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I need the following computational results for proving something. Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$, i.e. $\gamma_0\sim 14.134...$.

1) what is the residue of $1/\zeta(s)$ at $1/2 + i\gamma_0$

2)* what is the best lower bound we can give on $|\zeta(1/4 + i\gamma_0/2 \pm i\theta)|$, considering $\theta$ as a variable, varying in a small range(we may consider $\theta\in [-2\pi, 2\pi]$). (Here |.| means the absolute value of a complex number.)

  • For (2), we assume Riemann Hypothesis.

My above two questions are on very specific values of $\zeta(s)$, but I would be thankful if you can provide any general reference and results on these questions.

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    $\begingroup$ For the residue I get $1.245099646452575526742084314268271-0.1982183139843996762745694064662671i$. $\endgroup$
    – Kirill
    Commented Dec 3, 2014 at 6:03
  • $\begingroup$ @Kirill can you please mention, how you calculated it. I find it difficult to do. $\endgroup$
    – Kamalakshya
    Commented Dec 3, 2014 at 7:47
  • $\begingroup$ Using sage and mpmath: quad(lambda t: 0.1 * exp(2*pi*j*t)/zeta(zetazero(1) + 0.1 * exp(2*pi*j*t)), [0,1], verbose=True) $\endgroup$
    – Kirill
    Commented Dec 3, 2014 at 21:22
  • $\begingroup$ For 2), since $\theta$ is a variable, aren't you just asking for the minimum of $|\zeta(s)|$ on the vertical line $\text{Re}(s)=1/4$? Isn't the value of what you call $\gamma_0$ (everyone else calls $\gamma_1$) irrelevant? $\endgroup$
    – Stopple
    Commented Dec 10, 2014 at 21:11
  • $\begingroup$ @Stopple It seems $\gamma_0$(or $\gamma_1$) is irrelevant in 2) unless I provide the bound for $\theta$. It was a mistake of mine. Thank you for pointing it out. I wanted to understand the value of $\zeta(s)$ near $1/4 + \gamma_0/2$. The value of \theta may be assumed to vary between $[0, \delta_0]$, when $\delta_0\leq 2\pi$ (The bound $2\pi$ is from the context I am working on). Insted of an absolute(independent of \theta) lower bound of $\zeta(s)$, is it possible to get a lower bound in terms of $\theta$? $\endgroup$
    – Kamalakshya
    Commented Dec 13, 2014 at 5:56

1 Answer 1

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Using mpmath (included in sage), it is possible to calculate the first part numerically. The residue of $1/\zeta(s)$ at $s=s_1$ is

quad(lambda t: 0.1 * exp(2*pi*j*t)/zeta(zetazero(1) + 0.1 * exp(2*pi*j*t)), [0,1])

$$ \Rightarrow 1.245099646452575526742084314268271301603643880256669121683500170887942217\\ - 0.198218313984399676274569406466267138704018723703324456572853010894602099i. $$

The second part is not so tractable because the optimization domain is infinite, but there is one minimum near the origin:

f = lambda x: abs(zeta(0.25 + 0.5*j*im(zetazero(1)) + j*x))**2
x = findroot(lambda x: diff(f, x), -4.89219)

$$ \Rightarrow x = -4.892186470173845287564557049161362131897213434898606653165015263 $$ $$ \Rightarrow f(x) = 0.2126539487357932480452048269992908227779811024986163781739610312 $$

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  • $\begingroup$ Thank you Kirill. But I don't understand the answer to second part. In your notation I am looking for value of f in neighbourhood of $0$. Is it that $f$ has a minimum near $-4.89$? up to what range are you considering the value of $x$, for $-4.89$ to be minimum? $\endgroup$
    – Kamalakshya
    Commented Dec 13, 2014 at 7:11
  • $\begingroup$ @Kamalakshya No, $x=-4.89$ is the location of the local minimum, $|\zeta(\cdots)|^2 = 0.213$ is the value. $\endgroup$
    – Kirill
    Commented Dec 13, 2014 at 15:19

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