= Solution
Fix $p_0\in X$ on a connected <compact Riemann surface>, and construct its <Jacobian variety> by periods as in the first solution. For a <divisor on an algebraic curve> $E=\sum_p n_p p$ of degree zero, define
$$
\operatorname{AJ}(E)=\left[\omega\longmapsto\sum_p n_p\int_{p_0}^p\omega\right]\in\operatorname{Jac}(X).
$$
Changing the paths adds a <period lattice> element; changing $p_0$ adds a common integral multiplied by $\sum n_p=0$. The <Abel theorem for divisors> says
$$
\boxed{\operatorname{AJ}(E)=0\quad\Longleftrightarrow\quad E=\operatorname{div}(f)\text{ for a nonzero meromorphic function }f.}
$$
Equivalently, effective <divisors on an algebraic curve> of the same degree have the same Abelian sum exactly when they are equivalent under <linear equivalence of divisors>.
We need a meromorphic version of the <Riemann bilinear relations>. Let $\eta$ be a <differential of the third kind>, with <residues> $n_p$ at its simple poles, and let $\omega$ be a <holomorphic differential form>. Choose representatives of the cycles and paths avoiding the poles in the same cut polygon, and let $F(p)=\int_{p_0}^p\omega$ there. Then
$$
\sum_i\left(A_i(\omega)B_i(\eta)-B_i(\omega)A_i(\eta)\right)=2\pi i\sum_p n_pF(p).
$$
Here the equality is an equality of the chosen complex numbers, before taking any quotient. To prove it, remove small disks about the poles. Since $d(F\eta)=\omega\wedge\eta=0$ on the remaining polygon, its boundary integral is zero. Pairing opposite outer edges gives the left side, just as in the first solution. The clockwise boundary about $p$ contributes $-2\pi i n_pF(p)$ by the <residue theorem>. Moving these terms to the other side proves the formula and its sign.
Such an $\eta$ exists whenever $\sum n_p=0$. Here is the required existence argument. Let $S$ be the reduced support of $E$, and consider
$$
0\longrightarrow K_X\longrightarrow K_X(S)\xrightarrow{\operatorname{res}}\bigoplus_{p\in S}\mathbb C_p\longrightarrow0.
$$
Locally the last map takes $c\,dz/z$ to $c$. By <Serre duality>, the obstruction in $H^1(K_X)\cong\mathbb C$ is dual to restriction of constant functions, so it is the sum of the <residues>. Thus the kernel of this sum is exactly the image of global meromorphic differentials. Alternatively, the <Riemann-Roch theorem> gives $h^0(K_X(S))-h^0(K_X)=|S|-1$, and the <residue theorem> identifies that image with the same codimension-one subspace. Subtracting a linear combination of the normalized <holomorphic differential forms> makes $A_i(\eta)=0$ for every $i$.
Suppose now that $\operatorname{AJ}(E)=0$. Let $u_i=\sum_p n_p\int_{p_0}^p\omega_i$ with the paths used above. There are integer columns $m,n$ such that $u=m+\tau n$. The normalized meromorphic bilinear formula gives $B_i(\eta)=2\pi i u_i$. Therefore
$$
\eta'=\eta-2\pi i\sum_j n_j\omega_j
$$
has $a$-periods $-2\pi i n_i$ and $b$-periods $2\pi i m_i$. Its integrals around the poles are also in $2\pi i\mathbb Z$. These cycles generate the homology of the punctured surface, so
$$
f(q)=\exp\left(\int_{q_*}^q\eta'\right)
$$
is a single-valued, nowhere-zero <holomorphic function> away from the support of $E$. Near $p$, write $\eta'=n_p\,dz/z+h(z)\,dz$. Then $f=z^{n_p}\exp(H(z))$ times a nonzero constant, with $H'=h$. It extends meromorphically with order exactly $n_p$. Hence $\operatorname{div}(f)=E$.
Conversely, if $E=\operatorname{div}(f)$, take $\eta=df/f$. It has <residues> $n_p$ and every period lies in $2\pi i\mathbb Z$: continuation of a local logarithm of $f$ changes it by an integral multiple of $2\pi i$. Write $A_i(\eta)=2\pi i r_i$ and $B_i(\eta)=2\pi i s_i$. Applying the meromorphic bilinear formula to $\omega_i$ gives
$$
u_i=s_i-\sum_j\tau_{ij}r_j.
$$
Thus $u\in\Lambda$ and $\operatorname{AJ}(E)=0$, completing both directions without assuming the conclusion as a property of the <Jacobian variety>.
For completeness, the effective degree-$d$ version lives on the <symmetric product of a curve> $X^{(d)}=X^d/\mathfrak S_d$. The sum of integrals on $X^d$ is invariant under the finite <symmetric group> and descends to the <Abelian sum map> $u_d$. The quotient is locally described by elementary symmetric coordinates, so repeated points are included. The proof above identifies its fibre over $u_d(D)$ with
$$
|D|=\mathbb P H^0(X,\mathcal O_X(D)).
$$
Explicitly, a nonzero <global section> gives its effective zero <divisor>, and two sections give the same <divisor> precisely when their quotient is constant. These maps are holomorphic in local symmetric coordinates. More intrinsically, a family of effective <divisors on an algebraic curve> in this fixed class determines a line of sections of $\mathcal O_X(D)$, after locally trivializing any <line bundle> pulled back from the parameter space. Conversely such a line of sections gives its family of zero <divisors on an algebraic curve>. These operations are inverse and unaffected by that trivialization, proving the identification with the <projective space> in families as well as on points.
Finally this description identifies the analytic <Jacobian variety> with $\operatorname{Pic}^0(X)$. Every degree-zero <line bundle> has a meromorphic section: after twisting by a sufficiently large effective <divisor>, the <Riemann-Roch theorem> supplies a nonzero <global section>. Its <divisor> represents the original <line bundle>. The correspondence to the period quotient is well-defined and injective by the theorem. It is surjective because one may choose $g$ distinct points with independent evaluations of <holomorphic differential forms>; the derivative of the sum of their Abel integrals is then an isomorphism. Hence the subgroup generated by point differences contains an open subset of the connected <complex torus>, and must be the whole group. Local integration coordinates, or the exponential sequence for $\mathcal O_X$, give the same holomorphic identification. This justifies the subsequent use of degree-$d$ <line bundles> as points of a translate of the <Jacobian variety>.
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