Related papers: On the constant in the Mertens product for arithme…
We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.
A 1976 result from Norton may be used to give an asymptotic (but not explicit) description of the constant in Mertens' second theorem for primes in arithmetic progressions. Assuming the Generalized Riemann Hypothesis, we give an effective…
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give…
We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.
The aim of this paper is to present new upper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the…
We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
We give some experimental observations on the growth of the norm of certain matrices related to the Mertens function. The results obtained in these experiments convince us that linear algebra may help in the study of Mertens function and…
This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
In this paper we explore a family of congruences over $\N^\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of…
Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.
In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for…
We establish a function field analogue of Mertens' formula for Euler products restricted to primes in arithmetic progressions over the polynomial ring F_q[t]. Our results are in direct correspondence with those of Languasco and Zaccagnini…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
The main goal of this paper is to present new bounds for certain inner products in Hilbert spaces, with applications to the numerical radius and the operator norm. The obtained results significantly improve earlier results in this…
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary…
We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We…
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…