On the sign changes of coefficients of general Dirichlet series

2011

In this paper we study the mean values and zeroes of Dirichlet series of a view $\sum_{n}a_n n^{-s}$ with complex coefficients. There was introduced some class of Dirichlet series including such widely used series as the Riemann zeta-function, Dirichlet L-functions and ets. A new point of view is introduced in defining of a half plane of mean values. It was proven that in the half plane of mean values any natural degree of the series of an inroduced class, being regular in this half plane,has a mean value. In particular, the analog of Lindelöf Hypothesis is true. If, in addition, the Dirichlet series f(s) belongs to this class with the function f(s)^{-1} then the half plane of mean values was proved to be free from the zeroes.

Journal of Number Theory, 2014

For any periodic function f : N → C with period q, we study the Dirichlet series L(s, f) := n≥1 f (n)/n s. It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1, where it has a simple pole with residue ρ := q −1 1≤a≤q f (a). Thus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L(1, f) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L(1, f).

Ukrainian Mathematical Journal, 2005

Mathematics of Computation, 2008

The Liouville function λ(n) is the completely multiplicative function whose value is −1 at each prime. We develop some algorithms for computing the sum T (n) = n k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T (n) < 0. This answers a question originating in some work of Turán, who linked the behavior of T (n) to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum L(n) = n k=1 λ(k), and we determine some new local extrema for this function, including some new positive values.

Number Theory Related to Modular Curves, 2018

We address the question of non-vanishing of L(1, f) where f is an algebraicvalued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions f for which L(1, f) = 0. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.

We present a brief and informal account on the so called Bohr's absolute convergence problem on Dirichlet series, from its statement and solution in the beginings of the 20th century to some of its recent variations.

arXiv (Cornell University), 2017

For the Dirichlet series of the form F (z, ω) = +∞ k=0 f k (ω)e zλ k (ω) (z ∈ C, ω ∈ Ω) with pairwise independent real exponents (λ k (ω)) on probability space (Ω, A, P) an estimates of abscissas convergence and absolutely convergence are established.

2016

This paper investigates the analytic properties of the Liouville function&#39;s Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a novel method of summing the series by casting it as an infinite number of sums over sub-series that exhibit a certain symmetry and rapid convergence. In this procedure, which heavily invokes the prime factorization theorem, each sub-series has the property that it oscillates in a predictable fashion, rendering the analytic properties of the Dirichlet series determinable. With this method, the paper demonstrates that, for every integer with an even number of primes in its factorization, there is another integer that has an odd number of primes (multiplicity counted) in its factorization. Furthermore, by showing that a sufficient condition derived by Littlewood (1912) is satisfied, the paper demonstrates that the function F(s) is analytic over t...

Complex Analysis and Operator Theory, 2008

We consider Dirichlet series ζg,α(s) = P ∞ n=1 g(nα)e −λns for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series P ∞ n=1 g(nα)z n . We prove that a Dirichlet series ζ(s) = P ∞ n=1 g(nα)/n s has an abscissa of convergence σ0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence is smaller or equal than 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ(s) has an analytic continuation to the entire complex plane.