Let $m\geq 2$ be a positive integer and consider the Kloosterman sums $$ \mathrm{Kl}(a,b,m)=\sum_{\substack{1\leq x\leq m\\ \gcd(x,m)=1}}\exp\left(\frac{2\pi i}{m}(ax+b\bar{x})\right), $$ where $\bar{x}$ is the inverse of $x$ modulo $m$. The following upper bound is well-known $$ \mathrm{Kl}(a,b,m)\leq \tau(m)\gcd(a,b,m)^{1/2}m^{1/2},\qquad\qquad\qquad (1) $$ where $\tau(m)=\sum_{d\mid m}d$ is the divisor function.

However it seems that Estermann proved slightly better upper bound (see Kaisa Matomaki's paper). When $32\nmid m$ we have $$ \mathrm{Kl}(a,b,m)\leq 2^{\omega(m)}\gcd(a,b,m)^{1/2}m^{1/2},\qquad \qquad\qquad(2)$$ where $\omega(m)$ is the number of distinct prime divisors of $m$. When $32\mid m$ the bound holds with an additional factor $\sqrt{2}$ on the right hand side. Clearly we have $2^{\omega(m)}\leq d(m)$ and so (2) is much better than (1). The surprising point for me is that even Estermann in his paper formulated (1) rather than (2). My questions now are the following

1. First I would like to know whether (2) is correct and if it is correct, I would like to have a reference stating (2) as a theorem.
2. If (2) is correct why in most litterateur authors use (1). Is there any technical reason why (1) is used most of the time.

Let $m\geq 2$ be a positive integer and consider the Kloosterman sums $$ \mathrm{Kl}(a,b,m)=\sum_{\substack{1\leq x\leq m\\ \gcd(x,m)=1}}\exp\left(\frac{2\pi i}{m}(ax+b\bar{x})\right), $$ where $\bar{x}$ is the inverse of $x$ modulo $m$. The following upper bound is well-known $$ \mathrm{Kl}(a,b,m)\leq \tau(m)\gcd(a,b,m)^{1/2}m^{1/2},\qquad\qquad\qquad (1) $$ where $\tau(m)=\sum_{d\mid m}d$ is the divisor function.

However it seems that Estermann proved slightly better upper bound (see Kaisa Matomaki's paper). When $32\nmid m$ we have $$ \mathrm{Kl}(a,b,m)\leq 2^{\omega(m)}\gcd(a,b,m)^{1/2}m^{1/2},\qquad \qquad\qquad(2)$$ where $\omega(m)$ is the number of distinct prime divisors of $m$. When $32\mid m$ the bound holds with an additional factor $\sqrt{2}$ on the right hand side. Clearly we have $2^{\omega(m)}\leq \tau(m)$ and so (2) is much better than (1). The surprising point for me is that even Estermann in his paper formulated (1) rather than (2). My questions now are the following

1. First I would like to know whether (2) is correct and if it is correct, I would like to have a reference stating (2) as a theorem.
2. If (2) is correct why in most litterateur authors use (1). Is there any technical reason why (1) is used most of the time.