A Dirichlet character analogue of Ramanujan's formula for odd zeta values

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$\sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1},$$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for $\zeta(2m+1)$ while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely, $$\sum_{r=1}^{q}\sum_{n=1}^{\infty} \frac{\chi(r)n^{N-2h}{\exp\left(-\frac{r}{q}n^N x\right)}}{1-\exp({-n^N x})},$$ where $\chi$ denotes a Dirichlet character modulo $q$, $N\in 2\mathbb{N}$ and with some restriction on the variable $h$. In the current paper, we investigate the above series for {\it any} $N \in \mathbb{N}$ and $h \in \mathbb{Z}$. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for $\zeta(2m+1)$. Moreover, we establish a new identity for $L(1/3, \chi)$ analogous to Ramanujan's famous identity for $\zeta(1/2)$.Comment: 24 pages, comments are welcome

The impact of surgical delay on resectability of colorectal cancer: An international prospective cohort study

AIM: The SARS-CoV-2 pandemic has provided a unique opportunity to explore the impact of surgical delays on cancer resectability. This study aimed to compare resectability for colorectal cancer patients undergoing delayed versus non-delayed surgery. METHODS: This was an international prospective cohort study of consecutive colorectal cancer patients with a decision for curative surgery (January-April 2020). Surgical delay was defined as an operation taking place more than 4 weeks after treatment decision, in a patient who did not receive neoadjuvant therapy. A subgroup analysis explored the effects of delay in elective patients only. The impact of longer delays was explored in a sensitivity analysis. The primary outcome was complete resection, defined as curative resection with an R0 margin. RESULTS: Overall, 5453 patients from 304 hospitals in 47 countries were included, of whom 6.6% (358/5453) did not receive their planned operation. Of the 4304 operated patients without neoadjuvant therapy, 40.5% (1744/4304) were delayed beyond 4 weeks. Delayed patients were more likely to be older, men, more comorbid, have higher body mass index and have rectal cancer and early stage disease. Delayed patients had higher unadjusted rates of complete resection (93.7% vs. 91.9%, P = 0.032) and lower rates of emergency surgery (4.5% vs. 22.5%, P < 0.001). After adjustment, delay was not associated with a lower rate of complete resection (OR 1.18, 95% CI 0.90-1.55, P = 0.224), which was consistent in elective patients only (OR 0.94, 95% CI 0.69-1.27, P = 0.672). Longer delays were not associated with poorer outcomes. CONCLUSION: One in 15 colorectal cancer patients did not receive their planned operation during the first wave of COVID-19. Surgical delay did not appear to compromise resectability, raising the hypothesis that any reduction in long-term survival attributable to delays is likely to be due to micro-metastatic disease

Mortality from gastrointestinal congenital anomalies at 264 hospitals in 74 low-income, middle-income, and high-income countries: a multicentre, international, prospective cohort study

