N. Kalinin

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math.NT2026

Evaluating lattice sums via telescoping on $S L_{+} (2, Z)$ : a short proof of $\sum \frac{1}{∣ x ∣ ^{2} ∣ y ∣ ^{2} ∣ x + y ∣ ^{2}} = \frac{I ¨}{} 4$ and Zagier's identity

Nikita Kalinin

We study lattice sums $\sum \frac{1}{( ∥ x ∥∥ y ∥∥ x + y ∥ ) ^{s}}$ taken over $S L_{+} (2, Z)$ , i.e.\ the set of pairs $(x, y)$ of primitive lattice vectors in $Z_{\geq 0}^{2}$ w…

math.NT2025

A half-shift reflection identity for the digamma function

Nikita Kalinin

We prove the identity \[ 2W_1(x) + \log 4 + Ï\left(\tfrac{1}{2} + x\right) + Ï\left(\tfrac{3}{2} - x\right) = 0, \] where $I ¨ $ is the digamma function and \[ W_1(x) = 2\int_0^\in…

Evaluating lattice sums via telescoping on SL+​(2,Z): a short proof of ∑∣x∣2∣y∣2∣x+y∣21​=I¨​4 and Zagier's identity

A half-shift reflection identity for the digamma function

Evaluating lattice sums via telescoping on $S L_{+} (2, Z)$ : a short proof of $\sum \frac{1}{∣ x ∣ ^{2} ∣ y ∣ ^{2} ∣ x + y ∣ ^{2}} = \frac{I ¨}{} 4$ and Zagier's identity