Dependencies

\(\sigmafunc (n)\) is the sum of the divisors of \(n\).

A positive integer \(N\) is called highly abundant (HA) if

for all positive integers \(m {\lt} N\), where \(\sigmafunc (n)\) denotes the sum of the positive divisors of \(n\).

For each integer \(n \ge 1\), define

We call \((L_n)_{n \ge 1}\) the least common multiple sequence.

We say that \(L_n\) is highly abundant if \(L_n\) is a highly abundant integer in the sense above, i.e.

We have \(4 p_1 p_2 p_3 {\lt} q_1 q_2 q_3\).

There exist integers \(m \ge 0\) and \(r\) satisfying \(0 {\lt} r {\lt} 4 p_1 p_2 p_3\) and

With \(m,r\) as above, define the competitor

For \(n \ge X_0^2\), define \(\varepsilon := 1/n\). Then

Let \(M{\gt}0\). Let \(f\) be analytic on the closed ball \(|z|\leq R\) such that \(f(0)=0\) and suppose that \(2M - f(z)\neq 0\) for all \(|z|\leq R\). Then, with \(f_{M}\) defined as in Definition 21, \(f_{M}\) is analytic on \(|z|\leq R\).

Let \(f\) be a complex function analytic on an open set \(s\) containing \(0\) such that \(f(0)=0\). Then, with \(g\) defined as in Definition 20, \(g\) is analytic on \(s\).

Let \(f\) be a complex function analytic on the closed ball \(|z|\leq R\) such that \(f(0)=0\). Then, with \(g\) defined as in Definition 20, \(g\) is analytic on \(|z|\leq R\).

One has

for all \(x \geq 1\).

If \(f\) is bounded above in a punctured neighborhood of \(p\), then \(f\) is \(O(1)\) in that neighborhood.

Let \(g : \mathbb {C}\to \mathbb {C}\) be a holomorphic function on a rectangle, then \(g\) is bounded above on the rectangle.

The Fourier transform is a bijection on the Schwartz class. [Note: only surjectivity is actually used.]

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(B_f:\overline{\mathbb {D}_R}\to \mathbb {C}\) as follows.

Let \(0 {\lt} r {\lt} R{\lt}1\) and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\). Then \(B_f(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_r}\).

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). Then

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

for all \(|z|\leq r\).

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(C_f:\overline{\mathbb {D}_R}\to \mathbb {C}\) as follows. This function is constructed by dividing \(f(z)\) by a polynomial whose roots are the zeros of \(f\) inside \(\overline{\mathbb {D}_r}\).

where \(h_z(z)\) comes from Lemma 105.

For any \(a\) coprime to \(m\),

One has

If \(a\ (q)\) is a primitive residue class, then one has

The (second) Chebyshev Psi function is defined as

where \(\Lambda (n)\) is the von Mangoldt function.

Let \(M{\gt}0\) and let \(x\) be a complex number such that \(\Re x\leq M\). Then, \(|x|\leq |2M - x|\).

Conjugation symmetry of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

Conjugation symmetry of the Riemann zeta function in the half-plane of convergence. Let \(s \in \mathbb {C}\) with \(\Re (s) {\gt} 1\). Then \(\overline{\zeta (\overline{s})} = \zeta (s)\).

For \(x{\gt}1\) and \(\sigma {\lt}-3/2\), we have

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 \(x {\gt} 0\).

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 \(\Re (s)=1\).

\(\zeta _L\) has a simple pole at \(s=1\).

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have that

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 \(\epsilon {\gt}0\), we have

Let \(\delta _t=E/\log |t|\) where \(E\) is the constant coming from Theorem 41.

Let \(f : \mathbb {C} \to \mathbb {C}\) be a function at \(p \in \mathbb {C}\) with derivative \(a\). Then the derivative of the function \(g(z) = \overline{f(\overline{z})}\) at \(\overline{p}\) is \(\overline{a}\).

Conjugation symmetry of the derivative of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

Let \(R,\, M{\gt}0\) and \(0 {\lt} r {\lt} r' {\lt} R\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then we have that

for all \(|z|\leq r\).