Summary Background Congenital anomalies are the fifth leading cause of mortality in children younger than 5 years globally. Many gastrointestinal congenital anomalies are fatal without timely access to neonatal surgical care, but few studies have been done on these conditions in low-income and middle-income countries (LMICs). We compared outcomes of the seven most common gastrointestinal congenital anomalies in low-income, middle-income, and high-income countries globally, and identified factors associated with mortality. Methods We did a multicentre, international prospective cohort study of patients younger than 16 years, presenting to hospital for the first time with oesophageal atresia, congenital diaphragmatic hernia, intestinal atresia, gastroschisis, exomphalos, anorectal malformation, and Hirschsprung’s disease. Recruitment was of consecutive patients for a minimum of 1 month between October, 2018, and April, 2019. We collected data on patient demographics, clinical status, interventions, and outcomes using the REDCap platform. Patients were followed up for 30 days after primary intervention, or 30 days after admission if they did not receive an intervention. The primary outcome was all-cause, in-hospital mortality for all conditions combined and each condition individually, stratified by country income status. We did a complete case analysis. Findings We included 3849 patients with 3975 study conditions (560 with oesophageal atresia, 448 with congenital diaphragmatic hernia, 681 with intestinal atresia, 453 with gastroschisis, 325 with exomphalos, 991 with anorectal malformation, and 517 with Hirschsprung’s disease) from 264 hospitals (89 in high-income countries, 166 in middleincome countries, and nine in low-income countries) in 74 countries. Of the 3849 patients, 2231 (58·0%) were male. Median gestational age at birth was 38 weeks (IQR 36–39) and median bodyweight at presentation was 2·8 kg (2·3–3·3). Mortality among all patients was 37 (39·8%) of 93 in low-income countries, 583 (20·4%) of 2860 in middle-income countries, and 50 (5·6%) of 896 in high-income countries (p<0·0001 between all country income groups). Gastroschisis had the greatest difference in mortality between country income strata (nine [90·0%] of ten in lowincome countries, 97 [31·9%] of 304 in middle-income countries, and two [1·4%] of 139 in high-income countries; p≤0·0001 between all country income groups). Factors significantly associated with higher mortality for all patients combined included country income status (low-income vs high-income countries, risk ratio 2·78 [95% CI 1·88–4·11], p<0·0001; middle-income vs high-income countries, 2·11 [1·59–2·79], p<0·0001), sepsis at presentation (1·20 [1·04–1·40], p=0·016), higher American Society of Anesthesiologists (ASA) score at primary intervention (ASA 4–5 vs ASA 1–2, 1·82 [1·40–2·35], p<0·0001; ASA 3 vs ASA 1–2, 1·58, [1·30–1·92], p<0·0001]), surgical safety checklist not used (1·39 [1·02–1·90], p=0·035), and ventilation or parenteral nutrition unavailable when needed (ventilation 1·96, [1·41–2·71], p=0·0001; parenteral nutrition 1·35, [1·05–1·74], p=0·018). Administration of parenteral nutrition (0·61, [0·47–0·79], p=0·0002) and use of a peripherally inserted central catheter (0·65 [0·50–0·86], p=0·0024) or percutaneous central line (0·69 [0·48–1·00], p=0·049) were associated with lower mortality. Interpretation Unacceptable differences in mortality exist for gastrointestinal congenital anomalies between lowincome, middle-income, and high-income countries. Improving access to quality neonatal surgical care in LMICs will be vital to achieve Sustainable Development Goal 3.2 of ending preventable deaths in neonates and children younger than 5 years by 2030

Preoperative nasopharyngeal swab testing and postoperative pulmonary complications in patients undergoing elective surgery during the SARS-CoV-2 pandemic.

BACKGROUND: Surgical services are preparing to scale up in areas affected by COVID-19. This study aimed to evaluate the association between preoperative SARS-CoV-2 testing and postoperative pulmonary complications in patients undergoing elective cancer surgery. METHODS: This international cohort study included adult patients undergoing elective surgery for cancer in areas affected by SARS-CoV-2 up to 19 April 2020. Patients suspected of SARS-CoV-2 infection before operation were excluded. The primary outcome measure was postoperative pulmonary complications at 30 days after surgery. Preoperative testing strategies were adjusted for confounding using mixed-effects models. RESULTS: Of 8784 patients (432 hospitals, 53 countries), 2303 patients (26.2 per cent) underwent preoperative testing: 1458 (16.6 per cent) had a swab test, 521 (5.9 per cent) CT only, and 324 (3.7 per cent) swab and CT. Pulmonary complications occurred in 3.9 per cent, whereas SARS-CoV-2 infection was confirmed in 2.6 per cent. After risk adjustment, having at least one negative preoperative nasopharyngeal swab test (adjusted odds ratio 0.68, 95 per cent confidence interval 0.68 to 0.98; P = 0.040) was associated with a lower rate of pulmonary complications. Swab testing was beneficial before major surgery and in areas with a high 14-day SARS-CoV-2 case notification rate, but not before minor surgery or in low-risk areas. To prevent one pulmonary complication, the number needed to swab test before major or minor surgery was 18 and 48 respectively in high-risk areas, and 73 and 387 in low-risk areas. CONCLUSION: Preoperative nasopharyngeal swab testing was beneficial before major surgery and in high SARS-CoV-2 risk areas. There was no proven benefit of swab testing before minor surgery in low-risk areas