Twisted Hecke L-values and period polynomials

Let $f_1,...,f_d$ be an orthogonal basis for the space of cusp forms of even weight $2k$ on $\Gamma_0(N)$. Let $L(f_i,s)$ and $L(f_i,\chi,s)$ denote the $L$-function of $f_i$ and its twist by a Dirichlet character $\chi$, respectively. In this note, we obtain a ``trace formula'' for the values $L(f_i,\chi,m)\overline{L(f_i,n)}$ at integers $m$ and $n$ with $0<m,n<2k$ and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisly the value of the ratio $L(f,\chi,m)/L(f,n)$ for a Hecke eigenform $f$.Comment: 20 page

Relationship between serum calcium or phosphate levels and mortality stratified by parathyroid hormone level: an analysis from the MBD-5D study

Introduction: There is limited evidence about the association between calcium and phosphate levels and mortality stratified by intact parathyroid hormone (iPTH) level. Methods: We investigated whether differences in iPTH level affect the relationship between calcium and phosphate levels and all-cause mortality in hemodialysis patients with secondary hyperparathyroidism (SHPT). Calcium and phosphate levels were categorized as low (< 8.5 mg/dL,  < 4.0 mg/dL), medium (≥ 8.5–< 9.5 mg/dL,  ≥ 4.0–< 7.0 mg/dL), and high (≥ 9.5 mg/dL,  ≥ 7.0 mg/dL), respectively. iPTH levels were grouped into < 300 or ≥ 300 pg/mL. Adjusted incidence rate ratios (aIRRs) were analyzed by weighted Poisson regression. Results: For calcium, patients with higher iPTH (≥ 300 pg/mL) had significantly higher all-cause mortality rates in the high than in the medium category (aIRR 1.99, 95% confidence interval [CI] 1.16–3.42), and tended to have a higher mortality rate in the low category (aIRR 2.04, 95% CI 0.94–4.42). Patients with lower iPTH (< 300 pg/mL) had higher mortality rates in the high than in the medium category (aIRR 1.65, 95% CI 1.39–1.96). For phosphate, the mortality rate was significantly higher in the high than in the medium category in patients with higher and lower iPTH (aIRR 3.23, 95% CI 1.63–6.39 for iPTH ≥ 300 pg/mL; aIRR 1.58, 95% CI 1.06–2.36 for iPTH < 300 pg/mL). Conclusion: High calcium and phosphate levels were associated with increased risk of mortality irrespective of iPTH level

Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

First-line pembrolizumab vs chemotherapy in metastatic non-small-cell lung cancer: KEYNOTE-024 Japan subset

This prespecified subanalysis of the global, randomized controlled phase Ill KEYNOTE-024 study of pembrolizumab vs chemotherapy in previously untreated metastatic non-small-cell lung cancer without EGFR/ALK alterations and a programmed death-ligand 1 (PD-L1) tumor proportion score of 50% or greater evaluated clinical outcomes among patients enrolled in Japan. Treatment consisted of pembrolizumab 200 mg every 3 weeks (35 cycles) or platinum-based chemotherapy (four to six cycles). The primary end-point was progression-free survival; secondary end-points included overall survival and safety. Of 305 patients randomized in KEYNOTE-024 overall, 40 patients were enrolled in Japan (all received treatment: pembrolizumab, n = 21; chemotherapy, n = 19). The hazard ratio (HR) for progression-free survival by independent central review (data cut-off date, 10 July 2017) was 0.25 (95% confidence interval [CI], 0.10-0.64; one-sided, nominal P = .001). The HR for overall survival (data cut-off date, 15 February 2019) was 0.39 (95% CI, 0.17-0.91; one-sided, nominal P = .012). Treatment-related adverse events occurred in 21/21 (100%) pembrolizumab-treated and 18/19 (95%) chemotherapy-treated patients; eight patients (38%) and nine patients (47%), respectively, had grade 3-5 events. Immune-mediated adverse events and infusion reactions occurred in 11 patients (52%) and four patients (21%), respectively; four patients (19%) and one patient (5%), respectively, had grade 3-5 events. Consistent with results from KEYNOTE-024 overall, first-line pembrolizumab improved progression-free survival and overall survival vs chemotherapy with manageable safety among Japanese patients with metastatic non-small-cell lung cancer without EGFRIALK alterations and a PD-L1 tumor proportion score of 50% or greater