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

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-1-c-i*c\) and \(-1+c+i*c\) (except at \(s=-1\)).

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-c-i*c\) and \(c+i*c\) (except at \(s=0\)).

Any primitive residue class contains an infinite number of primes.

Let \(B{\gt}1\) and \(0 {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(|f(z)|\leq B\) for \(|z|\leq R\), then \(|B_f(z)|\leq B\) for \(|z|\leq R\) also.

Given a complex function \(f\), we define the function

Let \(f\) be a complex function and let \(z\neq 0\). Then, with \(g\) defined as in Definition 20,

For \(\sigma _0 {\gt} 1\), there exists a constant \(C {\gt} 0\) such that

If \(f\) is differentiable on a set \(s\) except at \(c\in s\), and \(f\) is bounded above on \(s\setminus \{ c\} \), then there exists a differentiable function \(g\) on \(s\) such that \(f\) and \(g\) agree on \(s\setminus \{ c\} \).

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(z\in \overline{\mathbb {D}_{R'}}\setminus \mathcal{K}_f(R')\) we have

For any arithmetic function \(f\) and real number \(x\), one has

and

If \(\psi : \mathbb {R}\to \mathbb {C}\) is integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

For \(x{\gt}1\) and \(\sigma {\gt}0\), we have

For \(x{\gt}0\), \(\sigma {\gt}0\), and \(x{\lt}1\), we have

Let \(t\in \mathbb {R}\) with \(|t|\geq 3\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\). Additionally, let \(\rho \in \mathcal{Z}_t\). Then we have that

For all \(s\in \mathbb {C}\) with \(|s|\leq 1\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\), we have that

Let \(f : \mathbb {C} \to \mathbb {C}\) be a complex differentiable function at \(p \in \mathbb {C}\) with derivative \(a\). Then the function \(g(z) = \overline{f(\overline{z})}\) is complex differentiable at \(\overline{p}\) with derivative \(\overline{a}\).

If \(f\) is holomorphic on a rectangle \(z\) and \(w\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is \(0\).

For any \(N\ge 1\), the function \(\zeta _0(N,s)\) is holomorphic on \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \).

We have that

Same with \(I_9\).

Let

We have that

Assuming a bound of the form of Lemma 82 we have that

Let

We have that

Assuming a bound of the form of Lemma 82 we have that

Same with \(I_7\).

Let

We have that

We have that

Same with \(I_6\).

Let

We have that

We have that

Let

We have that

We have that

Symmetry between \(I_2\) and \(I_8\):

Given a natural number \(k\) and a real number \(X_0 {\gt} 0\), there exists \(C \geq 1\) so that for all \(X \geq X_0\),

For every \(n\) there is some absolute constant \(C{\gt}0\) such that

The integral

is positive (and hence convergent - since a divergent integral is zero in Lean, by definition).

The conjugation symmetry of the interval integral. Let \(f : \mathbb {R} \to \mathbb {C}\) be a measurable function, and let \(a, b \in \mathbb {R}\). Then

Let \(x{\gt}0\). Then the function \(f(s) = x^s/(s(s+1))\) is holomorphic on the half-plane \(\{ s\in \mathbb {C}:\Re (s){\gt}0\} \).

Let \(x{\gt}0\) and \(\sigma \in \mathbb {R}\). Then

is integrable.

The set \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \) is path-connected.

Let \(B{\gt}1\) and \(0 {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\), define \(L_f(z)=J_{B_f}(z)\) where \(J\) is from Theorem 34 and \(B_f\) is from Definition 25.

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(|z|\leq r'\)

Let \(x\in \mathbb {R}\) and \(s \ne 0, -1\). Then

We have \(\sum _{n \leq x} \lambda (n) = o(x)\).

Let \(n \ge X_0^2\). Set \(x := \sqrt{n}\). Then, by repeated application of Theorem 50, there exist primes \(p_1,p_2,p_3\) with

and \(p_1 {\lt} p_2 {\lt} p_3\). Moreover, \(\sqrt{n} {\lt} p_1\) (for \(n\) sufficiently large).

