On the cubic $L$-function

Abstract

The cubic L-function is related to the cubic Kubota-Patterson theta function via the Mellin transformation. The cubic L-function obeys a functional equation of the Riemann type (with two gamma factors), but admits no expansion in an Euler product. In the paper, the cubic L-function is studied, and the distribution problem for the real parts of its zeros is considered. Some conjectures based on calculations are stated.

Throughput Maximization for Wireless Powered Buffer-Aided Successive Relaying Networks

“…On one hand, the convex-concave optimization algorithm [59] can be utilized. On the other hand, as P 2 ðtÞ is the only optimization variable, we can find out that the molecular term of the P 2 ðtÞ is a cubic function [60], and there are many existing methods 12 Wireless Communications and Mobile Computing to derive the stationary points, namely, the optimal transmit power P * 2 ðtÞ. In WIT-II mode with R 2 reception, the corresponding power allocation problem can be rewritten as…”

On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients

“…Another arithmetic example of a function from S that does not possess an Euler product is the cubic L-function L c (s), defined and studied by Proskurin in [15], who proved that the function L c (s) possesses both zeros off the critical strip 0 Re(s) 1 and a zero-free region.…”

“…, r, in axiom (iii). A subclass S R is chosen for two reasons: first, it arises naturally in many cases of number-theoretical interest and contains all arithmetic L-functions with real coefficients, including those with non-multiplicative coefficients in the Dirichlet series representation (such as cubic L-functions [15], Davenport-Heilbronn L-functions [5], Davenport-Heilbronn-type L-functions [2] etc); second, in this class, due to the reflection principle, zeros are symmetric with respect to the real line and hence it is possible to relate the series * ρ∈Z(F ) (1/(s − ρ)) to ξ F (s)/ξ F (s) for s / ∈ Z(F ). We investigate analytically and numerically the behavior of the coefficients λ F (n, τ ) for large sets of positive integers n and for different real values of τ .…”

On the zeros of the zeta function of the quadratic form $x^2+y^2+z^2$

“…One of such functions is ζ 3 . Some other functions were considered by the author earlier, see [1,2]. The distribution of zeros on the complex plane C and that of their real parts on R can be characterized by various methods.…”

“…Also, it is difficult to explain why similar rules are satisfied by the zeros of the function ζ 3 and those of the L-function associated with the cubic Kubota-Patterson theta function [1]. These functions admit no expansions like (1.3), (1.4).…”