Room 326

Linear independence of Briggs-Euler-Lehmer constants over number fields

Ekata Saha

IMSc

In 1731, Euler introduced the constant
$\gamma:=\lim_{x\to \infty}(\sum_{n\le x}\frac{1}{n} - \log x}$
while studying harmonic sums. This constant is known as Euler's
constant. He computed $\gamma$ upto five decimal places and wrote
it in terms of a conditionally convergent series involving zeta
values. Till today the arithmetic nature of this constant is not
known. Lehmer and Briggs independently studied this constant for
arithmetic progressions to unravel the mystery around it. In 2008,
Diamond and Ford generalised the constant over the finite sets of
primes and linked it to the Riemann hypothesis. In 2014, in a work
with S. Gun and S.B. Sinha, we considered the generalised Euler's
constant in arithmetic progressions and showed that in the infinite
list at most one number can be algebraic. In this talk we explore a
recent work with S. Gun and V.K. Murty where we study the nature of
these constants in more detail and establish linear independence
results about them.