Selberg Class Research Articles

A method for proving the completeness of a list of zeros of certain L-functions

When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of Littlewood on the average of the argument of the Riemann zeta function. In a previous paper we suggested an alternative method based on the Weil-Barner explicit formula. This method asymptotically sacrifices fewer zeros in order to prove the completeness of a list of zeros with imaginary part in a given interval. In this paper, we prove a general version of this method for an extension of the Selberg class including Hecke and Artin L-series, L-functions of modular forms, and, at least in the unramified case, automorphic L-functions. As an example, we further specify this method for Hecke L-series and L-functions of elliptic curves over the rational numbers.

On the Speiser equivalent for the Riemann hypothesis

Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the Riemann zeta-function left of the critical line. We investigate the relationship between the non-trivial zeros of the functions belonging to the extended Selberg class and of their derivatives left of the critical line. Every element of this class satisfies a functional equation of the Riemann type, but it contains zeta-functions for which the Riemann hypothesis is not true. As an example, we study the relationship between the trajectories of zeros of a certain linear combination of Dirichlet $$L$$ -functions and of its derivative computationally. In addition, we examine Speiser type equivalent for Dirichlet $$L$$ -functions with imprimitive characters for which the Riemann hypothesis is not true and which do not satisfy the Riemann type functional equation.

An inequality of Hardy–Littlewood type for Dirichlet polynomials

The Lq norm of a Dirichlet polynomial F(s)=∑n=1Nann−s is defined as‖F‖q:=(limT→∞⁡1T∫0T|F(it)|qdt)1/q for 0<q<∞. It is shown that(∑n=1N|an|2|μ(n)|[d(n)]log⁡qlog⁡2−1)1/2≤‖F‖q when 0<q<2; here μ is the Möbius function and d the divisor function. This result is used to prove that the Lq norm of DN(s):=∑n=1Nn−1/2−s satisfies ‖DN‖q≫(log⁡N)q/4 for 0<q<∞. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality ‖DN‖q≪(log⁡N)q/4 is shown to be valid in the range 1<q<∞. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of L-functions.

On zeros of certain Dirichlet polynomials

In this article we establish the zero-free region of certain Dirichlet polynomials $L_{F, X}$ arising in approximate functional equation for functions in the Selberg class and we prove an asymptotic formula for the number of zeros of $L_{F, X}$.

On a class of functions that satisfies explicit formulae involving the Möbius function

A class of functions that satisfies intriguing explicit formulae of Ramanujan and Titchmarsh involving the zeros of an $$L$$ -function in the reduced Selberg class of degree one and its associated Mobius function is studied. Moreover, a sufficient and necessary condition for the truth of the Riemann hypothesis due to Riesz is generalized.

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.

Algebraic independence of logarithmic singularities of some complex functions

We show that algebraic independence of some complex functions of one variable over regular functions implies their algebraic independence over a larger ring, containing complex powers of regular functions. Based on this we obtain a generalization of a special case of the theorem of Kaczorowski and Perelli on functional independence of logarithms of functions in the Selberg class. As an application we state a new result on oscillations of arithmetical functions.

On uniqueness in the extended Selberg class of Dirichlet series

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros.

EXTREME VALUES OF L-FUNCTIONS FROM THE SELBERG CLASS

We prove estimates for extreme values of L-functions from the Selberg class under assumption of the corresponding analogue of the Riemann hypothesis. The method of proof combines Montgomery's approach with an effective version of Kronecker's diophantine approximation theorem due to Weber.

Results on value distribution of L‐functions

AbstractWe will establish a theorem concerning value distribution of L‐functions in the Selberg class, which shows how an L‐function and a meromorphic function are uniquely determined by their c‐values and which, as a consequence, proves a result on the uniqueness of the Riemann zeta function. The results in this paper also extend the corresponding ones in Li 6.

Value Distribution of L-Functions with Rational Moving Targets

We prove some value-distribution results for a class of L-functions with rational moving targets. The class contains Selberg class, as well as the Riemann-zeta function.

Joint value distribution for zeta functions in disjoint strips

In this paper we investigate joint functional distribution for a set of zeta functions which belong to the Selberg class. Especially we establish a similar property to the joint universality theorem for the zeta functions.

A remark on the uniqueness of the Dirichlet series with a Riemann-type function equation

We show that if for a nonzero complex number c the inverse images L1−1(c) and L2−1(c) of two functions in the extended Selberg class are the same, then L1(s) and L2(s) must be identical.

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).

On the value distribution of L-functions of the Selberg class

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.

Analytic properties of the Hasse–Weil L-function

We prove that the Hasse–Weil L -function associated to an elliptic curve E over is of growth order one and belongs to the Selberg class with degree two. Moreover, we show that has an infinite product representation associated with the conductor .

The Saddle-Point Method and the Li Coefficients

AbstractIn this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund–Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any functionFin the Selberg classand under the Generalized Riemann Hypothesis, we havewithwhere γ is the Euler's constant and the notation is as below.

ON INTERPOLATION FUNCTIONS FOR GENERALIZED Li COEFFICIENTS IN THE SELBERG CLASS

We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class [Formula: see text] that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GL n(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function [Formula: see text].

On the representation of H-invariants in the Selberg class

We prove that H-invariants and the conductor of the function F belonging to the Selberg class can be represented as special values of (a meromorphic continuation of) a ”superzeta” function arising from non-trivial zeros of F .

On the zeros of degree one L-functions from the extended Selberg class

On the zeros of degree one L-functions from the extended Selberg class