Let \(n \ge X_0^2\). Then there exist primes \(q_1 {\lt} q_2 {\lt} q_3\) with

for \(i = 1,2,3\), and \(q_1 {\lt} q_2 {\lt} q_3 {\lt} n\).

If

then

Suppose

Then \(\sigmafunc (M) \ge \sigmafunc (L_n)\).

Let \(k\) be the largest integer such that \(2^k \le n\). Then the exponent of \(2\) in \(L'\) is at least \(k\), and the exponent of \(2\) in \(M\) is at least \(k+2\). Consequently,

Fix \(i \in \{ 1,2,3\} \). Suppose that in passing from \(L'\) to \(M\) we increase the exponent of \(p_i\) by exactly \(1\). Then the normalised divisor sum is multiplied by a factor

For all \(n \ge X_0^2 = 89693^2\) we have

For \(0 \le \varepsilon \le 1/89693^2\), we have

There exists a positive integer \(L'\) such that

and each prime \(q_i\) divides \(L_n\) exactly once and does not divide \(L'\).

With notation as above, we have:

1. \(0 {\lt} r {\lt} 4 p_1 p_2 p_3\).

2. \(M {\lt} L_n\).

3. \[ 1 {\lt} \frac{L_n}{M} = \Bigl(1 - \frac{r}{q_1 q_2 q_3}\Bigr)^{-1} {\lt} \Bigl(1 - \frac{4 p_1 p_2 p_3}{q_1 q_2 q_3}\Bigr)^{-1}. \]

The extra factor \(m\) in the definition of \(M\) can only increase the normalised divisor sum:

With \(p_i\) as in Lemma 139, we have for large \(n\)

For \(0 \le \varepsilon \le 1/89693^2\), we have

and

With \(p_i,q_i\) as in Lemmas 139 and 140, we have

With \(p_i,q_i\) as in Lemmas 139 and 140, we have

Let \(L'\) be as in Lemma 129. Then

With notation as above,

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported and \(x \geq 1\), then

With the hypotheses as above, we have

as \(x \to \infty \).

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 \(\sigma {\gt}0\) 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 \(a(\sigma )=0\).

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 that, for \(\Re (s){\gt}1\),

If \(f\) is holomorphic in a neighborhood of \(p\), and there is a simple pole at \(p\), then \(f'/ f\) has a simple pole at \(p\) with residue \(-1\):

There is an \(A{\gt}0\) so that for \(1-A/\log ^9 |t| \le \sigma {\lt} 1+A/\log ^9 |t|\) and \(3 {\lt} |t|\),

There exist \(A, C {\gt} 0\) such that

whenever \(|t|{\gt}3\) and \(\sigma {\gt} 1 - A/\log |t|^9\).

Let \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f(z)=\zeta (z+3/2+it)\), then for all \(z\in \overline{\mathbb {D}_R'}\setminus \mathcal{K}_f(R')\) we have that

There is an \(A{\gt}0\) so that for all \(T{\gt}3\), the function \( \frac{\zeta '}{\zeta }(s) \) is holomorphic on \(\{ 1-A/\log ^9 T \le \Re s \le 2, |\Im s|\le T \} \setminus \{ 1\} \).

There is a \(\sigma _2 {\lt} 1\) so that the function

is holomorphic on \(\{ \sigma _2 \le \Re s \le 2, |\Im s| \le 3 \} \setminus \{ 1\} \).

For \(T{\gt}0\) and \(\sigma '=1-\delta _T/3=1-F/\log T\), if \(|t|\leq T\) then we have that

There exists a constant \(F\in (0,1/2)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

There exists a constant \(F\in (0,1/2)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

Let \(0 {\lt} r {\lt} R{\lt}1\). Let \(B:\overline{\mathbb {D}_R}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_R}\) with \(B(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_R}\). Then there exists \(J_B:\overline{\mathbb {D}_r}\to \mathbb {C}\) that is analytic on neighborhoods of points in \(\overline{\mathbb {D}_r}\) such that

• \(J_B(0)=0\)

• \(J_B'(z)=B'(z)/B(z)\)

• \(\log |B(z)|-\log |B(0)|=\Re J_B(z)\)

for all \(z\in \overline{\mathbb {D}_r}\).

