Geometric Moments and Rational Lambert Values: Proofs and Further Results

The cyclotomic limit

We prove the cyclotomic limits of Zudilin [8] by summing over reciprocal intervals, following [7]. The summatory totient estimate handles each finite set of intervals. A bound on the remaining small indices permits passage to their infinite union.

Write ${\mathrm{\Sigma }}_n=\sum _{2\le l\le N}{\nu }_{n,l}\phi (l)$. The elementary summatory estimate is $\sum _{l\le y}\phi (l)=3{\pi }^{-2}{y}^2+O(y\log (2y))$. Fix one half-open interval $[u,v)$ of Lemma 2.1. The condition $\{n/l\}\in [u,v)$ holds exactly on the blocks

so the summatory estimate gives, for each fixed $k$, $n^{-2}\sum _{l\text{ in block }k}\phi (l)\to 3{\pi }^{-2}((k+u{)}^{-2}-(k+v{)}^{-2})$. For $L\ge 1$, all blocks with $k\ge L$ lie below $n/(L+u)$. For every $y\ge 0$,

For $y<1$ the sum is empty; otherwise it is at most $\lfloor y\rfloor (\lfloor y\rfloor +1)/2\le y^2$. Hence the normalised contribution of the omitted blocks is at most $(L+u{)}^{-2}$, uniformly in $n$. This is the bound needed to pass from finitely many blocks to all of them. Taking $n\to \mathrm{\infty }$ first and then $L\to \mathrm{\infty }$ justifies the infinite block sum, and summing the thirteen half-open intervals gives $n^{-2}{\mathrm{\Sigma }}_n\to 3J/{\pi }^2$. With $\sum _{l\le 15n}\phi (l)=3{\pi }^{-2}225n^2+O(n\log n)$, equation (8) yields

and hence $W_n/n^2\to C_1-C_0$. These are the rational-base degree limits built on Zudilin’s cyclotomic limits [8].

For the real size estimate, fix $x>1$. The Möbius product formula and reindexing the divisors give

The bound $-\log (1-x^{-d})\le x^{-d}/(1-x^{-1})$ shows that the series on the right is finite for each fixed $x>1$. Since ${\nu }_{n,l}\in \{0,1\}$, this proves the bound

Here $N=15n$, so the error is $O_x(n)=o(n^2)$. The bracket equals $W_n-K_n+M_n$ by (8). No uniform constant as $x\downarrow 1$ is asserted.