Does there exist positive rational $s$ for which $\zeta(s)$ is a positive integer?

Question

Does there exist positive rational $s$ for which the Riemann Zeta function $\zeta(s) \in N$ or equivalently, does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\ell}{m}\right) = n$$

Update: 23-Mar-2023: New and faster code using hash map. Using this code and the method described in my answer below, I have been able to show that if there is a solution then $l > 1.7\times 10^5$.

def get_n_value(a,b):
    c_b = (b/a).n(prec = prec)
    return max(2, -1 + floor(0.07281584548367672486058637587/(eg - c_b)))

def get_c_value(m):
    i   = 1
    c_m = 0
    while (i <= itr):
        c_m = (m + c_m - zeta(1 + 1/(m - 1 + c_m))).n(prec = prec)
        i = i + 1
    return c_m

def get_next_b(c_n1,b):
    b_prev = b
    b = 1 + floor(c_n1*a)
    if b <= b_prev:
        b = b_prev + 1
    if gcd(a,b) > 1:
        b = b + 1 
    return b

a = 1 
step = 10^1
target = a + step - 1
sd = 0

prec = 1500
n_max = 0 
eg = (1 - euler_gamma).n(prec = prec)
itr = 50
# c_dict = {}
c_2 = get_c_value(2)
 

while True:
    c_n1 = c_2
    b = 1 + floor(c_2*a)
    b_max = floor(eg*a)
    depth = 0
    
    while(b <= b_max):
        if(gcd(b,a) == 1):
            found = False
            test = b/a.n(prec = prec)
            n = get_n_value(a,b)
            
            if n in c_dict:
                c_n = c_dict.get(n)
            else:
                c_n = get_c_value(n)
                c_dict[n] = c_n
                
            while found == False:
            
                if (n+1) in c_dict:
                    c_n1 = c_dict.get(n+1)
                else:
                    c_n1 = get_c_value(n+1)
                    c_dict[n+1] = c_n1

                if c_n < test and test < c_n1:
                    found = True
                    depth  = depth + 1

                    if (n > n_max):
                        n_max = n
                        # print("Maximum n is at:", a, b, n_max)
                        
                    # print('found',a,b,'witness =',n)
                    break
                else:
                    c_n = c_n1
                    n = n + 1
                    
            b = get_next_b(c_n1,b)
            
            if(b > b_max):
                break
        else:
            b = get_next_b(c_n1,b)
            if(b > b_max):
                break

    sd = sd + depth
    
    if a == target:
        l = len(c_dict)
        print(a,'dict', l,l/a.n(), 'dep', depth, 'sd',sd, sd/a.n())
        target = target + step

    a = a + 1

Answer 1

I misread $l$ as $1$, but in any case, as a partial result, here's a resolution for the case $l=1$.

Fix $s\in\mathbb{R}$, with $s > 1$.

On the interval $(0,\infty)$, let $f(x)={\small{{\displaystyle{\frac{1}{x^{\large{s}}}}}}}$.

It's easily verified that $ {\displaystyle{ \int_{1}^\infty \!f(x)\,dx = {\small{\frac{1}{s-1}}} }} $.

Consider the infinite series $ {\displaystyle{ \sum_{k=1}^\infty \frac{1}{k^s} }} $.

Since $f$ is positive, continuous, and strictly decreasing, we get \begin{align*} \int_{1}^\infty \!f(x)\,dx < \;&\sum_{k=1}^\infty \frac{1}{k^s} < 1+\int_{1}^\infty \!f(x)\,dx\\[4pt] \implies\;{\small{\frac{1}{s-1}}} < \;&\sum_{k=1}^\infty \frac{1}{k^s} < 1+{\small{\frac{1}{s-1}}}\\[4pt] \end{align*} If $m$ is a positive integer, then letting $s=1+{\large{\frac{1}{m}}}$, we have ${\large{\frac{1}{s-1}}}=m$, hence \begin{align*} {\small{\frac{1}{s-1}}} < \;&\sum_{k=1}^\infty \frac{1}{k^s} < \;1+{\small{\frac{1}{s-1}}}\\[4pt] \implies\;m < \;\,&\zeta\bigl(1+{\small{\frac{1}{m}}}\bigr) < \;m + 1\\[4pt] \end{align*} so $\zeta\bigl(1+{\large{\frac{1}{m}}}\bigr)$ is not an integer.