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 \(\mathbb {R}\) or \(\mathbb {C}\), for \(x\neq 0\),

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) such that

is absolutely integrable on \([0,\infty )^2\). Then

The Mellin transform of \(1_{(0,1]}\) is

For any \(\epsilon {\gt}0\), the Mellin transform of \(\nu _\epsilon \) is

For any \(\epsilon {\gt}0\), we have

As \(\epsilon \to 0\), we have

The Mellin transform of \(\nu \) is

as \(|s|\to \infty \) with \(\sigma _1 \le \Re (s) \le 2\).

Fix \(\epsilon {\gt}0\). Then the Mellin transform of \(\widetilde{1_{\epsilon }}\) is

Given \(0{\lt}\sigma _1\le \sigma _2\), for any \(s\) such that \(\sigma _1\le \mathcal Re(s)\le \sigma _2\), we have

At \(s=1\), we have

We have \(\sum _{n \leq x} \mu (n) = o(x)\).

We have \(\sum _{n \leq x} \mu (n)/n = o(1)\).

We have \(|\sum _{n \leq x} \frac{\mu (n)}{n}| \leq 1\).

If a function \(f\) has a simple pole at a point \(p\) with residue \(A \neq 0\), then \(f\) is nonzero in a punctured neighborhood of \(p\).

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

One has

as \(x \to \infty \).

There exists a function \(c(x)\) such that \(c(x) = o(1)\) as \(x \to \infty \) and

for all \(x\) large enough.

One has

as \(n \to \infty \).

We have \(p_{n+1} - p_n = o(p_n)\) as \(n \to \infty \).

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 \(u \in \mathbb {R}\).

For every \(\epsilon {\gt}0\), there is a prime between \(x\) and \((1+\epsilon )x\) for all sufficiently large \(x\).

We have

For every integer \(n \ge X_0^2 = 89693^2 \approx 8.04\times 10^9\), the primes \(p_i,q_i\) constructed in Lemmas 139 and 140 satisfy the inequality 1.

A Rectangle’s border, given corners \(z\) and \(w\) is the union of the four sides.

A RectangleIntegral of a function \(f\) is one over a rectangle determined by \(z\) and \(w\) in \(\mathbb {C}\). We will sometimes denote it by \(\int _{z}^{w} f\). (There is also a primed version, which is \(1/(2\pi i)\) times the original.)

If \(f\) is holomorphic on a rectangle \(z\) and \(w\) except at a point \(p\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is the same as the integral of \(f\) over a small square centered at \(p\).

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

If \(f\) has a simple pole at \(p\) with residue \(A\), and \(g\) is holomorphic near \(p\), then the residue of \(f \cdot g\) at \(p\) is \(A \cdot g(p)\). That is, we assume that

near \(p\), and that \(g\) is holomorphic near \(p\). Then

If a function \(f\) is holomorphic in a neighborhood of \(p\) and \(\lim _{s\to p} (s-p)f(s) = A\), then \(f(s) = \frac{A}{s-p} + O(1)\) near \(p\).

For \(x{\gt}1\) (of course \(x{\gt}0\) would suffice) and \(\sigma {\gt}0\), we have

For \(x{\gt}1\), we have

The rectangle (square) integral of \(f(s) = 1/s\) with corners \(-1-i\) and \(1+i\) is equal to \(2\pi i\).

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 \(A\).

For any natural \(N\ge 1\), we define

Conjugation symmetry of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

The log derivative of the Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(-1\):

The Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(1\). In particular, the function

is bounded in a neighborhood of \(s=1\).

Given a complex function \(f\) and a real number \(M\), we define the function

where \(g\) is defined as in Definition 20.

The previous corollary also holds for functions \(\psi \) that are assumed to be in the Schwartz class, as opposed to being \(C^2\) and compactly supported.

