[Paper Review] On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

А. П. Веселов, John Ward|arXiv (Cornell University)|May 16, 2002

Functional Equations Stability Results4 citations

TL;DR

This paper investigates the real zeroes of the Hurwitz zeta-function ζ(σ,a) for large negative σ, showing that they asymptotically align with lines σ + 4a + 2m = 0 in the region 0 < a < -σ/(2πe). It establishes that the number of real zeroes of ζ(-p,a) grows asymptotically as N(p)/p → 1/(πe), providing a simple proof of Inkeri’s result on the asymptotic number of real roots of Bernoulli polynomials B_n(x), which is ~2n/(πe).

ABSTRACT

The behaviour of real zeroes of the Hurwitz zeta function $$ζ(s,a)=\sum_{r=0}^{\infty}(a+r)^{-s}\qquad\qquad a &gt; 0$$ is investigated. It is shown that $ζ(s,a)$ has no real zeroes $(s=σ,a)$ in the region $a &gt;\frac{-σ}{2πe}+\frac{1}{4πe}\log (-σ) +1$ for large negative $σ$. In the region $0 &lt; a &lt; \frac{-σ}{2πe}$ the zeroes are asymptotically located at the lines $σ+ 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $ζ(-p,a)$ with given $p$ then $$\lim_{p o\infty}\frac{N(p)}{p}=\frac{1}{πe}.$$ As a corollary we have a simple proof of Inkeri's result that the number of real roots of the classical Bernoulli polynomials $B_n(x)$ for large $n$ is asymptotically equal to $\frac{2n}{πe}$.

Motivation & Objective

• To investigate the distribution of real zeroes of the Hurwitz zeta-function ζ(σ,a) for large negative real values of σ.
• To extend the validity of Hurwitz's Fourier representation of ζ(σ,a) beyond the standard interval 0 < a ≤ 1 to a wider range.
• To derive asymptotic estimates for the number of real zeroes of ζ(-p,a) as p → ∞.
• To provide a new, elementary proof of Inkeri’s result on the asymptotic number of real roots of Bernoulli polynomials B_n(x).

Proposed method

• Utilizes Hurwitz’s integral representation of ζ(s,a) for Re(s) < 0 and 0 < a ≤ 1, extending its applicability to 0 < a < -σ/(2πe) for large negative σ.
• Applies the asymptotic expansion ζ(σ,a)/Q(σ) = sin(2πa + ½πσ) + o(1) as σ → -∞, where Q(σ) = 2Γ(1−σ)/(2π)^{1−σ}.
• Employs the functional relation ζ(s,a) = ζ(s,n+a) + ∑_{r=0}^{n-1} (r+a)^{-s} to analyze analytic continuation and zero distribution.
• Uses the derivative identity ∂/∂a ζ(s,a) = -s ζ(s+1,a) to relate the k-th derivative of ζ(-p,a) to ζ(-p+k,a), enabling asymptotic analysis of root behavior.
• Applies a lemma on the maximum number of roots of a function with constant-sign n-th derivative to bound the number of real zeroes in specific intervals.
• Combines asymptotic analysis with bounds on the interval of validity to estimate the number of real zeroes in regions where ζ(σ,a) is non-zero or oscillatory.

Experimental results

Research questions

• RQ1 Where are the real zeroes of ζ(σ,a) located for large negative σ and 0 < a < -σ/(2πe)?
• RQ2 What is the asymptotic growth rate of the number of real zeroes of ζ(-p,a) as p → ∞?
• RQ3 Can the asymptotic distribution of real roots of Bernoulli polynomials B_n(x) be re-derived using the Hurwitz zeta-function?
• RQ4 In what region does ζ(σ,a) have no real zeroes for large negative σ?
• RQ5 How do the derivatives of ζ(-p,a) behave asymptotically, and how does this affect root distribution?

Key findings

• For large negative σ, ζ(σ,a) has no real zeroes when a > -σ/(2πe) + (1/(4πe)) log(-σ) + 1.
• In the region 0 < a < -σ/(2πe), the real zeroes of ζ(σ,a) are asymptotically located on the lines σ + 4a + 2m = 0 for integer m.
• The number of real zeroes N(p) of ζ(-p,a) satisfies lim_{p→∞} N(p)/p = 1/(πe).
• The number of real roots of the Bernoulli polynomial B_n(x) satisfies lim_{n→∞} N(n)/n = 2/(πe), confirming Inkeri’s result.
• For large m, the largest real root A(m) of B_m(x) satisfies A(m)/m → 1/(2πe), and the total number of real roots N(m) satisfies N(m)/m → 2/(πe).
• The upper and lower bounds for A(m) and N(m) include logarithmic corrections, though these are not derived from the current elementary method.