Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is used to to show, in Titchmarsh's "The theory of the Riemann zeta function" (4.25) that "A necessary and sufficient condition for the Riemann hypothesis is to have $M(x)=O(x^{1/2+\epsilon})$ [for any $\epsilon >0$]". Are there tools analogous to Perron's formula to do something of the sort, but involving lower bounds? In particular, I am studying functions similar to $M$ and $L$ for which very likely, a lower bound exists. I would be interesting in proving something like the above statement in Titchmarsh, but in which you would have, instead of the upper bound condition on $M$, some lower bound condition on some other function. Or, putting it in another way, is there a way to turn Perron's formula into a statement about lower bounds?
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3$\begingroup$ Not Perron's formula, but rather Landau's lemma; see Chapter 15 of Montgomery-Vaughan. I answered some related questions using these techniques: mathoverflow.net/a/266415/3803, mathoverflow.net/a/368511/3803 $\endgroup$– Peter HumphriesCommented Apr 10, 2023 at 16:04
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$\begingroup$ @Peter Humphries Thank you. What you are commenting in those posts is quite relevant to what I am trying to do. I have a function for which I can prove, using Landau's lemma, that if it is of constant sign, RH follows and not(LI) follows (a similar case to L). However, the function which I have in mind hardly goes up and down, let alone oscillate. You can almost prove that it is of constant sign by induction, but this doesn't quite work; and there are other some such approaches, each with its own problem. I conjecture that it's bounded below the $x$-axis by $-\sqrt{x}.$ (contd.) $\endgroup$– EGMECommented Apr 10, 2023 at 16:34
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$\begingroup$ @Peter Humphries It is this function I need to bound away from the $x$-axis. An $\Omega$ type of bound won't do, however, since you want both the lim sup and the lim inf to be on the same side of the $x$-axis. You want the whole function to be bounded away from the axis. $\endgroup$– EGMECommented Apr 10, 2023 at 16:35
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$\begingroup$ I'm not sure I quite understand what you mean. Is there an explicit expression in terms of sums of zeroes? Then sometimes you can get quite good bounds conditional on standard conjectures. See for example my answer here: mathoverflow.net/a/260540/3803 $\endgroup$– Peter HumphriesCommented Apr 10, 2023 at 16:45
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$\begingroup$ @Peter Humphries. Yes, there are such expression, although I had not thought of using them thus far. What you just sent might be what I was hoping to find out about. I need to study it. Thank you. $\endgroup$– EGMECommented Apr 10, 2023 at 16:58
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