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Writing a function on $\mathbb{R}$ as a sum of two injections

The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Consider the Dirichlet series $$F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s},$$ where $f(n)$ is the completely multiplicative function which satisfies $f(2)=-1$ and $f(p)=(-1)^{(p-1)/2}$ for odd primes $p$. ...
• 95.1k

Writing a function on $\mathbb{R}$ as a sum of two injections

It works at least for (locally) absolutely continuous functions. Such a function is the integral of a locally $L^1$ function. This weak derivative can be written as a sum of a positive and negative ...
• 7,981

How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$

Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...
• 95.1k
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Are “most” bounded derivatives not Riemann integrable?

In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...
• 3,097

Fourier coefficients of the logarithm of a given function

What you stated is correct. The crude statement is $|d_n|\leq C(\mu)e^{-\mu n}$, for every $\mu<\mu_0$, where $$\mu_0=\min\{\lambda,\theta_0\},\; \theta_0=\min\{|\theta|: f(x+i\theta)=0\}.$$ Proof. ...
• 87.2k
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Uniqueness of the $J$ invariant

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...
• 35.7k