Polynomial Euler Product Research Articles

Estimates for L-functions in the Critical Strip Under GRH with Effective Applications

Assuming the generalized Riemann hypothesis, we provide explicit upper bounds for moduli of log {mathcal {L}(s)} and mathcal {L}'(s)/mathcal {L}(s) in the neighbourhood of the 1-line when mathcal {L}(s) are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood’s well-known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application, we provide conditional and effective estimates for the Mertens function.

Joint extreme values of L-functions

We consider L-functions \(L_1,\ldots ,L_k\) from the Selberg class which have polynomial Euler product and satisfy Selberg’s orthonormality condition. We show that on every vertical line \(s=\sigma +it\) with \(\sigma \in (1/2,1)\), these L-functions simultaneously take large values of size \(\exp \left( c\frac{(\log t)^{1-\sigma }}{\log \log t}\right) \) inside a small neighborhood. Our method extends to \(\sigma =1\) unconditionally, and to \(\sigma =1/2\) on the generalized Riemann hypothesis. We also obtain similar joint omega results for arguments of the given L-functions.

On a generalization of the Euler totient function

In a recent paper, J. Kaczorowski defined the generalized Euler totient function $\varphi(n, F)$ corresponding to a polynomial Euler product $F$. Let $E(x, F)$ denote the error term in the asymptotic formula of the summatory function of $\varphi(n, F).$ J.~Kaczorowski proved that the mean square of $E(x, F)$ has an asymptotic formula if $F$ satisfies GRH. In this paper we shall prove a sharper asymptotic formula under GRH.

REMARKS ON THE MIXED JOINT UNIVERSALITY FOR A CLASS OF ZETA FUNCTIONS

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

Zeros of combinations of Euler products for $$\sigma > 1$$ σ > 1

In this paper we consider Dirichlet series absolutely converging for $\sigma>1$ with an Euler product, natural bounds on the coefficients and satisfying orthogonality relations of Selberg type. Let $N\geq 1$, $F_1(s),...,F_N(s)$ be as above and $P(X_1,...,X_N)$ be a non-monomial polynomial with coefficients in the ring of $p$-finite Dirichlet series absolutely converging for $\sigma\geq 1$; then $P(F_1(s),\ldots,F_N(s))$ has infinitely many zeros for $\sigma>1$. Our result in particular applies to Artin $L$-functions, automorphic $L$-functions under the Ramanujan conjecture, and the elements of the Selberg class with polynomial Euler product under the Selberg orthonormality conjecture. This extends the work of Booker and Thorne, who proved the same result for automorphic $L$-functions under the Ramanujan conjecture. Our proof avoids to use the properties of twists by Dirichlet characters, a key point in Booker and Thorne's proof, replacing them by results on the Dirichlet density of non-zero coefficients of $L$-functions of the above type.

On the distribution of the a-points of a Selberg class L-function modulo one

Given a function F(s) from the Selberg class and an arbitrary fixed complex number a, we study the roots of the equation F(s) = a. We show that if F has a polynomial Euler product and satisfies the analogue of the Lindelof hypothesis, then the ordinates of these roots are uniformly distributed modulo one.

Multidimensional Shintani Zeta Functions and Zeta Distributions on $\mathbb{R}^d$

The class of Riemann zeta distribution is one of the classical classes of probability distributions on $\mathbb{R}$. Multidimensional Shintani zeta function is introduced and its definable probability distributions on $\mathbb{R}^d$ are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on $\mathbb{R}^d$, is also studied.

Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.

On a generalization of the Euler totient function

For a general polynomial Euler product F(s) we define the associated Euler totient function φ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of φ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).

Universality for L-functions in the Selberg class

We prove universality for L-functions \( \mathcal{L} \) from the Selberg class satisfying some mild condition on the Dirichlet coefficients (which might be considered as a prime number theorem for \( \mathcal{L} \)). This generalizes a previous universality theorem by the second author, where the L-function was assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis.

A note on the degree conjecture for the Selberg class

The degree conjecture for the Selberg class Open image in new window of L-functions states that the degree dF of every F ∈ Open image in new window is an integer. Moreover, it is expected that every F ∈ Open image in new window has polynomial Euler product, and that the degree ∂F of such an Euler product coincides with dF. In this note we prove that a suitable continuity assumption on the degree dF implies that ∂F = dF for all F ∈ Open image in new window with polynomial Euler product.

Joint value distribution of the matsumoto zeta-functions

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In this article, a joint limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-functions is proved.

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.

Strong multiplicity one for the Selberg class

Let S denote the Selberg class of L-functions. We prove the strong multiplicity one property for the subclass of functions F∈ S with polynomial Euler product.