MathOverflow

Suppose we have a deterministic sequence$$R_n(s) = S_n(s) - C_n(s)$$where $S_n(s) = \sum_{k=1}^n k^{-s}$ is a discrete Dirichlet sum and $C_n(s) = \frac{n^{1-s}}{1-s}$ is its continuous analytic counterpart.
This residue $R_n(s)$ represents the structural discrepancy between the discrete summation and the continuous flow of its integral approximation. At a zero-state $\zeta(s)=0$, this discrepancy must be perfectly resolved such that the distance vanishes...

In Higher Algebra, Remark 1.2.2.3, Lurie notes that an object$F \in \operatorname{Gap}(\mathbb{Z},\mathcal{C})$ determines a chain complex in the homotopy category$\mathrm{h}\mathcal{C}$, with terms$$C_n \simeq F(n-1,n)[-n].$$Since the construction of...

Question: Is the following statement known, provable under standard conjectures, or false?
For every integer $n ≥ 2$, there exist primes $p < n^2$ and $q > n^2$ such that $n^2 = \frac{p+q}{2}$, i.e., $p + q = 2n^2$.
Verification: Verified computationally up to $n=20$ (e.g., $n=2$: $p=3,q=5$; $n=5$: $p=19,q=31$; $n=10$: $p=97,q=103$). Pairs always exist within small $k$ where $p=n^2-k$, $q=n^2+k$....

Let $\mu$ be a probability measure on $ \mathbb{R}$ with $\mu(\{ 0\})<1$. Otherwise, no further assumptions (e.g. on the existence of moments) are imposed on $\mu$. Let $X_n$ and $Y_n$ denote two independent random walks driven by $\mu$. I am interested...