Polynomial growth of L-function
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Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|lambda(n)|leq sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in vertical direction in any strip, especially the critical strip $0leq Re(s)leq 1$? My idea was to use the Mellin transform and get
$$|L(f,s)|=int_0^infty |L(f,t)|t^{s-1} ,dtleq int_0^infty bigg|sum_{ngeq1}sigma_0(n)n^{-t}bigg|t^{Re(s)-1} ,dt.$$
complex-analysis modular-forms mellin-transform
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
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add a comment |
$begingroup$
Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|lambda(n)|leq sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in vertical direction in any strip, especially the critical strip $0leq Re(s)leq 1$? My idea was to use the Mellin transform and get
$$|L(f,s)|=int_0^infty |L(f,t)|t^{s-1} ,dtleq int_0^infty bigg|sum_{ngeq1}sigma_0(n)n^{-t}bigg|t^{Re(s)-1} ,dt.$$
complex-analysis modular-forms mellin-transform
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
$endgroup$
add a comment |
$begingroup$
Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|lambda(n)|leq sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in vertical direction in any strip, especially the critical strip $0leq Re(s)leq 1$? My idea was to use the Mellin transform and get
$$|L(f,s)|=int_0^infty |L(f,t)|t^{s-1} ,dtleq int_0^infty bigg|sum_{ngeq1}sigma_0(n)n^{-t}bigg|t^{Re(s)-1} ,dt.$$
complex-analysis modular-forms mellin-transform
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
$endgroup$
Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|lambda(n)|leq sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in vertical direction in any strip, especially the critical strip $0leq Re(s)leq 1$? My idea was to use the Mellin transform and get
$$|L(f,s)|=int_0^infty |L(f,t)|t^{s-1} ,dtleq int_0^infty bigg|sum_{ngeq1}sigma_0(n)n^{-t}bigg|t^{Re(s)-1} ,dt.$$
complex-analysis modular-forms mellin-transform
complex-analysis modular-forms mellin-transform
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
asked Nov 28 '18 at 12:40
Nodt GreenishNodt Greenish
29813
29813
add a comment |
add a comment |
1 Answer
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The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.
Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
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1 Answer
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1 Answer
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active
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$begingroup$
The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.
Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
add a comment |
$begingroup$
The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.
Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
add a comment |
$begingroup$
The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.
Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.
Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
answered Nov 28 '18 at 13:08
davidlowryduda♦davidlowryduda
74.4k7118252
74.4k7118252
add a comment |
add a comment |
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