[Paper Review] Majoration du nombre de zéros d'une fonction méromorphe en dehors d'une droite verticale et applications

Oswaldo Velásquez Castañón|ArXiv.org|Dec 8, 2007

Analytic Number Theory Research12 references4 citations

TL;DR

This paper establishes sufficient conditions under which all but finitely many zeros of functions of the form $ f(s) = h(s) \pm h(2a - s) $ lie on the critical line $ \Re s = a $, provided all but finitely many zeros of $ h(s) $ lie in $ \Re s < a $. The result generalizes the Hermite-Biehler theorem and provides effective bounds on the number of exceptional zeros outside the critical line, with applications to shifted Riemann zeta functions and $ L $-functions.

ABSTRACT

We study the distribution of the zeros of functions of the form $f(s)=h(s) \pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all but finitely many of the zeros of $f(s)$ lie on the line $\Re s = a$, called the {\it critical line} for the function $f(s)$, and be simple, given that all but finitely many of the zeros of $h(s)$ lie on the half-plane $\Re s &lt; a$. This results can be regarded as a generalization of the necessary condition of stability for the function $h(s)$, in the Hermite-Biehler theorem. We apply this results to the study of translations of the Riemann Zeta Function and $L$ functions, and integrals of Eisenstein Series, among others.

Motivation & Objective

• To study the distribution of zeros of functions $ f(s) = h(s) \pm h(2a - s) $, where $ h(s) $ is meromorphic and real on the real line.
• To relax the classical Hermite-Biehler stability condition (no zeros in $ \Re s \geq a $) by allowing finitely many such zeros.
• To derive effective upper bounds on the number of zeros of $ f(s) $ lying outside the critical line $ \Re s = a $.
• To apply the results to shifted Riemann zeta functions, $ L $-functions, and integrals of Eisenstein series.
• To establish conditions under which the non-critical zeros are simple and to quantify their number in vertical strips.

Proposed method

• Use the argument principle and contour integration to count zeros of $ f(s) $ in the strip $ 0 < \Im s < T $, denoted $ N(T) $.
• Compare $ N(T) $ with $ N_0(T) $, the number of zeros on the critical line $ \Re s = a $, to estimate the number of off-line zeros.
• Apply the functional equation $ f(2a - s) = \pm f(s) $ to exploit symmetry and relate zeros in $ \Re s > a $ to those in $ \Re s < a $.
• Use the assumption that $ f(s) \neq 0 $ for $ \Re s \geq \sigma_0 > a $ to confine all zeros to a vertical strip around $ \Re s = a $.
• Leverage known zero-free regions of $ h(s) $ and the structure of $ f(s) $ to derive effective bounds on the exceptional zero count.
• Utilize the theory of entire and meromorphic functions, including the use of the logarithmic derivative and Jensen's formula.

Experimental results

Research questions

• RQ1 Under what conditions does the function $ f(s) = h(s) \pm h(2a - s) $ have all but finitely many of its zeros on the critical line $ \Re s = a $?
• RQ2 How can the classical Hermite-Biehler condition of zero-free half-plane $ \Re s \geq a $ be relaxed while still ensuring most zeros lie on the critical line?
• RQ3 What effective upper bound can be established for the number of zeros of $ f(s) $ lying outside the critical line $ \Re s = a $?
• RQ4 Can the simplicity of the non-critical zeros of $ f(s) $ be guaranteed under the same conditions?
• RQ5 How do these results apply to the shifted Riemann zeta function and other $ L $-functions or Eisenstein series integrals?

Key findings

• All but finitely many zeros of $ f(s) = h(s) \pm h(2a - s) $ lie on the critical line $ \Re s = a $, provided all but finitely many zeros of $ h(s) $ lie in $ \Re s < a $, and $ f(s) \neq 0 $ for $ \Re s \geq \sigma_0 > a $.
• The number of zeros of $ f(s) $ outside the critical line is uniformly bounded in terms of the number of zeros of $ h(s) $ in $ \Re s \geq a $, with an effective upper bound derived via contour integration.
• The zeros of $ f(s) $ on the critical line are simple under the stated conditions, provided $ h(s) $ satisfies mild regularity and growth conditions.
• The method yields explicit estimates for $ N(T) - N_0(T) $, the number of non-critical zeros in $ 0 < \Im s < T $, in terms of the growth and zero distribution of $ h(s) $.
• The results are applied to show that certain shifted zeta functions and integrals of Eisenstein series have all but finitely many zeros on the critical line.
• The framework generalizes the Hermite-Biehler theorem by allowing a finite number of zeros of $ h(s) $ in $ \Re s \geq a $, while preserving the critical line alignment of $ f(s) $'s zeros.