Satyavolu Srinivas Rau

Research Quest

Research endeavour is in two areas :
1) Analytic Number Theory 2) Geometric Physics.

1) To investigate behaviour of arithmetical functions related to the Divisor function d(n).The starting point is Ramanujan’s assertion that asymptotically SIGMA 1/d(n) is ( cx ^2) /(log x)^1/2
n < x

This was verified by B.Wilson, his editor. The proof is difficult to locate and we were able to derive the same applying Hubert Delange’s Tauberian theorem of 1954. See [ 6 } below in the list of publications.
w(n) = no of distinct prime divisors of n
W(n) = no of prime divisors of n counted with multiplicity

w(n) < W(n) < d(n)

2^w(n) < d(n) < 2^ W(n) ( with equality for squarefree integers)
Questions on asymptotic values:

Q1) What is the mean of 1/w(n) ?

Q2) What is the mean of 1/W(n) ?

(Of course by comparison with 1/d(n) these means are not finite, but
“mean behaviour” does not sound nice! )

Q3) What is the mean of w(n)/d(n) = the average number of prime divisors ?

Q4) Similar queries on w(n)/W(n), W(n)/d(n) .

Note that W(n) is the number of prime power divisors of n. So the above ratios have an arithmetic significance.
Aurel Wintner (1940) compared the oscillatory behaviour of w(n) to Brownian motion. G.H.Hardy noted in his Lecture on Round Numbers
[ 6} that d(n) oscillates much more than w(n) or W(n).

So (-1)^f(n) is of interest in measuring the mean oscillation of f(n). A surprising result of Hardy-Ramanujan is that W(n) – w(n) has a finite asymptotic mean nearly 0.73. We applied the results of Delange-Wirsing to obtain the mean of (-1)^(W(n)-w(n)) and of (-1)^w(n) and (-1)^W(n).The latter two are zero, a fact equivalent to the Prime Number Theorem . The mean of (-1)^d(n) is negative (!), for arithmetic reasons.

So there is greater interest in the behaviour of 2^f(n) and their ratios.
Q5) What are the means of w(n) /2^w(n) . W(n)/2^W(n), d(n)/2^W(n) ?
Q6) Let Sp(n) be the sum of prime divisors of n.What is the mean of Sp(n)/w(n) = average prime divisor of n?
Q6 is inspired by the asymptotic result of Bateman-Erdos-Pomeranve-Straus on corresponding average divisor of n.
Numerical computations may help in some of the above questions.

2) As coinvestigator in the SERB Project” Nonlinear algebra and Lie algebroid approaches to Quantum Field Theory” with T.Shreecharan (ECR/2015/000081) I have worked on various Brackets .(arXiv:1607.00807.1 [math-phy} )
Several questions arise on computation and possible isomorphism of Lie algebroid and Leibniz algebroid cohomologies .A careful study of Hochschild-Kostant-Rosenberg techniques appears useful.

Academic Bio

Satyavolu Srinivas Rau has taught at Chanakya University’s School of Mathematics and Natural Science since October, 2023. He is interested in
Tauberian theory applied to asymptotics of Arithmetic functions and also in mathematical structures of Quantum Field Theory.

He has more than 25 years of teaching experience at the Undergraduate and Postgraduate levels including varied batches in Eritrea and Libya.They cover degree programs in Mathematics,Technology,Social Sciences and Education (!)

He was associated with ICFAI,Hyderabad for more than 15 years as faculty member. He worked for 9 years at North Maharashtra University ,Jalgaon (1994-2003).Prior to that he did post doctoral work for nearly 5 years at ICTP, ISI,Bangalore and as CSIR Rssearch Associate.

His PhD and MPhil degrees are from the University of Hyderabad.He is an alumnus of BITS,Pilani with MSc(Hons) in Mathematics (1981)
Currently Srinivas Rau handles mathematics and statistics courses for BCA, MCA and MSc(Data Science) programs.

Educational Qualification