#### 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.

Done