper we establish a new explicit upper and lower bound for the $n$-th prime number, which improve the currently best estimates given by Dusart in 2010. As the main tool we use some recently obtained explicit estimates for the prime…
We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative…
For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match…
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…
Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a generalization of a summation formula already proved by us [Advances in Applied Mathematics, 175 (2026) 103201], which…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums),…