If \(\psi : \mathbb {R}\to \mathbb {C}\) is continuous and compactly supported and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

Let \(R{\gt}0\) and \(f:\overline{\mathbb {D}_R}\to \mathbb {C}\). Define the set of zeros \(\mathcal{K}_f(R)=\{ \rho \in \mathbb {C}:|\rho |\leq R,\, f(\rho )=0\} \).

There exists \(C{\gt}0\) such that for all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 3\); if \(\zeta (\rho )=0\) with \(\rho =\sigma +it\), then

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(\delta \in (0,1)\) we have

If \(I\) is a closed interval contained in an open interval \(J\), then there exists a smooth function \(\Psi : \mathbb {R}\to \mathbb {R}\) with \(1_I \leq \Psi \leq 1_J\).

Let \(\epsilon {\gt}0\). Then we define the smooth function \(\widetilde{1_{\epsilon }}\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\). 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 \(y{\gt}0\).

If \(\nu \) is nonnegative and has mass one, then \(\widetilde{1_{\epsilon }}(x)\le 1\), \(\forall x{\gt}0\).

If \(\nu \) is nonnegative, then \(\widetilde{1_{\epsilon }}(x)\) is nonnegative.

Fix \(0{\lt}\epsilon {\lt}1\). There is an absolute constant \(c{\gt}0\) so that: if \(x\geq (1+c\epsilon )\), then

Fix \(\epsilon {\gt}0\). There is an absolute constant \(c{\gt}0\) so that: If \(0 {\lt} x \leq (1-c\epsilon )\), then

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

where we’ll take \(\sigma = 1 + 1 / \log X\).

We have that

We have that

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 \(\epsilon {\gt}0\), and \(\sigma \in (1, 2]\), the function

is integrable on \(\mathbb {R}\).

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 \(\epsilon {\gt}0\) and \(\sigma \in (1,2]\), the function \(x \mapsto \sum _{n=1}^\infty \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\) is equal to \(\sum _{n=1}^\infty \int _{(0,\infty )} \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\).

The smoothed Chebyshev integrand satisfies the conjugation symmetry

for all \(s \in \mathbb {C}\), \(X {\gt} 0\), and \(\epsilon {\gt} 0\).

We have that

The integrand

is integrable on the contour \(\sigma _0 + t i\) for \(t \in \mathbb {R}\) and \(\sigma _0 {\gt} 1\).

We have that

We have that

There exists a smooth (once differentiable would be enough), nonnegative “bumpfunction” \(\nu \), supported in \([1/2,2]\) with total mass one:

We have

Let \(a {\lt} b\), and let \(\phi \) be continuously differentiable on \([a, b]\). Then

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\), we have that

Let \(x{\gt}1\). Then

Let \(x{\gt}0\) and \(x{\lt}1\). Then

Let \(x{\gt}0\) and \(\sigma ',\sigma ''\in \mathbb {R}\). Then

goes to \(0\) as \(t\to -\infty \).

Let \(x{\gt}0\) and \(\sigma ',\sigma ''\in \mathbb {R}\). Then

goes to \(0\) as \(t\to \infty \).

Let \(n \geq 1\). Suppose that primes \(p_1,p_2,p_3,q_1,q_2,q_3\) satisfy

and the key criterion

Then \(L_n\) is not highly abundant.

There exists a constant \(X_0\) (one may take \(X_0 = 89693\)) with the following property: for every real \(x \ge X_0\), there exists a prime \(p\) with

For every integer \(n \ge 89693^2\), the integer \(L_n\) is not highly abundant.

We have that

for all \(\theta \in \mathbb {R}\).

Let \(f\) be a function from \(\mathbb {C}\) to \(\mathbb {C}\), and let \(\sigma \) be a real number. Then we define

Let \(x{\gt}0\) and \(\sigma {\gt}1\). Then

Let \(x{\gt}1\) and \(\sigma {\lt}-3/2\). Then

We have

If \(q ≥ 1\) and \(a\) is coprime to \(q\), we have

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 \(\mathrm{Re}(s)=1\).

