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There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then

the bound

$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{1/2} + (t/a)^{1/2} \tag{1}$$

follows from

$$\sum_{a <n <b} n^{it} \ll t^{1/2} + at^{-1/2} \tag{2}$$

''by partial summation.'' Titchmarsh doesn't include any details of the proof, but from the partial (Abel) summation formula:

https://en.wikipedia.org/wiki/Abel%27s_summation_formula

it's not clear to me how (2) follows from (1). Can someone kindly fill in the details omitted by Titchmarsh?

There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then

the bound

$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{1/2} + (t/a)^{1/2} \tag{1}$$

follows from

$$\sum_{a <n <b} n^{-it} \ll t^{1/2} + at^{-1/2} \tag{2}$$

''by partial summation.'' Titchmarsh doesn't include any details of the proof, but from the partial (Abel) summation formula:

https://en.wikipedia.org/wiki/Abel%27s_summation_formula

it's not clear to me how (2) follows from (1). Can someone kindly fill in the details omitted by Titchmarsh?

There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then

the bound

$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{1/2} + (t/a)^{1/2} \tag{1}$$

follows from

$$\sum_{a <n <b} n^{it} \ll t^{1/2} + at^{-1/2} \tag{1}$$

''by partial summation.'' Titchmarsh doesn't include any details of the proof, but from the partial (Abel) summation formula

https://en.wikipedia.org/wiki/Abel%27s_summation_formula

it's not clear to me how (2) follows from (1). Can someone kindly fill in the details omitted by Titchmarsh?

There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then

the bound

$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{1/2} + (t/a)^{1/2} \tag{1}$$

follows from

$$\sum_{a <n <b} n^{it} \ll t^{1/2} + at^{-1/2} \tag{2}$$

''by partial summation.'' Titchmarsh doesn't include any details of the proof, but from the partial (Abel) summation formula:

https://en.wikipedia.org/wiki/Abel%27s_summation_formula

it's not clear to me how (2) follows from (1). Can someone kindly fill in the details omitted by Titchmarsh?