Answer 2

Can you resolve the problem for any other value of l, other than l=1? For example, can you resolve the case l=2?

Yes and in fact I can show that there is no solution for $l \le 3*10^4$. Here is the outline of my approach which I am posting as an answer since it is too long to be a comment.

Step 1: The first step was to derive the following result

For every real $x \ge 1$ there exists a positive real $c_x$ such that ${\displaystyle{ \zeta\Big(1+\frac{1}{x-1+c_x}\Big) = x. }}$

The first few terms of the asymptotic expansion of $c_x$ in terms of $n$ and the Stieltjes constants $\gamma_i$ are

$$ c_x = 1-\gamma_0 + \frac{\gamma_1}{x-1} + \frac{\gamma_2 + \gamma_1 - \gamma_0 \gamma_1}{(x-1)^2} + \frac{\gamma_2 +2\gamma_2 - 2\gamma_2 \gamma_0 + \gamma_1 - 2\gamma_1 \gamma_0 + \gamma_1 \gamma_0^2 - \gamma_1^2}{(x-1)^3} + O\Big(\frac{1}{x^4}\Big) $$

Step 2: I computed the first few values of $c_n$ but I did not use the above result. Instead I used the following recurrence formula.

Let $\alpha_0$ be any positive real and ${\displaystyle{ \alpha_{r+1} = n + \alpha_r - \zeta\Big(1+\frac{1}{n -1 + \alpha_r}\Big); }}$ then ${\displaystyle{ \lim_{r \to \infty}\alpha_r = c_n}}$.

Using this we obtained $$ c_2 \approx 0.3724062 $$ $$ c_3 \approx 0.3932265 $$ $$ \ldots $$ $$ c_{12} \approx 0.4164435 $$

Step 3: Show that $l \ge 5$

Let ${\displaystyle{ \zeta\Big(1+\frac{l}{m}\Big) \in N}}$ and let $m = lk+d$ where $\gcd(l,d) = 1$ and $1 \le d < l$.

Clearly, $c_2 \le c_n < 1-\gamma_0$ or $0.3724062 \le c_n < 0.422785$. Hence we must have ${\displaystyle{ 0.3724062 \le \frac{d}{l} < 0.422785}}$. The fraction with the smallest value of $l$ satisfying this condition is ${\displaystyle{\frac{2}{5} }}$ hence $l \ge 5$.

Extending this approach using numerical computations, I am able to show that there is no solution for $l < 3*10^4$.

Problems with this approach:

With this approach and with powerful computing, we can prove results like if ${\displaystyle{ \zeta\Big(1+\frac{l}{m}\Big) \in N }}$ then $l$ must be greater than some large positive integer but I don't see how this approach will solve the general problem.

Answer 3

Whilst not a solution, I thought it might be interesting to see What happens if we use the functional equation:

$$\zeta\left(1+\frac{m}{n}\right)=2^{\left(1+\frac{m}{n}\right)}\pi^{\frac{m}{n}}\sin\left(\frac{\pi}{2}\left(1+\frac{m}{n}\right)\right)\Gamma\left(-\frac{m}{n}\right)\zeta\left(-\frac{m}{n}\right).$$

$\zeta(s)$ is known to be rational at the negative integers,

$$\zeta(-a)=(-1)^a\frac{B_{a+1}}{a+1}.$$

You actually get $\zeta(-a)=0$ for $a$ even due to the trivial zeros.

But it appears that when you combine this with the functional equation, then in the limit, you get something non-zero and irrational, so that doesn't even give you an integer.

On the other hand if $m/n=2k-1$ is odd then we have

$$\zeta\left(2k\right)=\frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!},$$

by Euler's formula, which is "almost" rational except for the $\pi$ factor. Bugger. So no chance of being integral.

If you modify your problem slightly, then using the above you can produce positive integers, but that's no fun.