Tools on Mellin transforms

1. Explicit truncated Perron formula
Here is Theorem 7.1 of [Ramaré, 2007 †Ramaré, O. 2007
Eigenvalues in the large sieve inequality
Funct. Approximatio, Comment. Math., 37, 7--35.
].
Theorem (2007)
Let F(z)=nan/nz$F(z)=\sum_{n}a_n/n^z$ be a Dirichlet series that converges absolutely for Rz>κa$\Re z>\kappa_a$, and let κ>0$\kappa>0$ be strictly larger than κa$\kappa_a$. For x1$x\ge1$ and T1$T\ge1$, we have
nxan=12iπκ+iTκiTF(z)xzdzz+O1/T|log(x/n)|u|an|nκ2xκduTu2.
2. L2${}^2$-means
We follow the idea of Corollary 3 of [Montgomery & Vaughan, 1974 †Montgomery, H.L., & Vaughan, R.C. 1974
Hilbert's inequality
J. Lond. Math. Soc., II Ser., 8, 73--82.
] but rely on [Preissmann, 1984 †Preissmann, E. 1984
Sur une inégalité de Montgomery et Vaughan
Enseign. Math., 30, 95--113.
] to get the following.
Theorem (2013)
Let (an)n1$(a_n)_{n\ge1}$ be a series of complex numbers that are such that nn|an|2<$\sum_n n|a_n|^2 < \infty$ and n|an|<$\sum_n |a_n| < \infty$. We have, for T0$T\ge0$,
T0n1annit2dt=nN|an|2(T+O(2πc0(n+1))),
where c0=1+2365$c_0=\sqrt{1+\frac23\sqrt{\frac{6}{5}}}$. Moreover, when an$a_n$ is real-valued, the constant 2πc0$2\pi c_0$ may be reduced to πc0$\pi c_0$.
This is Lemma 6.2 from [Ramaré, 2014 †Ramaré, O. 2014
An explicit density estimate for Dirichlet L$L$-series
To appear in Math. Comp., 35pp.
].
Corollary 6.3 and 6.4 of [Ramaré, 2014 †Ramaré, O. 2014
An explicit density estimate for Dirichlet L$L$-series
To appear in Math. Comp., 35pp.
] contain explicit versions of a Theorem of [Gallagher, 1970 †Gallagher, P.X. 1970
A large sieve density estimate near σ=1$\sigma =1$
Invent. Math., 11, 329--339.
]
Theorem (2013)
Let (an)n1$(a_n)_{n\ge1}$ be a series of complex numbers that are such that nn|an|2<$\sum_n n|a_n|^2 < \infty$ and n|an|<$\sum_n |a_n| < \infty$. We have, for T0$T\ge0$,
qQqφ(q)χmodq,χ primitiveTTnanχ(n)nit2dt7n|an|2(n+Q2max(T,3)).

Theorem (2013)
Let (an)n1$(a_n)_{n\ge1}$ be a series of complex numbers that are such that nn|an|2<$\sum_n n|a_n|^2 < \infty$ and n|an|<$\sum_n |a_n| < \infty$. We have, for T0$T\ge0$,
qQqφ(q)χmodq,χ primitiveTTnanχ(n)nit2dtn|an|2(43n+338Q2max(T,70)).

Last updated on July 14th, 2013, by Olivier Ramaré