The present study has been conducted to investigate the relative changes of carbonaceous aerosols (CA) over a high altitude Himalayan atmosphere with and without (very low) anthropogenic emissions. Measurements of atmospheric organic (OC) and elemental carbon (EC) were conducted during the lockdown period (April 2020) due to global COVID 19 outbreak and compared with the normal period (April 2019). The interesting, unexpected and surprising observation is that OC, EC and the total CA (TCA) during the lockdown (OC: 12.1 ± 5.5 μg m −3 ; EC: 2.2 ± 1.1 μg m −3 ; TCA: 21.5 ± 10 μg m −3 ) were higher than the normal period (OC: 7.04 ± 2.2 μg m −3 ; EC: 1.9 ± 0.7 μg m −3 ; TCA: 13.2 ± 4.1 μg m −3 ). The higher values for OC/EC ratio too was observed during the lockdown (5.7 ± 0.9) compared to the normal period (4.2 ± 1.1). Much higher surface O 3 during the lockdown (due to very low NO) could better promote the formation of secondary OC (SOC) through the photochemical oxidation of biogenic volatile organic compounds (BVOCs) emitted from Himalayan coniferous forest cover. SOC during the lockdown (7.6 ± 3.5 μg m −3 ) was double of that in normal period (3.8 ± 1.4 μg m −3 ). Regression analysis between SOC and O 3 showed that with the same amount of increase in O 3 , the SOC formation increased to a larger extent when anthropogenic emissions were very low and biogenic emissions dominate (lockdown) compared to when anthropogenic emissions were high (normal). Concentration weighted trajectory (CWT) analysis showed that the anthropogenic activities over Nepal and forest fire over north-east India were the major long-distant sources of the CA over Darjeeling during the normal period. On the other hand, during lockdown, the major source regions of CA over Darjeeling were regional/local. The findings of the study indicate the immense importance of Himalayan biosphere as a major source of organic carbon.

No abstract

No abstract

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities," which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G. F. Voronoï, G. H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings.The aforementioned convergent series in Ramanujan's "identity" is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ( 1 2 ), and in 1925, S. Wigert obtained a generalization giving a formula for ζ( 1 k ) for any even integer k ≥ 2. We extend the work of Ramanujan and Wigert in this paper.The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J. R. Wilton and derive an analogue involving the sum of divisors function σs(n).The modified Bessel functions Ks(x) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σs(n) and modified Lommel functions. We also discuss certain obscure related work of N. S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

No abstract