If \(q ≥ 1\) and \(a\) is coprime to \(q\), and \(\mathrm{Re} s {\gt} 1\), we have

We have

For any closed interval \(I \subset (0,+\infty )\), we have

If \(\Psi : (0,\infty ) \to \mathbb {C}\) is smooth and compactly supported away from the origin, then,

as \(x \to \infty \).

We have

Let \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). For all \(\rho \in \mathcal{K}_f(1)\) there exists \(h_\rho (z)\) that is analytic at \(\rho \), \(h_\rho (\rho )\neq 0\), and \(f(z)=(z-\rho )^{m_f(\rho )}\, h_\rho (z)\).

There exists a constant \(0 {\lt} E{\lt}1\) such that for all \(\rho =\sigma +it\) with \(\zeta (\rho )=0\) and \(|t|\geq 2\), one has

Let \(0 {\lt} R{\lt}1\) and \(f:\mathbb {C}\to \mathbb {C}\) be analtyic on neighborhoods of points in \(\overline{\mathbb {D}_1}\). For any zero \(\rho \in \mathcal{K}_f(R)\), we define \(m_f(\rho )\) as the order of the zero \(\rho \) w.r.t \(f\).

Let \(B{\gt}1\) and \(0{\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for \(|z|\leq R\), then

Let \(\mathcal{Z}_t=\{ \rho \in \mathbb {C}:\zeta (\rho )=0,\, |\rho -(3/2+it)|\leq 5/6\} \).

For \(\Re (s){\gt}0\), \(s\ne 1\), and for any \(N\),

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:

For any \(t\ne 0\) (so we don’t pass through the pole), and \(\sigma _1 {\lt} \sigma _2\),

Let

We have that \(\zeta (s)=\zeta _0(s)\) for \(\sigma {\gt}1\).

We have that \(\zeta _0(s)\) is analytic for all \(s\in S\) where \(S=\{ s\in \mathbb {C}:\Re s{\gt}0,\, s\neq 1\} \).

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma =\in (0,2], 2 {\lt} |t|\),

For any \(0 {\lt} a {\lt} b\) and \(s \in \mathbb {C}\) with \(\sigma =\Re (s){\gt}0\),

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma {\gt} 0\),

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma =\in (0,2], 2 {\lt} |t|\),

Given \(n ≤ t\) and \(\sigma \) with \(1-A/\log t \le \sigma \), we have that

For all \(\epsilon {\gt} 0\) sufficiently close to \(0\), the rectangle integral over \([1-\delta ,2] \times _{ℂ} [-T,T]\) of the integrand in \(\psi _{\epsilon }\) is

where the implicit constant is independent of \(X\).

For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\), there is an \(A{\gt}0\) so that for \(1-A/\log t \le \sigma \), we have

We have that

for all \(s\in S\).

For all \(t\in \mathbb {R}\) one has

For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log ^9 |t| \le \sigma {\lt} 1+A/\log ^9 |t|\) and \(3 {\lt} |t|\), we have that:

For all \(\sigma {\gt}1\),

For \(\sigma {\gt}1\) (and \(\sigma \le 2\)),

as \(|t|\to \infty \).

For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log ^9 |t| \le \sigma {\lt} 1\) and \(3 {\lt} |t|\), we have that:

There exists a \(c{\gt}0\) such that for all \(1 {\lt} \sigma {\lt}= 2\) and \(3 {\lt} |t|\),

There exists a \(c{\gt}0\) such that for all \(1 {\lt} \sigma ≤ 2\),

As \(\sigma \to 1^+\),

For any \(T{\gt}0\), there is a constant \(\sigma {\lt}1\) so that

for all \(|t| \leq T\) and \(\sigma ' \ge \sigma \).

The zeta function does not vanish on the 1-line.

Let \(0 {\lt} a {\lt} b\) be natural numbers and \(s\in \mathbb {C}\) with \(s \ne 1\) and \(s \ne 0\). Then

Let \(N\) be a natural number and \(s\in \mathbb {C}\), \(\Re (s){\gt}1\). Then

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

There is an \(A{\gt}0\) so that for \(1-A/\log ^9 |t| \le \sigma {\lt} 1\) and \(3 {\lt} |t|\),