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For any \(a\) coprime to \(m\),
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, then there exists a constant \(B\) such that
for all \(x \geq 1\).
If \(\psi :\mathbb {R}\to \mathbb {C}\) is \(C^2\) and obeys the bounds
for all \(t \in \mathbb {R}\), then
for all \(u \in \mathbb {R}\), where \(C\) is an absolute constant.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then
for all \(u \in \mathbb {R}\).
For any non-principal character \(\chi \) of \(Gal(K/L)\),
We have
for \(\Re (s) {\gt} 1\), where \(\chi \) runs over homomorphisms from \(G\) to \(\mathbb {C}^\times \) and \(L\) is the Artin \(L\)-function.
For any non-principal character \(\chi \) of \(Gal(K/L)\), \(L(\chi ,s)\) does not vanish for \(\Re (s)=1\).
Let \(\nu \) be a bumpfunction supported in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), we define the delta spike \(\nu _\epsilon \) to be the function from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by
For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\), and any \(0 {\lt} A {\lt} 1\) sufficiently small, and \(1-A/\log |t| \le \sigma \), we have
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported and \(x \geq 1\), then
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, and \(x \geq 1\), then
Let \(a:\mathbb {R}\to \mathbb {C}\) be a function, and let \(\sigma {\lt}-3/2\) be a real number. Suppose that, for all \(\sigma , \sigma '{\gt}0\), we have \(a(\sigma ')=a(\sigma )\), and that \(\lim _{\sigma \to -\infty }a(\sigma )=0\). Then \(a(\sigma )=0\).
We have
Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\). Then we define the Mellin convolution of \(f\) and \(g\) to be the function \(f\ast g\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by
Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) such that
is absolutely integrable on \([0,\infty )^2\). Then
Let \(F\) be a function from \(\mathbb {C}\) to \(\mathbb {C}\). We define the Mellin inverse transform of \(F\) to be the function \(\mathcal{M}^{-1}(F)\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by
where \(\sigma \) is sufficiently large (say \(\sigma {\gt}2\)).
Let \(f, g\) be once differentiable functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) so that \(fg'\) and \(f'g\) are both integrable, and \(f\cdot g (x)\to 0\) as \(x\to 0^+,\infty \). Then
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable then
for all \(u \in \mathbb {R}\). where \(C\) is an absolute constant.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable and of bounded variation, and \(\psi '\) is bounded variation, then
for all non-zero \(u \in \mathbb {R}\).
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then
for all non-zero \(u \in \mathbb {R}\).
Suppose that \(f\) is a holomorphic function on a rectangle, except for a simple pole at \(p\). By the latter, we mean that there is a function \(g\) holomorphic on the rectangle such that, \(f = g + A/(s-p)\) for some \(A\in \mathbb {C}\). Then the integral of \(f\) over the rectangle is \(A\).
If \(\psi : \mathbb {R}\to \mathbb {C}\) is continuous and compactly supported and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)
Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), the function \(x \mapsto \int _{(0,\infty )} x^{1+it} \widetilde{1_{\epsilon }}(x) dx\) is continuous at any \(y{\gt}0\).
Fix \(\epsilon {\gt}0\), and a bumpfunction supported in \([1/2,2]\). Then we define the smoothed Chebyshev function \(\psi _{\epsilon }\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by
Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\), the function
is integrable on \(\mathbb {R}\). ** Conditions are overkill; can remove some assumptions... **
Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\), the function \(x \mapsto \sum _{n=1}^\infty \frac{\Lambda (n)}{n^{2+it}} \mathcal{M}(\widetilde{1_{\epsilon }})(2+it) x^{2+it}\) is equal to \(\sum _{n=1}^\infty \int _{(0,\infty )} \frac{\Lambda (n)}{n^{2+it}} \mathcal{M}(\widetilde{1_{\epsilon }})(2+it) x^{2+it}\). ** Conditions are overkill; can remove some assumptions...**
Let \(a {\lt} b\), and let \(\phi \) be continuously differentiable on \([a, b]\). Then
Let \(k \le a {\lt} b\le k+1\), with \(k\) an integer, and let \(\phi \) be continuously differentiable on \([a, b]\). Then
If \(q ≥ 1\) and \(a\) is coprime to \(q\), the Dirichlet series \(\sum _{n \leq x: n = a\ (q)} {\Lambda (n)}{n^s}\) converges for \(\mathrm{Re}(s) {\gt} 1\) to \(\frac{1}{\varphi (q)} \frac{1}{s-1} + G(s)\) where \(G\) has a continuous extension to \(\mathrm{Re}(s)=1\).
For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log t \le \sigma _1 {\lt} \sigma _2\le 2\) and \(3 {\lt} |t|\), we have that:
The rectangle integral over \([1-\delta ,2] \times _{ℂ} [-T,T]\) of the integrand in \(\psi _{\epsilon }\) is
Let \(0 {\lt} a {\lt} b\) be natural numbers and \(s\in \mathbb {C}\) with \(s \ne 1\) and \(s \ne 0\). Then