[Paper Review] On the multiplicity of zeros of the zeta-function

Aleksandar Ivić|ArXiv.org|Jan 25, 2005

Analytic Number Theory Research4 references7 citations

TL;DR

This paper establishes new upper bounds for the multiplicity of zeros of the Riemann zeta-function $ζ(s)$, particularly showing that zeros with large multiplicities must lie to the left of the critical line $σ=1$. Using Jensen's formula and integral representations involving the gamma function and zeta, it derives bounds that improve upon classical estimates, especially under the Lindel"of and Riemann Hypotheses, and introduces a zero-density counting function incorporating multiplicities.

ABSTRACT

Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $ζ(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $σ= 1$. A zero-density counting function involving multiplicities is also discussed.

Motivation & Objective

• To derive improved unconditional and conditional upper bounds for the multiplicity $m(\beta+i\gamma)$ of nontrivial zeros of the Riemann zeta-function.
• To investigate the location of high-multiplicity zeros, showing they must lie to the left of $\sigma=1$.
• To introduce and analyze a zero-density counting function $N^{(r)}(\sigma,T)$ that accounts for multiplicities.
• To explore the implications of these bounds under the Lindel"of Hypothesis (LH) and the Riemann Hypothesis (RH), particularly regarding the distribution of multiple zeros.
• To compare the new bounds with existing results, including the classical $N(T)$ formula and moment estimates for $S(T)$.

Proposed method

• Applies Jensen's formula to the zeta-function on a disk centered at $1+i\gamma$, using the regularity of $\zeta(s)$ and known lower bounds for $|\zeta(\sigma+i\gamma+it)|$.
• Uses the integral representation of the function $f_r(x)$ defined via the gamma function and exponential decay, which models the behavior of multiple zeros.
• Employs the Dirichlet series $M_X(s) = \sum_{n\leq X} \mu(n)n^{-s}$ and its coefficients $a(n)$ to relate the multiplicity to sums over arithmetic functions.
• Derives bounds via contour integration and the residue theorem, particularly using the rectangle $\mathcal{D}$ in the complex plane to isolate the zero of multiplicity $r$.
• Applies Stirling's formula and estimates for $\Gamma(s)$ and $\zeta(s)$ to control the size of the integrals and extract multiplicity bounds.
• Uses iterative integration by parts on the function $f_r(x)$ to derive asymptotic estimates and establish monotonicity and decay properties.

Experimental results

Research questions

• RQ1 What are the best known upper bounds for the multiplicity $m(\beta+i\gamma)$ of a zero of $\zeta(s)$, and how do they depend on $\gamma$?
• RQ2 Under what conditions can the multiplicity of a zero be bounded more tightly, especially under the Lindel"of or Riemann Hypotheses?
• RQ3 Can the location of high-multiplicity zeros be restricted, and does it force them to lie off the critical line?
• RQ4 How does the zero-density function $N^{(r)}(\sigma,T)$, counting zeros of multiplicity $r$, behave asymptotically as $T \to \infty$?
• RQ5 Can the methods used for $N^{(r)}(\sigma,T)$ yield sharper results than existing moment-based bounds like $N_r(T) \ll N(T)e^{-C\sqrt{r}}$?

Key findings

• The paper establishes the bound $m(\beta+i\gamma) \leq \frac{1}{\log \frac{1}{2-2\beta}} \left( \max_{\sigma \geq \frac{1}{2}, |t| \leq \frac{1}{2}} \log|\zeta(\sigma + i\gamma + it)| + O(\log\log\gamma) \right)$ for $\frac{1}{2} < \beta < 1$, which improves upon trivial estimates.
• Under the Lindel"of Hypothesis, the bound improves to $m(\beta+i\gamma) = o(\log\gamma)$ as $\gamma \to \infty$, indicating that multiple zeros are rare under this assumption.
• Under the Riemann Hypothesis, the bound becomes $m(\beta+i\gamma) \ll \frac{1}{\log\log\gamma} \log\gamma$, showing that even under RH, multiple zeros are sparse.
• The paper introduces a zero-density function $N^{(r)}(\sigma,T)$ that counts zeros of multiplicity $r$, and shows that $\lim_{T\to\infty} \frac{N^{(r)}(\sigma,T)}{N(\sigma,T)} = 0$ for $r \geq 2$, suggesting that multiple zeros are negligible in density.
• The method yields a bound that is smaller than classical estimates by a factor of $\log^A T$ for fixed $r \geq 2$, indicating a significant improvement in the density estimate for multiple zeros.
• The conjecture $m(\beta+i\gamma) \ll_\varepsilon (\log\log\gamma)^{1+\varepsilon}$ is discussed as a plausible but unproven improvement, and the paper shows that current methods cannot reach such a bound via pointwise $S(T)$ estimates.